MATH1052 Multivariate Calculus and Ordinary Differential Equations

MATH1052 Multivariate Calculus and Ordinary Differential Equations
MATH1052
Multivariate Calculus
and
Ordinary Differential Equations
WORKBOOK
First Semester, 2013
c School of Mathematics and Physics, The University of Queensland, Brisbane QLD 4072, Aus
tralia.
2
How to use this workbook
This workbook is designed for use during lectures. It contains copies of the slides
that will be shown in lectures, giving you time to listen and think in class, rather
than spending the entire time writing.
There are exercises, definitions and examples in the workbook for you to fill in.
These will be covered in lectures, and you should write down all the information
given. The lecturer will make it clear when they are covering something that you
need to complete. Take particular note of any highlighted formulae or results such
as
1 + 1 = 2.
These are formulae or results we expect you to remember for your exams.
The completed workbook will act as a study guide to assist you in working through
assignments and preparing for the mid-semester and final exams. For this reason,
it is very important to attend lectures.
For further information about the course, including copies of assignments and worksheets, please go to Blackboard at http://blackboard.elearning.uq.edu.au
3
Overview
MATH1052 is a second course in calculus. It will enable you to use and visualise
functions of two or more variables. Most importantly it extends your knowledge of
calculus so you can differentiate multivariable functions and find their maxima and
minima. These ideas are basic to the complex optimisation problems that occur
when mathematics is used in economics, finance, and engineering.
MATH1052 also introduces ordinary differential equations (ODEs); one of the basic
tools in mathematical modelling. ODEs are used to describe how things change
with time. In engineering and the physical sciences ODEs are used to work out the
motion of particles and satellites, the rates of chemical reactions, or to model electric
fields. In biology they are used to model populations (to determine safe harvesting
levels, for example) and to model epidemics.
The textbook for the course is Calculus (7th Edition) (the references in the parentheses corresponds to the 6th Edition) by James Stewart (Brooks/Cole). This is the
only text you need for the course.
4
CONTENTS
Contents
1 Functions of Several Variables
1.1
1.2
1.3
1.4
1.5
1.6
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.1.1
Visualising functions . . . . . . . . . . . . . . . . . . . . . . . 12
1.1.2
Main points . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Conic Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.2.1
Parabolas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.2.2
Circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.2.3
Ellipses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.2.4
Hyperbolas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.2.5
Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.2.6
Main points . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Scalar Equation for a Plane . . . . . . . . . . . . . . . . . . . . . . . 26
1.3.1
Horizontal planes . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.3.2
Vertical planes . . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.3.3
Arbitrary planes . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.3.4
Main points . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
Vector Equation of a Plane I . . . . . . . . . . . . . . . . . . . . . . . 32
1.4.1
Revision of vectors . . . . . . . . . . . . . . . . . . . . . . . . 32
1.4.2
Vector equation of a plane I . . . . . . . . . . . . . . . . . . . 36
1.4.3
Main points . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
Equations for a Line . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
1.5.1
Parallel, skew and orthogonal Lines. . . . . . . . . . . . . . . . 45
1.5.2
Vector equation of a plane II . . . . . . . . . . . . . . . . . . . 48
1.5.3
Distance from a Point to a Plane . . . . . . . . . . . . . . . . 50
1.5.4
Main points . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
Contour Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
1.6.1
1.7
11
Main points . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
Cross-sections of a Surface . . . . . . . . . . . . . . . . . . . . . . . . 62
1.7.1
Main points . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
CONTENTS
5
2 Partial Derivatives and Tangent Planes
2.1
2.2
2.3
2.4
2.5
67
Limits and Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . 67
2.1.1
Review of the 1-variable case
. . . . . . . . . . . . . . . . . . 67
2.1.2
Multivariable limits . . . . . . . . . . . . . . . . . . . . . . . . 69
2.1.3
Multivariable continuity . . . . . . . . . . . . . . . . . . . . . 73
2.1.4
Main points . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
Partial Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
2.2.1
Slope in the x-direction . . . . . . . . . . . . . . . . . . . . . . 74
2.2.2
Slope in the y-direction . . . . . . . . . . . . . . . . . . . . . . 75
2.2.3
Partial derivatives for f (x, y, z) . . . . . . . . . . . . . . . . . 77
2.2.4
Higher order derivatives . . . . . . . . . . . . . . . . . . . . . 79
2.2.5
Main points . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
The Tangent Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
2.3.1
Review for f (x) . . . . . . . . . . . . . . . . . . . . . . . . . . 80
2.3.2
Equation for a tangent plane . . . . . . . . . . . . . . . . . . . 81
2.3.3
Smoothness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
2.3.4
Main points . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
Linear Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . 87
2.4.1
Review for f (x) . . . . . . . . . . . . . . . . . . . . . . . . . . 87
2.4.2
Linear approximations for f (x, y) . . . . . . . . . . . . . . . . 88
2.4.3
Estimating small changes . . . . . . . . . . . . . . . . . . . . . 90
2.4.4
Estimating error . . . . . . . . . . . . . . . . . . . . . . . . . 91
2.4.5
Extra reading: Differentials . . . . . . . . . . . . . . . . . . . 92
2.4.6
Main points . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
Gradients and Directional Derivatives . . . . . . . . . . . . . . . . . . 93
2.5.1
Directional derivatives . . . . . . . . . . . . . . . . . . . . . . 93
2.5.2
The gradient vector ∇f
2.5.3
Properties of the gradient vector . . . . . . . . . . . . . . . . . 97
. . . . . . . . . . . . . . . . . . . . . 95
6
CONTENTS
2.5.4
2.6
Main points . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
The Chain Rule and Applications . . . . . . . . . . . . . . . . . . . . 103
2.6.1
Review for f (x) . . . . . . . . . . . . . . . . . . . . . . . . . . 103
2.6.2
The chain rule for f (x, y)
2.6.3
Larger Chain Rules . . . . . . . . . . . . . . . . . . . . . . . . 105
2.6.4
Implicit Differentiation . . . . . . . . . . . . . . . . . . . . . . 107
2.6.5
Main points . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
. . . . . . . . . . . . . . . . . . . . 104
3 Max and Min Problems on Surfaces
3.1
3.2
3.3
Quadratic Approximation . . . . . . . . . . . . . . . . . . . . . . . . 110
3.1.1
Review for f (x) . . . . . . . . . . . . . . . . . . . . . . . . . . 110
3.1.2
Quadratic approximations of f (x, y) . . . . . . . . . . . . . . . 112
3.1.3
Main points . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
Critical Points and Optimisation . . . . . . . . . . . . . . . . . . . . 117
3.2.1
Review for f (x) . . . . . . . . . . . . . . . . . . . . . . . . . . 117
3.2.2
Critical points for f (x, y) . . . . . . . . . . . . . . . . . . . . . 119
3.2.3
Global maxima and minima . . . . . . . . . . . . . . . . . . . 127
3.2.4
The Matlab command fminsearch . . . . . . . . . . . . . . . 129
3.2.5
Main points . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
Constrained Optimisation and Lagrange Multipliers . . . . . . . . . . 130
3.3.1
Lagrange multipliers . . . . . . . . . . . . . . . . . . . . . . . 130
3.3.2
Main points . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
4 Ordinary Differential Equations
4.1
110
138
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
4.1.1
Examples of ODEs . . . . . . . . . . . . . . . . . . . . . . . . 138
4.1.2
How do ODEs arise? . . . . . . . . . . . . . . . . . . . . . . . 139
4.1.3
Solution to an ODE
4.1.4
Initial value problems . . . . . . . . . . . . . . . . . . . . . . . 141
. . . . . . . . . . . . . . . . . . . . . . . 140
CONTENTS
7
4.1.5
The order of an ODE . . . . . . . . . . . . . . . . . . . . . . . 141
4.1.6
Riding your bike at constant speed . . . . . . . . . . . . . . . 142
4.1.7
Motion of Projectiles . . . . . . . . . . . . . . . . . . . . . . . 143
4.1.8
Extra reading: Realistic models . . . . . . . . . . . . . . . . . 145
4.1.9
Analytical and numerical solutions . . . . . . . . . . . . . . . 147
4.1.10 Main points . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
4.2
4.3
4.4
4.5
Slope Fields and Equilibrium Solutions . . . . . . . . . . . . . . . . . 148
4.2.1
Example: the slope field of y 0 = 2y . . . . . . . . . . . . . . . 149
4.2.2
Equilibrium solutions . . . . . . . . . . . . . . . . . . . . . . . 149
4.2.3
Stability of equilibrium solutions . . . . . . . . . . . . . . . . 153
4.2.4
Existence and uniqueness of solutions . . . . . . . . . . . . . . 155
4.2.5
Long-term behaviour . . . . . . . . . . . . . . . . . . . . . . . 155
4.2.6
Main points . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
Euler’s Method for Solving ODEs Numerically . . . . . . . . . . . . . 157
4.3.1
Euler’s method using tangent lines . . . . . . . . . . . . . . . 157
4.3.2
Euler’s method using Matlab; an example . . . . . . . . . . . 160
4.3.3
Demonstrating errors in Euler’s method . . . . . . . . . . . . 161
4.3.4
Main points . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
Separable ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
4.4.1
Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
4.4.2
Solving separable ODEs . . . . . . . . . . . . . . . . . . . . . 163
4.4.3
Singular solutions . . . . . . . . . . . . . . . . . . . . . . . . . 166
4.4.4
Main points . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
Applications: Law of Cooling, Population Growth . . . . . . . . . . . 169
4.5.1
Newton’s law of cooling . . . . . . . . . . . . . . . . . . . . . 169
4.5.2
Malthus: “. . . doomed to misery and vice” . . . . . . . . . . . 172
4.5.3
Verhulst to the rescue; down with the Malthusian law . . . . . 174
4.5.4
Extra reading: numerically solving ODEs . . . . . . . . . . . . 177
8
CONTENTS
4.5.5
4.6
4.7
4.8
4.9
Main points . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
Solving Linear First Order Equations . . . . . . . . . . . . . . . . . . 178
4.6.1
Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
4.6.2
Worked-out example: RL circuit . . . . . . . . . . . . . . . . . 185
4.6.3
Main points . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
Linear Second-Order Differential Equations with Constant Coefficients 187
4.7.1
Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
4.7.2
The superposition principle . . . . . . . . . . . . . . . . . . . 188
4.7.3
Reduction of order . . . . . . . . . . . . . . . . . . . . . . . . 189
4.7.4
Homogeneous linear second-order ODEs with constant coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
4.7.5
Extra reading: numerical solutions . . . . . . . . . . . . . . . 199
4.7.6
Main points . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
Simple Harmonic Motion and Damped Oscillations . . . . . . . . . . 200
4.8.1
The undamped spring . . . . . . . . . . . . . . . . . . . . . . 200
4.8.2
The damped spring . . . . . . . . . . . . . . . . . . . . . . . . 205
4.8.3
Underdamping . . . . . . . . . . . . . . . . . . . . . . . . . . 206
4.8.4
Critical damping . . . . . . . . . . . . . . . . . . . . . . . . . 209
4.8.5
Overdamping . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
4.8.6
Summary of motion of a spring: . . . . . . . . . . . . . . . . . 213
4.8.7
The pendulum
4.8.8
Main points . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
. . . . . . . . . . . . . . . . . . . . . . . . . . 214
Inhomogeneous Linear Second-Order ODEs . . . . . . . . . . . . . . . 216
4.9.1
Inhomogeneous linear second-order ODEs with constant coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
4.9.2
Main points . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
CONTENTS
9
5 Parametrisation of Curves and Line Integrals
5.1
5.2
5.3
5.4
5.5
224
Parametrisation of Curves . . . . . . . . . . . . . . . . . . . . . . . . 224
5.1.1
Sketching parametric curves . . . . . . . . . . . . . . . . . . . 225
5.1.2
Parametrising circles and ellipses . . . . . . . . . . . . . . . . 228
5.1.3
Parametric curves in R3 . . . . . . . . . . . . . . . . . . . . . 231
5.1.4
Piecewise linear paths . . . . . . . . . . . . . . . . . . . . . . 232
5.1.5
Plotting parametrised equations in Matlab . . . . . . . . . . . 232
5.1.6
Main points . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
Position Vectors, Velocity and Acceleration . . . . . . . . . . . . . . . 234
5.2.1
Position vector . . . . . . . . . . . . . . . . . . . . . . . . . . 234
5.2.2
Parametrising straight lines . . . . . . . . . . . . . . . . . . . 235
5.2.3
Velocity vector . . . . . . . . . . . . . . . . . . . . . . . . . . 238
5.2.4
Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
5.2.5
Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
5.2.6
Extra reading: stone on a string . . . . . . . . . . . . . . . . . 241
5.2.7
Main points . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
Line Integrals and Work Done . . . . . . . . . . . . . . . . . . . . . . 242
5.3.1
Length of a curve . . . . . . . . . . . . . . . . . . . . . . . . . 242
5.3.2
Work done by a constant force . . . . . . . . . . . . . . . . . . 248
5.3.3
Work done over a curve . . . . . . . . . . . . . . . . . . . . . 249
5.3.4
Main points . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
Gradient Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
5.4.1
Conservative fields . . . . . . . . . . . . . . . . . . . . . . . . 255
5.4.2
Main points . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
Work Done or Path Integrals for Non-conservative Fields . . . . . . . 263
5.5.1
Main points . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266
10
CONTENTS
6 MATLAB Guide
6.1
267
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
6.1.1
Getting started . . . . . . . . . . . . . . . . . . . . . . . . . . 267
6.1.2
Getting help . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
6.2
The Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268
6.3
M-files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
6.4
6.5
6.3.1
If statements . . . . . . . . . . . . . . . . . . . . . . . . . . . 270
6.3.2
For and while loops . . . . . . . . . . . . . . . . . . . . . . . 271
Visualising Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
6.4.1
Ezplot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
6.4.2
Ezcontour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
6.4.3
Ezsurf and ezmesh . . . . . . . . . . . . . . . . . . . . . . . . 276
6.4.4
Functions of three variables . . . . . . . . . . . . . . . . . . . 276
Visualising Data
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276
6.5.1
Plot and plot3 . . . . . . . . . . . . . . . . . . . . . . . . . . 277
6.5.2
Contour, surf and mesh . . . . . . . . . . . . . . . . . . . . . 277
6.6
Animations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
6.7
Optimisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280
6.8
6.9
6.7.1
Using fminsearch . . . . . . . . . . . . . . . . . . . . . . . . 280
6.7.2
Lines of best fit . . . . . . . . . . . . . . . . . . . . . . . . . . 281
Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
6.8.1
Slope fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
6.8.2
Euler’s method . . . . . . . . . . . . . . . . . . . . . . . . . . 285
6.8.3
Runge–Kutta method . . . . . . . . . . . . . . . . . . . . . . . 285
6.8.4
Systems of differential equations . . . . . . . . . . . . . . . . . 287
Parametric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
6.10 Common commands and operations . . . . . . . . . . . . . . . . . . . 290
6.10.1 Common numerical operations and constants . . . . . . . . . . 290
6.10.2 Vector and matrix operations . . . . . . . . . . . . . . . . . . 291
6.10.3 Special matrices . . . . . . . . . . . . . . . . . . . . . . . . . . 291
11
1
Functions of Several Variables
1.1
Introduction
From MATH1051 or earlier mathematics courses, you should be familiar with functions f of one variable. The tools of calculus were useful because you could:
• Sketch the graph y = f (x) of f .
• Find the minima and maxima of f .
• Analyse the slope of f by calculating f 0 .
• Find approximations of f using Taylor series.
• Find solutions to f (x) = 0.
In this first module of MATH1052, we shall undertake these same basic ideas, but
in the framework of functions of several variables (multivariate functions). Most of
the theory for this section is covered in Stewart, Chapter 14 (Chapter 15).
Many familiar formulae are essentially just functions of more than one variable.
For example, the volume V of a box is a function of its width, height and depth:
V (w, h, d) = whd, and the profile of a guitar string is a function of time and position
along the string: f (x, t) = A sin x cos t.
5
0
/&%
-,.
() *,+'
&$ %
%
3
1
−1
−3
0.8
−5
10
0.6
8
0.4
6
!"
#!
4
0.2
2
0
0
12
1. FUNCTIONS OF SEVERAL VARIABLES
1.1.1
Visualising functions
Consider the volume of a cylinder as a function of two variables:
V (r, `) = πr2 `.
A first method of visualising this function is to plot V for fixed ` and varying r.
Using Matlab, the ezplot function is the simplest graphing aid:
ezplot(’pi*r^2*1’,[0,5])
hold on; ezplot(’pi*r^2*2’,[0,5])
hold on; ezplot(’pi*r^2*0.5’,[0,5])
Alternatively, we could keep r fixed and plot V as a function of `:
ezplot(’pi*1^2*l’,[0,5])
hold on;
ezplot(’pi*0.5^2*l’,[0,5])
ezplot(’pi*2^2*l’,[0,5])
1.1. INTRODUCTION
13
Better still is visualising the surface by sketching the graph of z = f (x, y). In
Matlab, the ezsurf function will prove to be our quickest tool:
ezsurf(’pi*r^2*l’,[0,2,0,2])
!"#$&%('*)+-,.0/'
200
C
[email protected] ? A
<>9;: =
150
100
50
0
4
12(4357628'
2
0
0
0.5
1
1.52
2.5
3
3.5
4
Note how you can change the point of view of the plot with the ‘rotate’ button.
In Matlab we can plot contour lines for the volume function using the ezcontour
function:
ezcontour(’pi*r^2*l’,[0,2,0,2])
Using these tools, you should be able to replicate the plots in Stewart, Section 14.1
(Section 15.1).
We shall start the course by looking at quadratic functions of two variables. These
have contours which are circles, ellipses, parabolas or hyperbolas.
1.1.2
Main points
• What is a function of several variables and why are they useful?
• Methods to visualise a function of two variables.
• Matlab commands ezplot, ezsurf and ezcontour.
14
1. FUNCTIONS OF SEVERAL VARIABLES
1.2
Conic Sections
The goal of this section is for you to become an expert in graphing parabolas,
circles, ellipses and hyperbolas (without use of a graphics calculator, of course).
The key-trick to this is “completing the square”. See also Stewart, Section 10.5
(Section 11.5).
1.2.1
Parabolas
Example: Sketch the following parabolas:
y = x2 ,
y = x2 + 3,
y = (x + 2)2 ,
y = 2x2 ,
y = 2(x + 2)2 + 3.
1.2. CONIC SECTIONS
15
Putting a quadratic y = ax2 + bx + c in the form
y = a(x − h)2 + k,
is called completing the square.
Question: What is the interpretation of the point (h, k)?
Completing the square makes use of the identity
x2 + 2xh + h2 = (x + h)2 .
Example: Sketch the parabola y = x2 + 6x + 8.
16
1. FUNCTIONS OF SEVERAL VARIABLES
Example: Sketch the parabola y = −3x2 − 6x + 1.
Example: Prove the quadratic formula. In other words, given ax2 + bx + c = 0,
by completing the square, show that
x=
−b ±
√
b2 − 4ac
.
2a
1.2. CONIC SECTIONS
17
Parabolas, circles, ellipses and hyperbolas are all examples of conic sections; they
all arise by intersecting a double cone such as z 2 = x2 + y 2 , with a plane.
See the diagrams in Stewart p. 694 (p. 690). Slice an upright cone horizontally and
you get a circle. Tip your knife a bit and you will get an ellipse. Keep tipping and
you will get a parabola. Continue tipping and you will cut out an hyperbola.
18
1.2.2
1. FUNCTIONS OF SEVERAL VARIABLES
Circles
A circle centered at the origin (0, 0) with radius r is given by the equation
x2 + y 2 = r 2 .
A circle with radius r and center (h, k) corresponds to
(x − h)2 + (y − k)2 = r2 .
1.2. CONIC SECTIONS
Example: Sketch the circle x2 + 10x + y 2 − 4y + 20 = 0.
19
20
1.2.3
1. FUNCTIONS OF SEVERAL VARIABLES
Ellipses
In standard form, an ellipse centered at the origin is given by the equation
x2 y 2
+ 2 = 1.
a2
b
An ellipse may also be defined as the set of points in a plane the sum of whose distances from two fixed focal points F1 and F2 is a constant, and you should definitely
ask your lecturer to use two pins and a piece of string to demonstrate this. If a ≥ b,
then the foci are at (±c, 0) where c2 = a2 − b2 . These lie on the major axis (x-axis
here). The ellipse intersects the x axis at the vertices (±a, 0), and the y-axis at the
vertices (0, ±b).
Question: What is the equation of an ellipse centered at the point (h, k)?
1.2. CONIC SECTIONS
Example: Sketch the ellipse x2 − x + 9y 2 = 0.
We could check this in Matlab with the command
ezplot(’x^2-x+9*y^2=0’,[-1,2],[-1,1])
21
22
1.2.4
1. FUNCTIONS OF SEVERAL VARIABLES
Hyperbolas
A hyperbola centered at the origin, with asymptotes y = ±bx/a, is given by the
equation
x2 y 2
− 2 =1
a2
b
or by
x2 y 2
− 2 + 2 = 1.
a
b
A hyperbola may also be defined as the set of points in a plane, the difference of
whose distances from two fixed focal points F1 and F2 is a constant. For such a
hyperbola, the foci are at (±c, 0) and (0, ±c) respectively, where c2 = a2 + b2 .
Question: What is the equation of a hyperbola centered at the point (h, k)?
1.2. CONIC SECTIONS
Example: Sketch the hyperbola x2 − 4y 2 = 9.
Another kind of hyperbola is the very familiar y = 1/x.
23
24
1.2.5
1. FUNCTIONS OF SEVERAL VARIABLES
Quiz
Can you match each equation with one of the graphs?
(a) x2 − y 2 − 1 = 0
(d) x2 + 2x − y + 1 = 0.
(b) x + y 2 − 1 = 0
(e) x2 − y 2 + 4 = 0.
(c) x2 − 2x + y 2 = 0
(f) 4x2 − 8x + y 2 = 0.
y
y
2
1
x
1
x
−1
−2
y
y
1
−1
1
x
1
x
−1
y
y
2
1
−1
1
x
−2
Double-check each answer by putting each equation in standard form.
x
1.2. CONIC SECTIONS
1.2.6
25
Main points
• By completing the square, you should be able to
(i) sketch any curve of the form ax2 + by 2 + cx + dy + e = 0,
(ii) identify this curve as a straight line (or lines), circle, parabola, ellipse or
hyperbola, and
(iii) point out its main features such as turning points, asymptotes and intercepts with the x- and/or y-axes.
• You should also be able to obtain the equation of a curve of the above form
from a sketch with features given.
26
1.3
1. FUNCTIONS OF SEVERAL VARIABLES
Scalar Equation for a Plane
The equation of a plane can be expressed using scalars or vectors. In this section
we will sketch planes in R3 and determine their scalar equations.
This material is covered in Stewart, Section 12.5 (Section 13.5).
1.3.1
Horizontal planes
The x- and y-axes lie in the horizontal plane z = 0. All other horizontal planes are
parallel to z = 0 and are given by the equation z = c.
Example: Sketch the horizontal plane z = 2.
1.3. SCALAR EQUATION FOR A PLANE
1.3.2
27
Vertical planes
A vertical plane has the form ax + by = d; it depends on x and y only and z does
not appear. If you are not told this is an equation of a plane, or, equivalently, an
equation in R3 , then you cannot distinguish it from the equation of a line in R2 .
Example: Sketch the plane x + y = 1.
First sketch the line x + y = 1 in the xy-plane; the plane x + y = 1 contains this
line and is parallel to the z axis.
1.3.3
Arbitrary planes
The general equation of a plane in R3 is given by
ax + by + cz = d
with a, b, c, d fixed real numbers. If the plane is not vertical, i.e., c 6= 0, this equation
can be rearranged so that z is expressed as a function of x and y:
z = F (x, y) = −(a/c)x − (b/c)y + (d/c) = mx + ny + z0 .
To plot the plane in Matlab, simply plot F (x, y) using ezsurf. The easiest way to
sketch the plane by hand is to use the triangle method: If all of a, b, c 6= 0 the plane
ax + by + cz = d intercepts each axis at precisely one point. These three points
make up a triangle which fixes the plane.
28
1. FUNCTIONS OF SEVERAL VARIABLES
Example: The plane x + 2y + z = 4 intersects the x-axis at x = 4, the y-axis at
y = 2 and the z-axis at z = 4.
"!
5
0
−5
−10
0
1
2
3
4
5
4
3
2
1
0
The triangle method is based on the simple fact that any three points that lie in a
plane uniquely determine this plane provided these three points do not lie on a single
straight line.
1.3. SCALAR EQUATION FOR A PLANE
29
Example: Use the triangle method to sketch the plane 2x + 3y − z + 6 = 0.
In Matlab, the graphing command is simply:
ezsurf(’2*x+3*y+6’,[-2,2,-5,5])
20
10
0
−10
−2
−1
−2
−1
0
0
1
1
2
2
30
1. FUNCTIONS OF SEVERAL VARIABLES
Important remark: It is customary to say the equation of a plane, even though
it is not unique. Multiplying the equation of a plane by a nonzero constant gives
another equation for the same plane. For example, x − 2y + 3z = 4 and −2x + 4y −
6z = −8 are equations of the same plane.
Example: Find the equation of the plane through (0, 0, 5), (1, 3, 2) and (0, 1, 1).
50
0
−50
−5
−2
−1
0
0
1
5
2
The Matlab command to produce a plot of this plane would be:
ezsurf(’9*x-4*y+5’,[-2,2,-5,5])
1.3. SCALAR EQUATION FOR A PLANE
31
The following picture shows the intersection of the planes 2x + 3y − z + 6 = 0 and
z = 9x − 4y + 5. Note that the intersection is a line.
"!$#&%(' )*+ ,#.-"!$#&/
50
0
−50
−5
−3
−2
−1
−1
1
!
0
3
1
5
2
To produce two or more plots on the same figure, in Matlab use the hold command:
hold on;
1.3.4
Main points
• You should be able to draw horizontal, vertical and general planes by hand.
• You should be able to plot non-vertical planes in Matlab.
• You should be able to find the equation of a plane through three given points
that are not on a straight line.
• You should know the scalar equation of a plane.
32
1. FUNCTIONS OF SEVERAL VARIABLES
1.4
Vector Equation of a Plane I
For this section see also Stewart, Sections 12.1–12.5 (Sections 13.1–13.5).
1.4.1
Revision of vectors
Pythagoras gives the distance d between the points (x0 , y0 ) and (x1 , y1 ) as
p
d = (x0 − x1 )2 + (y0 − y1 )2 .
In three dimensions, the distance d between (x0 , y0 , z0 ) and (x1 , y1 , z1 ) is
p
d = (x0 − x1 )2 + (y0 − y1 )2 + (z0 − z1 )2 .
Example: Find the distance between the points (1, 2, 3) and (0, 0, 0).
The norm of a vector is its length. In Matlab, you can directly call the norm of a
vector:
v=[1 2 3]
v =
1
2
norm(v)
ans =
3.7417
3
Recall that the dot product of two vectors a and b is given by
a · b = (a1 , a2 , a3 ) · (b1 , b2 , b3 ) = a1 b1 + a2 b2 + a3 b3 .
1.4. VECTOR EQUATION OF A PLANE I
33
It is also true that
a · b = kak kbk cos θ,
where θ is the angle between a and b, and kak is the norm of a:
q
√
kak = a · a = a21 + a22 + a23 .
If a · b = 0, then a and b are orthogonal.
The projection of b onto a is “the component of b in the direction of a”:
proja b = kbk cos θ ×
a
a·b
=
a.
kak
kak2
34
1. FUNCTIONS OF SEVERAL VARIABLES
Recall that the cross product a × b of two vectors in R3 is a vector orthogonal to
both a and b given by
a × b = (a1 , a2 , a3 ) × (b1 , b2 , b3 ) = (a2 b3 − a3 b2 , a3 b1 − a1 b3 , a1 b2 − a2 b1 ).
This is best memorised using a 3 by 3 determinant.
The cross product points in the direction given by the right-hand-rule.
1.4. VECTOR EQUATION OF A PLANE I
35
It is also true that
ka × bk = kak kbk sin θ.
This means that ka × bk gives the area of the parallelogram spanned by the vectors
a and b.
Example: Compute the area of the triangle with vertices (1, 1), (2, 3), (4, 2).
36
1.4.2
1. FUNCTIONS OF SEVERAL VARIABLES
Vector equation of a plane I
We have seen in Section 1.3 that any three points in R3 which do not lie on a straight
line determine a plane. There is another geometric way to view a plane, which gives
rise to the vector equation of a plane.
Let P be a plane and n be a vector perpendicular to P . Such a vector is called a
normal to P . Let r 0 be a vector from the origin to a point r 0 in P and projn r 0 its
projection onto n. Then the length of this projection vector, i.e., kprojn r 0 k is the
distance of the plane to the origin.
n
projn r 0
r0
0
If we had taken another vector, say r, from the origin to a point r in P then
projn r = projn r 0 .
In other words, the plane P is given by the collection of all points r whose corresponding vectors r have the same orthogonal projection onto n as r 0 . Hence
n · (r − r 0 ) = 0.
This is the vector equation for a plane.
The vector and scalar equations are in fact the same, as shown next.
1.4. VECTOR EQUATION OF A PLANE I
37
We start with the vector equation, where n = (a, b, c) is a normal, r 0 = (x0 , y0 , z0 )
and r = (x, y, z). Then
n · (r − r 0 ) = 0
(a, b, c) · (x − x0 , y − y0 , z − z0 ) = 0
a(x − x0 ) + b(y − y0 ) + c(z − z0 ) = 0
ax + by + cz = d
where d = n · r 0 = ax0 + by0 + cz0 .
Important remark: Given a plane ax + by + cz = d we now have a geometric
interpretation of the vector (a, b, c): it is a normal to the plane. Obviously the
normal is not unique since any vector of the form k(a, b, c), where k is a nonzero
scalar, is also a normal.
Example: Find the vector and scalar equations for the plane passing through the
point (2, 4, −1) with normal n = 2i + 3j + 4k = (2, 3, 4).
38
1. FUNCTIONS OF SEVERAL VARIABLES
Our understanding of normal vectors allows us to compute angles between planes,
see also Stewart Section 12.5 (Section 13.5).
Example: Find the angle between the planes x+4y−3z = 1 and −3x+6y+7z = 0.
Important remark: Contrary to popular belief, the angle between two planes
cannot exceed π/2. It is a cardinal offense to ever hand in work that claims otherwise.
1.4.3
Main points
• You should be able to use the dot product, projections and the cross product.
• You should be able to derive the scalar equation from the vector equation of
a plane.
• You should be able to find the equation of a plane given sufficient information.
1.5. EQUATIONS FOR A LINE
1.5
39
Equations for a Line
There are three common ways to represent a line:
(a) the vector representation;
(b) the parametric representation;
(c) the symmetric equations, obtained by eliminating parameters in (b).
The representation that is best depends on the particular problem at hand.
(a) Vector representation
Let P0 = (x0 , y0 , z0 ) be a point on the line L with corresponding position vector r 0
and let v be a vector parallel to L, known as a direction vector. For an arbitrary
point P = (x, y, z) lying on L we have
−→
P0 P = (x − x0 , y − y0 , z − z0 ).
P
r
P0
r0
v
0
−→
−→
Since P0 P , is parallel to v, we have P0 P = λv for some scalar λ. Hence
−→
r = r 0 + P0 P = r 0 + λv.
The equation
r = r 0 + λv,
λ∈R
is know as the vector equation of the line L.
Important remark: As with the equation of a plane, the vector equation of a line
is not unique. One can choose any point P0 on the line as “starting point”and one
can multiply the vector v by any nonzero constant.
40
1. FUNCTIONS OF SEVERAL VARIABLES
Example: Find (a) the vector representation of the line passing through (5, 1, 3)
and parallel to the vector (1, 4, −2) and (b) the vector representation of the line
passing through (4, −3, 5) and parallel to the vector (−2, −8, 4). What is the relation
between the two lines?
1.5. EQUATIONS FOR A LINE
41
(b) Parametric representation
The parametric representation of a line is a scalar representation of the vector equation r = r 0 + λv. Writing each vector in component form,
r = (x, y, z)
r 0 = (x0 , y0 , z0 )
v = (a, b, c)
gives
(x, y, z) = (x0 , y0 , z0 ) + λ(a, b, c).
Matching components results in three scalar equations



x = x0 + aλ
y = y0 + bλ


z = z + cλ
(∗)
0
known as the parametric equations of a line.
Example: Find the parametric equations for the line passing through (5, 1, 3) and
parallel to i + 4j − 2k.
42
1. FUNCTIONS OF SEVERAL VARIABLES
(c) Symmetric equations
The parameter λ can be eliminated from the parametric equations of a line. For
example, by eliminating λ from each of the three equations x = 5 + λ, y = 1 + 4λ,
z = 3 − 2λ for the line L, we obtain
λ=x−5=
y−1
z−3
=
.
4
−2
The equations
z−3
y−1
=
4
−2
are known as the symmetric equations of L.
x−5=
What these equation really are is a set of two non-identical, non-parallel planes
x−5=
y−1
4
and x − 5 =
z−3
,
−2
which, as we know, must intersect to give a line.
The general form of the symmetric equations of the line (∗) from the previous page
is given by
x − x0
y − y0
z − z0
=
=
,
a
b
c
but some care is needed when one (or two) of a, b or c is equal to zero. For example,
if a = 0 the above needs to be replaced by
x = x0 ,
and so on.
y − y0
z − z0
=
b
c
1.5. EQUATIONS FOR A LINE
43
Example: Find the symmetric equations of the line that passes through the points
(2, 4, −3) and (3, −1, 1).
44
1. FUNCTIONS OF SEVERAL VARIABLES
Example: Find the symmetric equations of the line r = (1, 2, 3) + λ(0, 1, 0).
To plot a line in three-dimensional space, use the parametric form. The Matlab
command is ezplot3. As input it takes three functions of a parameter, and the
range of the parameter.
Example: Plot the line



x = 100 + 200t
y = 200 + 300t


z = 300 − 10t
for t between −100 and 100. The Matlab command would be
ezplot3(’100+200*t’,’200+300*t’,’300-10*t’,[-100,100])
You can grab and rotate the resulting axes.
1.5. EQUATIONS FOR A LINE
1.5.1
45
Parallel, skew and orthogonal Lines.
Two lines are parallel if, when written as r = r 0 +λv, and r = s0 +µu, the direction
vectors u and v are linear multiples of each other, i.e., u = kv for some nonzero
scalar k.
Two straight lines in three dimensional space rarely intersect. Non parallel, nonintersecting lines are called skew lines.
Example: Show that the lines L1 and L2 given by




x
=
1
+
λ


x = 2µ

L2 : y = 3 + µ
L1 : y = −2 + 3λ




z = −3 + 4µ
z = 4 − λ
are skew lines.
46
1. FUNCTIONS OF SEVERAL VARIABLES
Two lines are orthogonal if their direction vectors are orthogonal. An easy test for
orthogonality is to take the dot product of the direction vectors; if and only if this
gives zero are the lines orthogonal.
Example: Show that the lines L1 and L2 given by
x−1
z+5
=y−3=
,
2
4
5−y
= z − 5.
L2 : 2 − x =
2
L1 :
are orthogonal.
Further theory
Unlike lines, planes can not be skew in 3 dimensions. Given two planes, the only
possibilities are that they intersect, or that they are parallel. If two planes are
parallel, then their normal vectors must be parallel. If two (distinct) planes intersect,
then they intersect in a line.
1.5. EQUATIONS FOR A LINE
47
Example: Find the parametric equations for the intersection of the planes
z =x+y
and 2x − 5y − z = 1.
48
1.5.2
1. FUNCTIONS OF SEVERAL VARIABLES
Vector equation of a plane II
Previously we discussed the scalar equation of a plane, which is an equation of the
form ax + by + cz = d, and vector equation a plane, which is an equation of the form
n · (r − r 0 ) = 0.
There is a second type of vector equation of a plane, which is similar in form to
the vector equation of a line. One fixes a point P0 on the plane with corresponding
position vector r 0 , but now not one, but two (non-parallel) vectors u and v parallel
to the plane are required to fully determine the plane:
r = r 0 + λu + µv,
λ, µ ∈ R.
Example: Sketch a figure explaining the above equation geometrically.
Important remark: In some sense the above vector equation of a plane is even
less unique than the vector equation of a line. Again one can take r 0 to be any
point on the plane and one can multiply both u and v by nonzero constants. But
unlike a line, we can also replace u and v by any other pair of (independent) vectors
parallel to the plane. For example, (x, y, z) = (1, 0, 1) + λ(0, 1, 2) + µ(3, 1, −1) and
(x, y, z) = (4, 1, 0) + λ(−1, 0, 1) + µ(−3, 1, 5) are different vector representations of
the same plane! So do not panic if in your exam the person next to you writes down
a very different-looking equation; they still might have the right answer. . .
1.5. EQUATIONS FOR A LINE
49
Example: Find the second type of vector equation of the plane through (0, 0, 5),
(1, 3, 2) and (0, 1, 1).
Example: Find the second type of vector equation for the plane 2x + y − z = 4.
50
1.5.3
1. FUNCTIONS OF SEVERAL VARIABLES
Distance from a Point to a Plane
A common problem that arises in applications is to find the distance from a point
to a plane, where, by distance, we always mean minimum distance.
If P = (x1 , y1 , z1 ) is a point, with corresponding position vector p, and Π is a plane
(with normal n = (a, b, c)) given by ax + by + cz = d, then the formula for the
distance between P and Π is
|n · p − d|
|ax1 + by1 + cz1 − d|
√
D=
=
.
knk
a2 + b 2 + c 2
Important remark: It is best not to try to memorise such a formula because,
oddly enough, under exam pressure small details often subtly change for the worse.
Also, it depends on one’s choice of representation of a plane. Stewart, for example
writes ax + by + cz + d = 0 in which case the numerator contains +d instead of −d.
Best is to understand the actual derivation given below.
One derivation of the above distance formula using orthogonal projections is given
in Stewart, Section 12.5 (Section 13.5). Below we present an alternative method,
which is not quite as slick, but which is very easy to carry out for explicit examples,
even under exam conditions.
First we construct the line L through the point P and orthogonal to the plane Π:
L:
r = p + λn,
λ ∈ R.
Next we determine the point, say Q, where L intersects Π. That is, we substitute
the equation for L into the equation r · n = d for Π:
(p + λn) · n = d.
Solving for λ yields
d−p·n
,
knk2
so that the point Q, with position vector q, is given by
λ=
q =p+
d−n·p
n.
knk2
The distance between P and Π is now the distance between P and Q. But
p−q =
so that
n·p−d
n
knk2
(n · p − d)2
(n · p − d)2
n
·
n
=
.
knk4
knk2
The formula now follows by taking the square root on both sides and remembering
that a distance can never be negative, whereas n · p − d can.
D2 = kp − qk2 =
1.5. EQUATIONS FOR A LINE
51
Example: Find the distance from the point (1, 2, 3) to the plane z = 4.
Important remark: In assignments or exams we do not provide boxes whose size
might indicate how hard or how trivial an actual question is. Hence it is important
to always think first before you start to calculate. Sometimes, just thinking will be
enough.
Example: Find the distance between the point (1, 2, 3) and the plane x+y +z = 1.
52
1. FUNCTIONS OF SEVERAL VARIABLES
1.5.4
Main points
• You should know the various representations for a line.
• Given one representation of a line, you should be able to find the other representations.
• You should be able to find the equation for a line given sufficient information.
• You should know how to find the intersection of two planes or a plane and a
line.
• You should be able to find the distance from a point to a plane, or between
two planes.
1.6. CONTOUR DIAGRAMS
1.6
53
Contour Diagrams
Geographical maps have curves of constant height above sea level, or curves of
constant air pressure (isobars), or curves of constant temperature (isothermals).
Drawing contours is an effective method of representing a 3-dimensional surface
in two dimensions. We now look at functions f of two variables. A contour is a
curve corresponding to the equation z = f (x, y) = C, see also Stewart, Section 14.1
(Section 15.1).
Consider the surface z = f (x, y) = x2 + y 2 sliced by horizontal planes z = 0, z =
1, z = 2, . . .
Plane
z=0
Contour
Description
z=1
z=2
z=3
z=4
Note that as the radius increases, the contours are more closely spaced.
54
1. FUNCTIONS OF SEVERAL VARIABLES
!"$#%'&)(* +-,!. & 0/213,4/
4
3.5
3
%
2.5
2
1.5
1
0.5
0
−2
−1
0
1
,
2
2
1
0
−1
−2
Example: Draw a contour diagram of f given by
p
f (x, y) = x2 + y 2 .
In Matlab, the command is
ezcontour(’sqrt(x^2+y^2)’,[-10,10,-10,10]);
If horizontal planes are equally spaced, say z = 0, c, 2c, 3c, . . . , it is not hard to
visualise the surface from its contour diagram. Spread-out contours mean the surface
is quite flat and closely spaced ones imply a steep climb.
1.6. CONTOUR DIAGRAMS
55
! "$#&% '!()#*'
4
3.5
3
2.5
2
1.5
1
0.5
0
−4
−2
0
2
4
#
2
4
−4
−2
0
Note that the contours of the last two functions were all circles. Such surfaces have
circular symmetry. When x and y only appear as x2 + y 2 in the definition of f , then
the graph of f has circularp
symmetry about the z axis. The height z depends only
on the radial distance r = x2 + y 2 .
Example: z = f (x, y) = e−x
2 −y 2
!#"$&%'"('%
)*+-,* * ./0 !#"$&%'"('%
2
1
1.5
0.2
0.3
1
0
0.4
0.6
0.1
−0.5
0.3
0.2
0.9
0.8
0.7
0.4
0.5
0.1
0.5
0.6
4
0. 5
0.
0.1
0.8
−1
0.2
0
2
−1.5
1
2
1
0
0
−1
−1
−2
−2
−2
−2
−1
0
1
2
56
1. FUNCTIONS OF SEVERAL VARIABLES
Example: Draw a contour diagram for z = f (x, y) = x2 + 4y 2 − 2x + 1.
Plane
z=1
z=2
z=3
Contour
Description
1.6. CONTOUR DIAGRAMS
57
Example: A saddle z = x2 − y 2 has hyperbolic contours.
z=0:
z=1:
z=2:
z = −1 :
z = −2 :
!"
2
−2
−2
−1
1.5
1
0
−1
1
0.5
2
0
2
0
−0.5
0
−1
1
1
1
−1.5
−1
0
−1
−2
−2
−2
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
58
1. FUNCTIONS OF SEVERAL VARIABLES
Example: Sketch the contour diagram for z = f (x, y) = 9x2 −4y 2 +2 with contours
at z = −2, 2, 6 and 10.
1.6. CONTOUR DIAGRAMS
59
Example: Sketch a contour diagram for z = f (x, y) = x2 and use this to sketch
the graph of f .
60
1. FUNCTIONS OF SEVERAL VARIABLES
Example: Sketch a contour diagram for z = f (x, y) = x − y 2 .
Example: Sketch contour diagrams of z = f (x, y) = x and z = g(x, y) = x + y.
Note: Contour diagrams of functions whose graphs are planes consist of equidistant
parallel lines. (Equidistant lines in R2 are in fact always parallel, unlike R3 . Why?)
1.6. CONTOUR DIAGRAMS
1.6.1
61
Main points
• You should be able to plot contour diagrams in Matlab using the ezcontour
function (and its variants).
• You should be able to recognise circular symmetry in an equation.
• You should be able to match contour diagrams with functions.
• You should be able to sketch simple contour diagrams.
62
1.7
1. FUNCTIONS OF SEVERAL VARIABLES
Cross-sections of a Surface
A cross-section is the intersection of a surface with a vertical plane such as y = C,
see also Stewart Section 12.6 (Section 13.6).
Example:
The height z of a vibrating guitar string can be expressed as a function of horizontal
distance x, and time t
z = f (x, t) = A sin(πx) cos(2πt)
where
0 < x < 1.
The snapshots where t is constant are cross-sections of the ‘surface’.
Varying time we get
t=0:
1
t= :
8
1
t= :
4
3
t= :
8
1
t= :
2
These represent sine curves, with
z = A sin(πx)
1√
z=
2A sin(πx)
2
z=0
z=−
1√
2A sin(πx)
2
z = −A sin(πx).
amplitudes between 0 and A.
We can also consider the cross-sections in x. For instance x = 21 (at the top of the
sine wave), then z = A cos(2πt) which equals the amplitude of the sine wave.
1.7. CROSS-SECTIONS OF A SURFACE
63
"!$#&%&'()+*,! %
1
0
−1
1
1
0.8
0.6
#
0.4
0.5
0.2
0
0
Matlab can be used to make a movie of the 2-dimensional surface by plotting crosssections at different t values in sequence. The sequence of plots can be stored in a
vector and played as a movie using the following code:
x=(0:0.25:1);
for j=1:100
t=j/25;
z=sin(pi*x)*cos(2*pi*t);
plot(x,z);axis([0,1,-1,1]);
M(j)=getframe;
end
Note: ezplot cannot be used to do this because Matlab gets confused about which
of t, x is a variable and which is a number.
64
1. FUNCTIONS OF SEVERAL VARIABLES
Example: Sketch the cross-sections of the surface V = 13 πr2 h (volume of a cone).
Here is the 3-dimensional picture from Matlab.
!"$#&%('*),+-/.0(1%
A
@> = ?
:<798 ;
8
6
4
2
2
0
0
1.5
1
0.5
1
2&34656%
0.5
1.5
2
0
1.7. CROSS-SECTIONS OF A SURFACE
65
Example: The cross-sections of a saddle z = x2 − y 2 are parabolas. For y = y0
they point up: z = x2 − (y0 )2 ; and for x = x0 they point down: z = −y 2 + x20 .
The surface is tricky to draw, unless you are an equestrian.
Here is the Matlab plot of the saddle z = x2 − y 2 and its contours.
4
2
0
−2
−4
2
1
2
1
0
0
−1
−1
−2
−2
66
1. FUNCTIONS OF SEVERAL VARIABLES
Example: Use cross-sections to sketch the graph of z = f (x, y) = x2 .
1.7.1
Main points
• You should be able to construct cross-sections of multivariate functions.
• Cross sections are 2-dimensional graphs.
• Animation of cross-sections is another way to visualise multivariate functions.
67
2
Partial Derivatives and Tangent Planes
We will need to consider derivatives of functions of more than one variable. To
do this, we first check how the familiar concepts of limits and continuity extend
to functions of more than one variable. This material is covered in Section 14.2
(Section 15.2) of Stewart.
2.1
2.1.1
Limits and Continuity
Review of the 1-variable case
Let f : D → R be a function with domain D an open subset of R. For a ∈ D we
say that the limit lim f (x) exists if and only if, (i) the limit from the left exists, (ii)
x→a
the limit from the right exists, and (iii) these two limits coincide, i.e.,
lim f (x) = lim+ f (x).
x→a−
x→a
Furthermore, if the limit exists and is equal the actual value of f at a, i.e., if
lim f (x) = lim+ f (x) = f (a),
x→a−
x→a
we say that f is continuous at x = a.
If f is continuous on all of D we say that f is a continuous function on D.
68
2. PARTIAL DERIVATIVES AND TANGENT PLANES
We can also consider the limit for points on the boundary of the domain D of a
function. For example, if f : (0, 2) → R is defined by f (x) = 1/x, then
lim− f (x) =
x→2
1
2
but
lim f (x)
x→0+
does not exist.
Another instructive example is f : R\{0} → R given by f (x) = x2 . The domain is
now the punctured real line, i.e., D = (−∞, 0) ∪ (0, ∞), but
lim f (x) = lim+ f (x) = 0.
x→0−
x→0
In this situation we also say that limx→0 f (x) exists and in fact one can fix the hole
by defining f (0) = 0, to extend f to a continuous function on all of R.
Important remark: Never, ever compute limx→a f (x) by blindly substituting
x = a in f . For example, if
(
x2 for x 6= 3
f (x) =
π for x = 3,
then limx→3 f (x) exists and is given by 9 which is not equal to π: the function f is
not continuous at x = 3.
y
9
π
x
3
As a second example, if f : D → R with D = R\{1} is given by
f (x) =
x2 − 1
,
x−1
then limx→1 f (x) = 2. Those (and there will be some) who write “This limit gives
0/0 which does not exist” should hang their heads in shame.
2.1. LIMITS AND CONTINUITY
2.1.2
69
Multivariable limits
When f is a function of more than one variable, the situation is more interesting.
There are more than two ways to approach a given point of interest. Consider the
function
x2
f (x, y) = 2
x + y2
with domain given by R2 \{(0, 0)}.
To see the graph of f in Matlab, type
ezsurf(’(x^2/(x^2+y^2)’)
! !#"
%" $ &'"
1
0.8
0.6
0.4
0.2
0
1
0.5
0
−0.5
−1
−1
−0.5
0
Next we consider the limit as (x, y) → (0, 0).
(i) Approaching the origin along y = 0:
0.5
1
70
2. PARTIAL DERIVATIVES AND TANGENT PLANES
(ii) Approaching the origin along x = 0:
Does
lim
f (x, y) exist?
(x,y)→(0,0)
In general, for the limit
lim
f (x, y) to exist, it is necessary that every path in
(x,y)→(a,b)
D approaching (a, b) (the point (a, b) itself may or may not be in D) gives the same
limiting value. This gives us the following method for finding if a limit does not
exist.
If
Test for showing no limit exists

f (x, y) → L1 as (x, y) → (a, b) along the path C1 ∈ D
f (x, y) → L
2
as (x, y) → (a, b) along the path C2 ∈ D
such that L1 6= L2 , then the limit
lim
f (x, y) does not exist.
(x,y)→(a,b)
Important remark: The above notation is somewhat deficient and perhaps one
should write
lim
f (x, y)
(x,y)→D (a,b)
to indicate that only paths in D terminating in (a, b) (which itself may or may not be
in D) are considered. For example, if f (x, y) = x2 + y 2 with D = {(x, y) : x2 + y 2 <
1} then lim(x,y)→(1,0) f (x, y) exists and is 1. However, if
(
x2 + y 2 for D = {(x, y) : x2 + y 2 < 1}
f (x, y) =
0
for D = {(x, y) : x2 + y 2 > 1}
then lim(x,y)→(1,0) f (x, y) does not exist.
2.1. LIMITS AND CONTINUITY
71
Example: Let D = R2 \{(0, 0)} and f : D → R2 be given by f (x, y) =
x2 − y 2
.
x2 + y 2
Show that lim(x,y)→(0,0) f (x, y) does not exist.
Example:
With the same D as above but now f (x, y) =
lim(x,y)→(0,0) f (x, y) does not exist.
x2
xy
, show that
+ y2
72
2. PARTIAL DERIVATIVES AND TANGENT PLANES
Important remark: There are infinitely many paths terminating in a given point,
say (a, b), in R2 , raising the question if one can ever prove that lim(x,y)→(a,b) f (x, y)
does exist. The good news is that there are methods that can deal with infinitely
many paths simultaneously. The bad news is that these methods (typically -δ
proofs) are not part of this course. See Stewart Sec 14.2 (Sec 15.2), Example 4 for
a rigorous -δ proof that
lim f (x, y) = 0,
(x,y)→(0,0)
where
f (x, y) =
3x2 y
(x2 + y 2 )
and D = R2 \{(0, 0)}.
Example: Give a non-rigorous proof that the above limit is indeed correct by
writing x = r cos θ and y = r sin θ.
2.1. LIMITS AND CONTINUITY
2.1.3
73
Multivariable continuity
Definition Given a function f : D → R2 , where D is an open subset of R2 . Let
(a, b) ∈ D. Then f (x, y) is continuous at (a, b) if
lim
f (x, y) = f (a, b),
(x,y)→(a,b)
i.e., the limit (x, y) → (a, b) of f (x, y) exists and is equal to f (a, b).
If a function is continuous on all of D we say simply that it is continuous on D.
Most of the functions we will consider are continuous. For example, polynomials in
x and y are continuous on R2 . As a rule of thumb, if a function with domain D is
defined by a single expression it will be continuous on D.
x2 − y 2
Example: Returning to the first example on page 71, where f (x, y) = 2
and
x + y2
D = R2 \{(0, 0)}, is f (x, y) a continuous function?
Example: If we edit the above example by instead defining f on all of R2 by taking
f (0, 0) = 0, then is f a continuous function?
2.1.4
Main points
• You should be able to show when a limit does not exist.
• You should understand continuity of multivariate functions.
74
2.2
2. PARTIAL DERIVATIVES AND TANGENT PLANES
Partial Derivatives
This material is covered in Stewart, Section 14.3 (Section 15.3).
2.2.1
Slope in the x-direction
Consider the surface z = f (x, y) = 1 − x2 − y 2 and the point P = (1, −1, −1) on
the surface. Use the “y-is-constant” cross-section through P to find the slope in the
x-direction at P .
"!$#%'&)(+*,.-/ 10 (32546-8794
7
0
&
−2
−4
−6
−8
−2
−1
2
1
0
-
0
1
−1
2
−2
The slope in the x-direction, with y held fixed, is called the partial derivative of f
with respect to x at the point (a, b)
∂f
f (a + h, b) − f (a, b)
(a, b) = fx (a, b) = lim
.
h→0
∂x
h
2.2. PARTIAL DERIVATIVES
2.2.2
75
Slope in the y-direction
Use the “x-is-constant” cross-section to find the slope at P = (1, −1, −1) in the y
direction, i.e., where x = 1.
"!#%$&!
$
0
−2
−4
−6
−8
−2
'"(*)+,.-0/./.12/.3+4.52-06
2
−1
1
0
0
1
−1
2
−2
Similarly, the slope in the y-direction, with x held fixed, is called the partial derivative of f with respect to y at the point (a, b)
∂f
f (a, b + h) − f (a, b)
(a, b) = fy (a, b) = lim
.
h→0
∂y
h
Important remark: Normal rules of differentiation apply, we simply think of the
variables being held fixed as constants when doing the differentiation.
76
2. PARTIAL DERIVATIVES AND TANGENT PLANES
Example: Find the partial derivatives
∂f
∂f
and
of f (x, y) = x sin y + y cos x.
∂x
∂y
Example: Given f (x, y) = xy 3 + x2 , find fx (1, 2) and fy (1, 2).
2.2. PARTIAL DERIVATIVES
2.2.3
77
Partial derivatives for f (x, y, z)
Example: The volume of a box V (x, y, z) = xyz.
If x changes by a small amount, say ∆x, denote the corresponding change in V by
∆V . We can easily visualise that ∆V = yz∆x.
Therefore,
∆V
= yz.
∆x
∂V
= yz.
∂x
For partial derivatives only one independent variable changes and all other independent variables remain fixed.
Letting ∆x → 0 we have
78
2. PARTIAL DERIVATIVES AND TANGENT PLANES
Example: Parallel resistance
In an electrical circuit, the combined resistance R, from three resistors R1 , R2 and
R3 connected in parallel, is
1
1
1
1
=
+
+
.
R
R1 R2 R3
What is the rate of change of the total resistance R with respect to R1 ?
2.2. PARTIAL DERIVATIVES
2.2.4
79
Higher order derivatives
The second order partial derivatives of f , if they exist, are written as
fxx =
∂ 2f
,
∂x2
fyx =
∂ ∂f ∂ 2f
=
,
∂x∂y
∂x ∂y
fyy =
∂ 2f
,
∂y 2
fxy =
∂ ∂f ∂ 2f
=
.
∂y∂x
∂y ∂x
If fxy and fyx are both continuous, then fxy = fyx .
Example: Returning to the example on page 76 for which f (x, y) = x sin y + y cos x,
∂ 2f
∂ 2f
calculate all of the second order partial derivatives of f and show that
=
.
∂x∂y
∂y∂x
2.2.5
Main points
• You should know the definition and meaning of partial derivatives.
• You should be able to evaluate partial derivatives of functions.
80
2.3
2. PARTIAL DERIVATIVES AND TANGENT PLANES
The Tangent Plane
This section is covered in Stewart, Section 14.4 (Section 15.4).
2.3.1
Review for f (x)
Recall that if y = f (x) then the tangent line at the point (a, f (a)) is given by
y − f (a) = f 0 (a)(x − a)
⇔
y = f (a) + f 0 (a)(x − a).
Example: Find the tangent line to y = f (x) = x2 at x = 1.
2.3. THE TANGENT PLANE
2.3.2
81
Equation for a tangent plane
In general, the equation of the tangent plane to a given surface z = f (x, y) at
(a, b, f (a, b)), is
z = f (a, b) + fx (a, b)(x − a) + fy (a, b)(y − b),
or, equivalently,
(x, y, z) = a, b, f (a, b) + λ 1, 0, fx (a, b) + µ 0, 1, fy (a, b) ,
λ, µ ∈ R.
Indeed, the first two components of this vector equation for the tangent plane imply
λ = x − a and µ = y − b. Substituting this into the third component gives
z = f (a, b) + λfx (a, b) + µfy (a, b) = f (a, b) + fx (a, b)(x − a) + fy (a, b)(y − b).
We also note that if we write ∆x = x − a, ∆y = y − b and ∆z = z − f (a, b) then
the vector equation for the tangent plane may be rewritten as
(∆x, ∆y, ∆z) = λ 1, 0, fx (a, b) + µ 0, 1, fy (a, b) ,
λ, µ ∈ R.
This shows that if ∆x = 1 and ∆y = 0 (i.e., λ = 1 and µ = 0) then ∆z = fx (a, b)
and if ∆x = 0 and ∆y = 1 (i.e., λ = 0 and µ = 1) then ∆z = fy (a, b), matching our
interpretation of fx and fy as the respective slopes of f in the x or y direction.
82
2. PARTIAL DERIVATIVES AND TANGENT PLANES
Example: Find the equation for the tangent plane to the surface z = 1 − x2 − y 2
at the point P = (1, −1, −1).
0
−2
−4
−6
−8
−2
2
−1
1
0
0
1
−1
2
−2
2.3. THE TANGENT PLANE
83
Example: What is the plane tangent to the surface z = f (x, y) = 4 − x2 + 4x − y 2
at (1, 1)?
!#"
"$%&"(')*+,.- 0&/ 1/ 10
5
0
−5
−10
4
−15
2
−20
−25
4
0
3
2
1
0
−1
−2
−3
−4
−2
84
2. PARTIAL DERIVATIVES AND TANGENT PLANES
Example: Find the tangent plane of z = f (x, y) = e−x
2 −y 2
at (x, y) = (1, 3).
2.3. THE TANGENT PLANE
2.3.3
85
Smoothness
Can we always find partial derivatives and tangent planes?
Example: Simple cusp-like functions are not smooth:
!#" "
$&%(' )*+,-./-.
3
2.5
2
1.5
1
2
1
2
1
0
0
0
−1
−2
−1
−2
A surface z = f (x, y) is smooth at (a, b) if f , fx and fy are all continuous at (a, b).
When you zoom in close enough to a smooth surface it looks like a plane.
One way to see this is to look at the contours. The contours of a plane are straight
parallel lines, the same perpendicular distance apart. As you zoom into a smooth
surface the contours straighten out. This means that close to (a, b) the surface is
approximated by a plane; in fact it can be approximated by the tangent plane. Do
straight contours imply smoothness?
86
2. PARTIAL DERIVATIVES AND TANGENT PLANES
Example: Brownian motion is not smooth. Look at the figure below. No matter
how much you zoom in, it always looks rough—in fact, Brownian motion is a fractal.
There are surface-analogues to Brownian motion, demonstrated with the fractal
surface below.
10
5
1500
1000
0
500
−5
0
−500
−10
−1000
−15
−1500
100
80
−20
100
60
40
−25
2.3.4
50
20
0
200
400
600
800
1000
0
0
Main points
• You should know how to find a tangent plane to a smooth surface, and recognise when a tangent plane or partial derivatives do not exist.
2.4. LINEAR APPROXIMATIONS
2.4
87
Linear Approximations
See Stewart, Section 14.4 (Section 15.4).
2.4.1
Review for f (x)
In science and engineering practice, a function f (x), of one variable, is sometimes
approximated by a straight line.
In particular, if we know an exact value (a, f (a)), then the tangent line through this
point is usually an accurate approximation close to the point. It is called a linear
or first-order approximation
f (x) ' f (a) + f 0 (a)(x − a).
Example: Estimate e0.1 .
88
2.4.2
2. PARTIAL DERIVATIVES AND TANGENT PLANES
Linear approximations for f (x, y)
The corresponding linear or first-order approximation for a function f of two variables, near a known point (a, b) is the tangent plane. The linear approximation (i.e.,
tangent plane) is accurate close to the known point, provided f is smooth.
The linear approximation to f at (a, b) is
f (x, y) ' f (a, b) + fx (a, b)(x − a) + fy (a, b)(y − b).
Example: The temperature in a region is given by T (x, y) = 100 − x2 − y 2 . Find
the linear approximation to T (x, y) near (0, 5).
!"$#%&'#
()+*
$,*
(-(.*0/12( 345,67
100
80
60
40
20
−2
0
2
4
6
8
4
2
0
−2
−4
2.4. LINEAR APPROXIMATIONS
89
2
Example: Find the tangent plane to z = e−x sin y at (1,
2
approximate value for e−(0.9) sin(1.5).
π
) and use it to find an
2
!#"%$&
')((*+,(.-/!10 3 24
1
0.5
0
−0.5
−1
0
1
2
3
$
4
5
2
1
0
−1
−2
90
2.4.3
2. PARTIAL DERIVATIVES AND TANGENT PLANES
Estimating small changes
We may use the equation for the tangent plane to infer that
∆z ≈ fx (a, b)∆x + fy (a, b)∆y,
where ∆x = x − a, ∆y = y − b and ∆z = z − f (a, b) represent a small change in
x, y and z respectively. This is useful for estimating small changes in z arising from
small changes in x and y.
Example: Electric power is given by P (E, R) = E 2 /R where E is the voltage and
R is the resistance. Find a linear approximation for P (E, R) if E ' 200 (in Volts)
and R ' 400 (in Ohms). Use this to find the effect that a change in E and R has
on P .
2.4. LINEAR APPROXIMATIONS
2.4.4
91
Estimating error
If the error in x is at most E1 and in y is at most E2 , then a reasonable estimate of
the worst-case error in the linear approximation of f at (a, b) is
|E| ≈ |fx (a, b)E1 | + |fy (a, b)E2 |.
Example: Suppose when making up a metal barrel of base radius 1 (in metres)
and height 2 (in metres), you allow for an error of 5% in radius and height. Estimate
the worst-case error in volume.
92
2. PARTIAL DERIVATIVES AND TANGENT PLANES
2.4.5
Extra reading: Differentials
If we reduce our small changes ∆x and ∆y down to infinitesimal changes dx and
dy, we may rewrite the linear approximation (tangent plane) as
dz = df = fx (a, b) dx + fy (a, b) dy.
This infinitesimal change dz is called the total differential. Note that this equation
is not just an approximation.
Example: Work done by a gas
Let the work done by an expanding gas be given by W = P V , where P denotes
pressure and V volume.
An expression for the total differential dW is then
dW =
∂W
∂W
dP +
dV
∂P
∂V
= V dP + P dV.
2.4.6
Main points
• You should be able to calculate finite small deviations of a function given a
small change in one of the variables. Note that this is a computationally more
efficient method of approximation than simply evaluating the function at x
and x + ∆x, and then taking the difference.
• You should be able to estimate error using the linear approximation.
2.5. GRADIENTS AND DIRECTIONAL DERIVATIVES
2.5
93
Gradients and Directional Derivatives
This material is covered in Stewart, Section 14.6 (Section 15.6).
2.5.1
Directional derivatives
The partial derivative fx (or fy ) corresponds to the slope of f (x, y) in the x-direction
(or y-direction). We now turn our attention to the question of slopes in arbitrary
directions, such as i + 2j or −j.
Let u = (u1 , u2 ) be an arbitrary unit vector in R2 , i.e., kuk = 1. We have already
seen that the equation of the tangent plane at (a, b) of a function f may be written
as
(∆x, ∆y, ∆z) = λ 1, 0, fx (a, b) + µ 0, 1, fy (a, b) , λ, µ ∈ R,
where ∆x = x − a, ∆y = y − a and ∆z = z − f (a, b). This allows us to easily
describe the slope of f at the point (a, b) in the direction of u: we must “measure”
∆z when ∆x = u1 and ∆y = u2 . Hence we take λ = u1 and µ = u2 in order to find
that
∆z = u1 fx (a, b) + u2 fy (a, b)
= fx (a, b), fy (a, b) · u.
This is usually denoted as fu (a, b) and known as the slope of f at the point (a, b) in
the direction of u, or the directional derivative of f at (a, b) in the direction of u.
Because it is not always convenient to work with unit vectors, we more generally
have that the directional derivative of f at (a, b) in the direction of an arbitrary
nonzero vector u is given by
u
fu (a, b) = fx (a, b), fy (a, b) ·
.
kuk
94
2. PARTIAL DERIVATIVES AND TANGENT PLANES
Example: Find the directional derivative of f (x, y) = 4 − x2 − 4y 2 at (a, b) in the
(1, 1) direction.
Example: If f (x, y) = x2 − 3y 2 + 6y, find the slope at (1, 0) in the direction i − 4j.
2.5. GRADIENTS AND DIRECTIONAL DERIVATIVES
2.5.2
95
The gradient vector ∇f
The gradient vector or simply gradient of f is a vector with the partial derivatives
as components
grad f = ∇f = (fx , fy ) = fx i + fy j.
Example: Find the gradient of f (x, y) = x2 − 3(y − 1)2 + 3.
Similarly, for a function f (x, y, z, w) of four variables,
grad f = ∇f = (fx , fy , fz , fw ).
Note that directional derivative can be conveniently expressed in terms of the gradient as
u
.
fu = ∇f ·
kuk
Example: Find f(1,−1) (0, 1) for f (x, y) = x − x2 y 2 + y.
96
2. PARTIAL DERIVATIVES AND TANGENT PLANES
2
Example: Find the directional derivative of g(x, y) = ex cos y at (1, π) in the
direction −3i + 4j.
2.5. GRADIENTS AND DIRECTIONAL DERIVATIVES
2.5.3
97
Properties of the gradient vector
Example: Consider the contour diagram of a plane z = f (x, y) = mx + ny + c.
For n 6= 0 the contours have slope −m/n in the xy-plane.
The gradient vector, mi + nj (m 6= 0), has slope n/m and so is perpendicular to the
contours. It points in the direction of increasing f . In fact, the direction in which
it points is the direction of greatest slope.
98
2. PARTIAL DERIVATIVES AND TANGENT PLANES
Example: Consider f (x, y) = x2 + y 2 . The contours are circles centered at the
origin, and ∇f = 2xi + 2yj points radially out. Once again, ∇f points in the
direction of greatest slope, perpendicular to the contour lines.
2.5. GRADIENTS AND DIRECTIONAL DERIVATIVES
In general, two important properties of the gradient of a function are:
The gradient ∇f (a, b) is perpendicular to the contour line through
(a, b) and points in the direction of increasing f . In fact, the direction and
magnitude of steepest slope at (a, b) are given by ∇f (a, b) and k∇f (a, b)k.
We can understand these two facts by considering the value of cos θ in
u
= k∇f k cos θ.
fu = ∇f ·
kuk
99
100
2. PARTIAL DERIVATIVES AND TANGENT PLANES
Example: T (x, y) = 20 − 4x2 − y 2 describes the temperature on the surface of a
metal plate.
In which direction away from the point (2, −3) does the temperature increase most
rapidly?
In which directions away from the point (2, −3) is the temperature not changing?
2.5. GRADIENTS AND DIRECTIONAL DERIVATIVES
101
Example: A team from a large British oil company is mapping the ocean floor to
assist in the plugging of a leaking oil well in the Gulf of Mexico. Using sonar, they
develop the model
πy D(x, y) = 1700 − 30x2 − 50 sin
,
2
where x and y are distance in kilometres, D is depth in metres, and −2 ≤ x ≤ 2
and −2 ≤ y ≤ 2.
(a) The well is located at (1, 0.5). What is its depth?
(b) Determine the slope of the ocean floor in the positive x-direction and in the
positive y-direction in the area considered by the clean-up team.
(c) Determine the magnitude and direction of greatest rate of change of depth at
the position of the well.
102
2. PARTIAL DERIVATIVES AND TANGENT PLANES
2.5.4
Main points
• You should be able to find the slope of a function in a given direction.
• You should be able to find the direction and rate of maximal increase of a
function.
• You should understand directional derivative and gradient.
2.6. THE CHAIN RULE AND APPLICATIONS
2.6
103
The Chain Rule and Applications
The chain rule is covered in Stewart, Section 14.5 (Section 15.5).
2.6.1
Review for f (x)
For related functions of one variable, such as y = f (u) and u = g(x), we may use
dy
the chain rule to find
:
dx
dy
dy du
=
or
f 0 (x) = f 0 (u) g 0 (x).
dx
du dx
Example: Find y 0 (x) where y = f (x) = (x2 + 1)5 .
Example: Suppose the radius of a cylinder decreases at a rate of r0 (t) = −2 cm/s.
How fast is the volume decreasing when r = 1 cm and h = 2 cm?
104
2.6.2
2. PARTIAL DERIVATIVES AND TANGENT PLANES
The chain rule for f (x, y)
Given f (x, y) with x and y functions of t. Then
df
∆f
f (x(t + ∆t), y(t + ∆t)) − f (x(t), y(t))
= lim
= lim
.
∆t→0 ∆t
∆t→0
dt
∆t
Now if f is smooth and ∆f is small, we can relate it to ∆x and ∆y through the
linear approximation
∆f ' fx ∆x + fy ∆y.
Hence
∆f
∆x
∆y
' fx
+ fy
.
∆t
∆t
∆t
Now we let ∆t → 0, and provided x(t) and y(t) are smooth,
dx
dy
df
= fx
+ fy
dt
dt
dt
which is the chain rule for f (x(t), y(t)).
Example: Continuing the previous example, suppose that not just the radius but
dh
also the height h is decreasing:
= −1 cm/s. What is the rate of change in
dt
volume?
The chain rule can be extended to any number of dimensions.
Example: If V (a(t)), b(t), c(t)) = abc is the volume of a box then find
dV
.
dt
2.6. THE CHAIN RULE AND APPLICATIONS
2.6.3
105
Larger Chain Rules
To keep check of what depends on what, you can draw a tree diagram for z =
f (x(t), y(t)) as follows:
∂z
∂x
z
∂z
∂y
y
x
dx
dt
dy
dt
t
t
Hence
dz
∂z dx ∂z dy
=
+
.
dt
∂x dt
∂y dt
If z = f u(x(t), y(t)), v(x(t), y(t)) then the tree diagram is
z
∂z
∂u
∂z
∂v
u
∂u
∂x
x
v
∂u
∂y
∂v
∂x
y
dx
dt
x
dy
dt
t
t
∂v
∂y
y
dx
dt
dy
dt
t
Hence
dz
∂z ∂u dx ∂u dy
∂z ∂v dx ∂v dy
=
+
+
+
dt
∂u ∂x dt
∂y dt
∂v ∂x dt
∂y dt
∂z ∂u ∂z ∂v dx
∂z ∂u ∂z ∂v dy
=
+
+
+
.
∂u ∂x ∂v ∂x dt
∂u ∂y ∂v ∂y dt
t
106
2. PARTIAL DERIVATIVES AND TANGENT PLANES
Example: Find the slope of z = f (x, y) = sin x cos y in the radial direction at
∂z
(π, 2π). That is, find
.
∂r
Example: Given h = f (x, y, z) and each of x, y and z depend on both s and t,
∂h
find an expression for
.
∂t
2.6. THE CHAIN RULE AND APPLICATIONS
2.6.4
107
Implicit Differentiation
Both y + sin(x + y) = 0 and x11 − y 7 + xy 2 + 1 = 0 correspond to curves in R2 .
In neither of these examples can we explicitly solve for y as a function of x (or vice
versa). This raises the following question: Can we compute y 0 (x)?
The answer to this question is affirmative and the method to do this goes by the
name of implicit differentiation.
First we will try to find y 0 (x) by “brute force”. It is only advisable to use this
method if you have a PhD in applications of the chain rule.
Example: Find y 0 (x) on the curve y + sin(x + y) = 0.
Important remark: Although we have found y 0 (x) it is almost never possible
to express the answer in terms of x only (as this requires y to be solved for x).
Extreme care has thus to be taken in using the above formula for y 0 (x). It only has
meaning for points (x, y) that lie on the curve y + sin(x + y) = 0, such as (0, 0) or
(−1 − π/2, 1).
108
2. PARTIAL DERIVATIVES AND TANGENT PLANES
So what if you do not have PhD in applications of the chain rule?
The idea is to view a given curve C in R2 (such as y + sin(x + y) = 0) as the
contour z = 0 of a function F (x, y). That is, we define F (x, y) = y + sin(x + y).
We already know that, for any given point (x, y) on C, ∇F (x, y) corresponds to a
vector perpendicular to C at that point.
(1, y 0 )
∇F
In other words,
∇F (x, y) · (1, y 0 (x)) = 0.
Solving this equation for y 0 (x) gives
dy
Fx
=− .
dx
Fy
Example: Find y 0 (x) on the curve y + sin(x + y) = 0.
2.6. THE CHAIN RULE AND APPLICATIONS
109
Example: The Folium of Descartes is given by x3 + y 3 = 6xy. Find y 0 (x).
Question: What happens at x = y = 0?
2.6.5
Main points
• You should be able to differentiate a multivariable function whose variables
depend on multivariable functions.
• You should be able to find
dy
from implicit forms.
dx
110
3
3. MAX AND MIN PROBLEMS ON SURFACES
Max and Min Problems on Surfaces
The content of this chapter is found in Stewart, Sections 14.7 and 14.8 (Sections
15.7 and 15.8).
3.1
Quadratic Approximation
In the previous chapter, we looked at the first-order, or linear approximation to a
function. In this section we introduce the second-order, or quadratic approximation
to a function.
3.1.1
Review for f (x)
We saw in Section 2.4.1 that the value of f (x) near x = a can be approximated
using a straight line:
f (x) ' f (a) + f 0 (a)(x − a).
Adding a second-order term will improve this approximation
f (x) ' f (a) + f 0 (a)(x − a) +
f 00 (a)
(x − a)2 .
2
This quadratic (or second-order) approximation corresponds to the approximation of
f by the parabola through (a, f (a)) with slope and second derivative at a matching
those of f . Unless f is very wild around x = a this is generally quite a good
approximation for x − a not too large.
3.1. QUADRATIC APPROXIMATION
111
Example: Compute e0.1 using a linear and a quadratic approximation.
Example: In the same graph, sketch f (x) = cos x as well as its linear and quadratic
approximations around 0.
112
3. MAX AND MIN PROBLEMS ON SURFACES
3.1.2
Quadratic approximations of f (x, y)
This material is covered in Stewart p. 980 (p. 969).
The quadratic or second-order approximation to f (x, y) around (a, b) is a function
of the form
Q(x, y) = h + mx + ny + ax2 + bxy + cy 2
such that Q(a, b) = f (a, b) and such that all first order and second order partial
derivatives of f and Q agree at (a, b).
It is not hard to verify that this gives
Q(x, y) =f (a, b) + fx (a, b)(x − a) + fy (a, b)(y − b)
1
1
+ fxx (a, b)(x − a)2 + fxy (a, b)(x − a)(y − b) + fyy (a, b)(y − b)2 .
2
2
Example: Check that Qxx (a, b) = fxx (a, b) and Qxy (a, b) = fxy (a, b).
3.1. QUADRATIC APPROXIMATION
113
Example: Find the quadratic approximation around (0, 0) of
f (x, y) = 1 − x2 + y 2 + xy + x3 + x2 y 2 .
If |x|, |y| < 0.1 then we are ignoring terms on the order of (0.1)3 = 0.001.
In this example the function is a polynomial in x and y and we can immediately
read off the quadratic approximation. In the next example we really need to use the
formula (or do we?).
114
3. MAX AND MIN PROBLEMS ON SURFACES
Example: Find the linear and quadratic approximations around (0, 0) of
f (x, y) = e−x
2 −y 2
.
3.1. QUADRATIC APPROXIMATION
115
Example: Consider, again, a barrel with base of radius 1 and height 2.
V (r, h) = πr2 h
⇒
V (1, 2) = 2π
∂V
= 2πrh
∂r
⇒
∂V
(1, 2) = 4π
∂r
∂V
= πr2
∂h
⇒
∂V
(1, 2) = π
∂h
∂ 2V
= 2πh
∂r2
⇒
∂ 2V
(1, 2) = 4π
∂r2
⇒
∂ 2V
(1, 2) = 2π
∂r∂h
∂ 2V
= 2πr
∂r∂h
∂ 2V
= 0.
∂h2
Hence the quadratic approximation around (1, 2) is
V (r, h) ' Q(r, h)
= 2π + 4π(r − 1) + π(h − 2) + 2π(r − 1)2 + 2π(r − 1)(h − 2).
Let ∆r = r − 1 and ∆h = h − 2. Use Q(r, h) to approximate the change ∆V in the
volume of the barrel when ∆r = 0.05 and ∆h = 0.1.
116
3. MAX AND MIN PROBLEMS ON SURFACES
3.1.3
Main points
• You should be able to find the quadratic approximation given a function of
two variables.
• In Matlab, you should be able to plot both the linear and quadratic approximations of a function, and make a comparison of the errors.
• The quadratic approximation is superior to the linear approximation.
3.2. CRITICAL POINTS AND OPTIMISATION
3.2
3.2.1
117
Critical Points and Optimisation
Review for f (x)
Smooth functions f of one variable have local maxima and minima where f (x) has
zero slope, i.e., where the first derivative of f is zero:
df
= 0.
dx
The second derivative test tells us whether f (x) is concave up (a minimum) or
concave down (a maximum):
d2 f
> 0 ⇒ local minimum,
dx2
d2 f
< 0 ⇒ local maximum.
dx2
d2 f
=0⇒?
dx2
Example: Find the local maxima and minima of f (x) = x2 − x4 .
118
3. MAX AND MIN PROBLEMS ON SURFACES
Note, however, that the local minimum in the above example is not a global minimum. At the local minimum, x = 0 and f (0) = 0. But for |x| > 1, we have
f (x) < 0.
If we define the domain as [−2, 2], the global minimum occurs on the boundary
f (±2) = 4 − 16 = −12.
1
1 1
The global maximum, however, is at the local maximum, i.e. f √
= − =
2 4
2
at
1
.
4
The global maximum or minimum of a continuous function f (x) occurs either
at a local minimum, a local maximum, or on the boundary of its domain.
In some circumstances there may be local maxima or minima at points where the
df
is not defined (at cusps, for instance).
derivative
dx
Example: f (x) = |x − 3|.
A more general definition of a local minimum or maximum is:
• A local minimum occurs at a point a where f (x) ≥ f (a) for all x sufficiently
close to a.
• A local maximum occurs at a point a where f (x) ≤ f (a) for all x sufficiently
close to a.
If f is continuous, a local maximum or minimum occurs
either when f 0 (x) = 0 or when f 0 (x) is undefined.
3.2. CRITICAL POINTS AND OPTIMISATION
3.2.2
119
Critical points for f (x, y)
In this section we will find and classify critical points of a function of two variables, see Stewart, Section 14.7 (Section 15.7). (There are also some useful Matlab
functions which perform minimisation, such as fmin and fminsearch).
First we define local maxima and local minima:
• f (x, y) has a local maximum at (a, b) if f (a, b) ≥ f (x, y) for all (x, y) sufficiently
close to (a, b)
• f (x, y) has a local minimum at (a, b) if f (a, b) ≤ f (x, y) for all (x, y) sufficiently
close to (a, b).
If f is differentiable, a local maximum or minimum can only occur in points where
the corresponding tangent plane is horizontal, i.e., where
∇f = 0.
Note that if either one of
∂f
∂f
or
is undefined then ∇f is undefined.
∂x
∂y
Definition: A critical point of f (x, y) is a point where either ∇f = (0, 0)
or where ∇f is undefined.
Local maxima and minima occur at critical points, but not all critical points are
local maxima or minima.
Remark: (For the mathematically inclined.) The rather vague description “sufficiently close” can be made much more precise. In the simplest case, when f : R2 →
R, it goes as follows. The function f has a local maximum at (a, b) if there exists
an > 0 such that f (a, b) ≥ f (x, y) for all (x, y) such that k(x, y) − (a, b)k < .
120
3. MAX AND MIN PROBLEMS ON SURFACES
Example: Find the critical points of f (x, y) = x2 + 3x + y 2 .
Example: Find the critical points of f (x, y) = x2 + 3x − y 2 .
Definition. A function f has a saddle point at P0 if P0 is a critical point of f and
if within any distance of P0 (no matter how small) there are points P1 and P2 with
f (P1 ) > f (P0 )
and
f (P2 ) < f (P0 ).
Important remark: Not all saddle points look like a “horse saddle”, and in one
of your Matlab sessions you will be asked to look at a so-called monkey saddle,
described by an equation of the form z = x3 − 3xy 2 .
3.2. CRITICAL POINTS AND OPTIMISATION
121
Example: Find all the critical points of
f (x, y) = x2 + y 2 (1 − x)3
and use a quadratic approximation to investigate the nature of the critical points.
122
3. MAX AND MIN PROBLEMS ON SURFACES
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3.2. CRITICAL POINTS AND OPTIMISATION
123
The plus or minus signs in front of x2 and y 2 in the quadratic approximation can
indicate whether f has a maximum, minimum or saddle at a given critical point.
However, when Q(x, y) has cross-terms such as xy one needs to be extremely careful.
Example: Examine the critical point (0, 0) of f (x, y) = x2 + xy + y 2 .
Next consider
y2 1 2 1
= x + y − y2.
8
2
8
This function has a saddle point at (0, 0).
f (x, y) = x2 + xy +
Question: Can we find a general rule to determine the nature of critical points?
124
3. MAX AND MIN PROBLEMS ON SURFACES
Let Q be the quadratic function
Q(x, y) = ax2 + bxy + cy 2 + dx + ey + f
by 2 b2 y 2 cy 2
=a x+
+ dx + ey + f
− 2 +
2a
4a
a
by 2 (4ac − b2 ) 2
=a x+
+
y + dx + ey + f.
2a
4a2
Then
• a > 0 and 4ac − b2 > 0 implies that f has a local minimum;
• a < 0 and 4ac − b2 > 0 implies that f has a local maximum;
• 4ac − b2 < 0 implies that f has a saddle.
Applying this argument to our quadratic approximation formula for f around (a, b):
Q(x, y) = f (a, b) + fx (a, b)(x − a) + fy (a, b)(y − b)
1
1
+ fxx (a, b)(x − a)2 + fxy (a, b)(x − a)(y − b) + fyy (a, b)(y − b)2
2
2
gives what is known as the second-derivative test.
Theorem: Assume that f and its first- and second-order partial derivatives are
all continuous at (a, b) and ∇f (a, b) = (0, 0). Define the 2 × 2 determinant
fxx (a, b) fxy (a, b) .
D = fyx (a, b) fyy (a, b) Then:
• fxx (a, b) > 0 and D > 0 implies that (a, b) is a local minimum;
• fxx (a, b) < 0 and D > 0 implies that (a, b) is a local maximum;
• D < 0 implies that (a, b) is a saddle point;
• if D = 0 the second-derivative test is inconclusive.
Remark: (For the connoisseurs.) The matrix
fxx fxy
fyx fyy
is known as the Hessian and its determinant as the discriminant.
3.2. CRITICAL POINTS AND OPTIMISATION
125
The beauty of this test is that we do not even have to write down the quadratic
approximation.
Example: Find and classify the critical points of f (x, y) = e−x
2 +y 2
.
126
3. MAX AND MIN PROBLEMS ON SURFACES
Example: Find and classify the critical points of f (x, y) = x2 + y 2 − 2x2 y 2 .
3.2. CRITICAL POINTS AND OPTIMISATION
3.2.3
127
Global maxima and minima
To find the global maximum or minimum values of f on a closed and bounded
domain D, we need to compare both
• the values of f at the critical points (where ∇f = (0, 0) or undefined), and
• the extreme values of f on the boundary of D.
Example: What are the global maximum and minimum values of the function
f (x, y) = x2 + y 2 − 2x2 y 2
on the domain [−1, 1] × [−1, 1]?
128
3. MAX AND MIN PROBLEMS ON SURFACES
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0.3
0.2
0
−1
0.2
0.4
0.8
0.4
0.5
0.30.4
0.5
0.3
0.2
0.6
0.7
0.5
0.8
0.3
0.4
1
1
) 0.6
1
3.2. CRITICAL POINTS AND OPTIMISATION
3.2.4
129
The Matlab command fminsearch
The Nelder–Mead algorithm is a search method built into the Matlab function fminsearch. One advantage of this method is that it works with non-smooth functions
as well, so it extends beyond traditional calculus methods.
The syntax is that fminsearch will try to minimise a function f that you must
store as a function in an M-file. Then you give fminsearch a starting point and let
it run.
For example, to minimise f (x, y, z) = (x − 1)2 + (y − 2)2 + (z − 3)2 , use the following
code:
function output = f(x)
output = (x(1)-1)^2 + (x(2)-2)^2 + (x(3)-3)^2;
Note that x(1) corresponds to x, x(2) to y and x(3) to z. This is because fminsearch requires that f takes a single vector as input (in this example the threedimensional vector x).
Now, to run the minimisation, call
fminsearch(’f’,[0,5,10])
where x = 0, y = 5, z = 10 is the initial guess.
For functions with several local minima, fminsearch can get stuck within a local
minimum. Starting it with different initial guesses or restarting it when it thinks
it has converged may help it to look for “better” minima. How would you use
fminsearch to find a maximum value of f ?
3.2.5
Main points
• You should know that the local minimum or maximum is obtained when either
all the partial derivatives are zero or some of the partial derivatives are not
defined.
• You should know how to find critical points and classify them using either the
quadratic approximation, or the second derivative test.
• You should be able to find global maxima and minima of a function on a
bounded domain.
• You should be competent finding extrema with fminsearch.
130
3.3
3.3.1
3. MAX AND MIN PROBLEMS ON SURFACES
Constrained Optimisation and Lagrange Multipliers
Lagrange multipliers
In practical problems one often needs to maximise or minimise a function subject
to certain constraints, see also Stewart, Section 14.8 (Section 15.8).
Example: Find the minimum value of x2 + y 2 subject to the constraint x + y = 1.
We want to minimise the function f given by f (x, y) = x2 + y 2 , subject to the
constraint x + y = 1.
In this baby-example we can explicitly solve the constraint-equation: y = 1 − x, so
F (x) := f (x, 1 − x) = x2 + (1 − x)2 = 2x2 − 2x + 1,
which is a function of x alone. The critical points of F occur when F 0 (x) = −2+4x =
0, yielding x = 1/2. Since F 00 (x) = 4 > 0 it follows that F has a minimum at
x = 1/2. Therefore, f subject to the constraint x + y = 1 achieves its minimum
value at x = y = 1/2, with actual value also 1/2.
There is a much better way to solve the above problem not requiring the explicit
solution of the constraint-equation. The trick is to define a second function, say
g, such that the constraint-equation corresponds to g(x, y) = 0. In our example,
g(x, y) = x + y − 1. Now graph the contour plot of f (x, y) = x2 + y 2 and add to this
graph the single contour g(x, y) = 0:
y
the circles are the contours
f = 21 , 1, 32 , 2, 25
x
g=0
3.3. CONSTRAINED OPTIMISATION AND LAGRANGE MULTIPLIERS
131
The minimum occurs where the contour g(x, y) = 0 touches one of the contours of
f , which means that ∇f and ∇g are parallel. Hence
∇f = λ∇g
for some λ, known as the Lagrange multiplier.
Example: Find the minimum value of x2 + y 2 subject to the constraint x + y = 1.
132
3. MAX AND MIN PROBLEMS ON SURFACES
Important remark: When solving a constrained optimisation problem with Lagrange multipliers, it is typically not necessary to find the value for λ. So, when
solving your equations obtained from ∇f = λ∇g and the constraint-equation, your
aim is really to eliminate λ in order to solve for the variables in your problem.
Important remark: If upon using the method of Lagrange multipliers you obtain
simple, linear equations to solve then consider yourself lucky. In some cases you
will in fact obtain nonlinear equations that you need to solve. If this happens we
suggest the first thing you should try is taking the ratio of your equations obtained
from ∇f = λ∇g. This allows you to immediately eliminate λ from your equations
and proceed from there. The next example nicely illustrates this point.
Example: Let A = 4xy describe the area of a rectangle centred on the origin
with width 2x and height 2y. Maximise the area of this rectangle subject to the
x2 y 2
constraint that it is inscribed within the ellipse 2 + 2 = 1.
a
b
3.3. CONSTRAINED OPTIMISATION AND LAGRANGE MULTIPLIERS
133
134
3. MAX AND MIN PROBLEMS ON SURFACES
The method of Lagrange multipliers can be extended to any number of dimensions.
Example: Let V (x, y, z) = xyz describe the volume of a box with vertices at
(0, 0, 0), (x, 0, 0), (0, y, 0), . . . , (x, y, z), with x, y, z > 0. Maximise the volume of the
box subject to the constraint that the vertex (x, y, z) lies in the plane z +2x+3y = 6.
3.3. CONSTRAINED OPTIMISATION AND LAGRANGE MULTIPLIERS
135
Example: The distance between a point and a plane.
We now return to a problem considered Chapter 1 — that of finding the distance
between a point and a plane. Recall that when we say distance, we always mean
minimum distance.
Let P = (x1 , y1 , z1 ) be a point and Π : ax + by + cz = d a plane. We want to
compute the distance between P and Π.
If we take an arbitrary point Q = (x, y, z) in R3 then the square of the distance
between P and Q is given by
D2 = (x − x1 )2 + (y − y1 )2 + (z − z1 )2 =: f (x, y, z).
We want to minimise this distance subject to the constraint that Q lies in the plane
Π, i.e., subject to the constraint ax + by + cz = d. Following the method of Lagrange
multipliers we thus define g(x, y, z) = ax + by + cz − d and compute ∇f = λ∇g to
get



2(x − x1 ) = λa
2(y − y1 ) = λb


2(z − z ) = λc.
(∗)
1
First we use (∗) to find
D2 = f (x, y, z) = (λa/2)2 + (λb/2)2 + (λc/2)2 =
λ 2
2
(a2 + b2 + c2 )
from which we still need to eliminate λ. Second we use (∗) to find
x = λa/2 + x1 ,
y = λb/2 + y1 ,
z = λc/2 + z1 .
Substituting this into the constraint equation ax + by + cz − d = 0 we find that
λ
d − ax1 − by1 + cz1
=
.
2
a2 + b 2 + c 2
Therefore,
2
D =
λ 2
2
(d − ax1 − by1 + cz1 )2
(a + b + c ) =
a2 + b 2 + c 2
2
2
2
as before.
If needed, we can also obtain the point on the plane that is closest to (x1 , y1 , z1 ).
136
3. MAX AND MIN PROBLEMS ON SURFACES
Example: Use the method of Lagrange multipliers to find the distance between
the point (1, 2, 3) and the plane x + y + z = 1.
3.3. CONSTRAINED OPTIMISATION AND LAGRANGE MULTIPLIERS
3.3.2
137
Main points
• You should be able to apply the method of Lagrange multipliers to solve
problems in constrained optimisation.
• You should understand the method of Lagrange multipliers geometrically in
terms of gradient vectors.
138
4. ORDINARY DIFFERENTIAL EQUATIONS
4
Ordinary Differential Equations
4.1
Introduction
This material is covered in Stewart, Section 9.1 (Section 10.1).
In almost all fields of scientific endeavour, quantifying time-dependent occurrences
almost inevitably comes down to a differential equation (DE). A DE is an equation
which contains one or more derivatives of an unknown function. There are two types
of DEs:
• ordinary differential equations (ODEs), where the unknown function is a function of only one variable, and
• partial differential equations (PDEs) where the unknown function is a function
of more than one variable. We will not consider PDEs in this course.
4.1.1
Examples of ODEs
• Unbounded population growth: P 0 (t) = kP .
• Bounded population growth: P 0 (t) = kP (1 − P/θ).
• Motion due to gravity: my 00 (t) = −mg.
• Spring system: mx00 (t) = −kx.
In ODEs, one often takes t instead of x for the independent variable, where t denotes
time. Also, derivatives, such as x0 (t) and x00 (t) with respect to time are often written
as ẋ and ẍ respectively.
4.1. INTRODUCTION
4.1.2
139
How do ODEs arise?
Population modelling. Assume a population grows at a constant rate proportional to the size of the population. If P = P (t) stands for the population at time
t, then the model states that
dP
∝ P,
dt
i.e.,
dP
= kP,
dt
where k is the growth constant. If k > 0 the population is growing (think humans)
and if k < 0 the population is decreasing (think tigers).
Question: What happens for k = 0?
Newton’s eureka moment. Newton’s second law of motion states that
mass × acceleration = force.
Let y = y(t) be the vertical displacement from ground at time t of an apple of mass
d2 y
m, soon to land on Newton’s head. Then the acceleration is 2 so that
dt
m
d2 y
= −mg,
dt2
where g is the constant of acceleration at the surface of the earth due to gravity.
140
4.1.3
4. ORDINARY DIFFERENTIAL EQUATIONS
Solution to an ODE
Suppose that we are given an ODE for y which is an unknown function of x. Then
y = f (x) is said to be a solution to the ODE if the ODE is satisfied when y = f (x)
and its derivatives are substituted into the equation.
Example: Show that y = y(x) = A exp(x2 /2) is a solution to the ODE y 0 = xy.
When asked to solve an ODE, you are expected to find all possible solutions. This
means that you need to find the general solution to the ODE. For an ODE that
involves only the unknown function y and its first derivative, the general solution
will involve one arbitrary constant.
You should already know how to solve ODEs of the form
dy
= f (x) or
dx
d2 y
= g(x).
dx2
Example: Find the general solution to the differential equation y 0 = x2 .
4.1. INTRODUCTION
4.1.4
141
Initial value problems
An initial value problem (IVP) is the problem of solving an ODE subject to some
initial conditions of the form y(t0 ) = a, y 0 (t0 ) = b, etc.
The solution to an initial value problem no longer contains arbitrary constants from
the general solution to the ODE — these are determined by the initial conditions of
the problem at hand.
Example: A flow-meter in a pipeline measures flow-through as 2+sin t litres/second.
How much fluid passes through the pipeline from time zero up to time T ?
4.1.5
The order of an ODE
The order of the highest-order derivative in an ODE defines the order of the ODE.
Exercise: What is the order of each of these ODEs?
•
dP
= kP
dt
• mẍ = −kx
• x(y 00 )2 + y 0 y 000 + 4y 5 = yy 0
• y 0 = xy
142
4.1.6
4. ORDINARY DIFFERENTIAL EQUATIONS
Riding your bike at constant speed
Find the position of your bike (at time t) if you are travelling at a constant speed
of 60 km/h along a perfectly straight road.
If x = x(t) is the distance you have travelled at time t then the corresponding ODE
is
dx
= 60.
(∗)
dt
We can directly integrate to find x(t):
Z
Z
dx
= 60
⇒
dx = 60dt.
dt
This yields the general solution to the ODE
x(t) = 60t + C.
To determine the constant C we need more information, such as your initial position
at time 0, in which case we would be solving an IVP.
For different values of C we get different solutions, and below we graph some of
these. If C = 0 then y = 60t. All the other solutions are parallel to this line.
600
500
*)
'(
&
400
300
200
100
0
0
1
2
3
4
5
!#"%$
6
7
8
9
10
The curves y = 60t + C are called solution curves to the ODE (∗). Note that in this
particular case all curves are straight lines with slope 60.
4.1. INTRODUCTION
4.1.7
143
Motion of Projectiles
Example: Consider an apple falling under gravity. Find an expression for the
height y of the apple at time t.
144
4. ORDINARY DIFFERENTIAL EQUATIONS
Example: Now suppose you are throwing apples over your fence to your neighbour.
Find an expression for the position (x(t), y(t)) of the apple if you assume the initial
position is x(0) = 0, y(0) = 0 and the initial velocity is ẋ(0) = u, ẏ(0) = v.
y
x
4.1. INTRODUCTION
4.1.8
145
Extra reading: Realistic models
More realistic models of projectile motion deal with air resistance.
• For fast moving objects a good model includes air resistance proportional to
square of the velocity
d2 y
dy dy = −g − f
.
dt2
dt dt • For slower objects, a good enough model has air resistance proportional to
velocity.
Realistic models may also include the fact that gravity diminishes as you move away
from the earth’s surface (Newton’s inverse square law).
Example: A model with air resistance
Consider the motion of a ball subject to air resistance proportional to the square of
the speed. Note that the air resistance vector is always directed against the direction
of motion.
Decompose the coordinates into x and y directions, and again apply Newton’s law
F = ma.
Drag force, FD
Direction of motion, (ẋ, ẏ)
θ
Gravity force, mg
146
4. ORDINARY DIFFERENTIAL EQUATIONS
Note that
sin θ = p
ẏ
ẋ2
+
and
ẏ 2
ẋ
cos θ = p
.
2
ẋ + ẏ 2
Hence F = ma in the y direction gives
mÿ = −mg − FD sin θ
⇒
⇒
⇒
mÿ = −mg − kv 2 sin θ
ẏ
mÿ = −mg − k(ẋ2 + ẏ 2 ) × p
ẋ2 + ẏ 2
k p
ÿ = −g − ẏ ẋ2 + ẏ 2 .
m
In much the same way
ẍ = −
k p 2
ẋ ẋ + ẏ 2 .
m
This is a coupled system of ODEs which is extremely difficult to solve analytically.
However, numerical solutions are easy to obtain, and you can undertake this a little
later in the semester. A sample trajectory in the xy-plane is shown here.
!"#$&%'!"()(*%(+
18
No Drag
Drag
16
14
12
10
8
6
4
2
0
0
5
10
15
20
25
30
35
4.1. INTRODUCTION
4.1.9
147
Analytical and numerical solutions
To solve an ODE (or IVP) analytically means to give a solution curve in terms of
continuous functions defined over a specified interval, where the solution is obtained
exactly by analytic means (e.g., by integration). The solution satisfies the ODE
(and initial conditions) on direct substitution.
To solve an ODE (or IVP) numerically means to use an algorithm to generate a
sequence of points which approximates a solution curve.
Important remark: As we have already seen, the ODE
dy
= f (t)
dt
can be solved analytically. The solution simply is
Z
y = f (t)dt.
Now this may seem like a cop-out because all we are saying is that the solution is
given by the anti-derivative of f (t), and the above solution is as informative as the
actual ODE. In practice one hopes to be able perform the above integral to get a
more explicit form of the solution. Depending on the form of f this may either be
done exactly or by numerical means.
4.1.10
Main points
• You should understand what a differential equation (DE) is and how they can
arise.
• You should understand what it means for a function to be a solution to an
ODE.
• You should know what a general solution is.
• You should know what an initial value problem (IVP) is.
• You should be able to solve simple differential equations of the form
dy
= f (x)
dx
d2 y
or
= g(x).
dx2
• You should understand what is meant by the order of a differential equation.
• You should be able to derive, using Newton’s second law, equations of motion
in a force field (such as gravity).
• You should understand the difference between an analytical and numerical
solution to an ODE.
148
4. ORDINARY DIFFERENTIAL EQUATIONS
4.2
Slope Fields and Equilibrium Solutions
This material is covered in Stewart, Section 9.2 (Section 10.2).
We have seen that in order to solve
dy
= f (t)
dt
we only need to integrate. However, for the more general first-order ODE
dy
= f (t, y)
dt
this no longer works. Nonetheless, the ODE gives a qualitative picture of the solution
by noting that at (y, t) = (a, b) the slope of y(t) is f (a, b). So what one can do is as
follows:
• In the ty-plane, at (t, y) = (a, b) draw a small straight line with slope f (a, b).
• Repeat the process for many different values of (a, b).
• The resulting diagram is called the slope field.
Note the slope field can be generated without having to solve the ODE.
4.2. SLOPE FIELDS AND EQUILIBRIUM SOLUTIONS
149
Example: the slope field of y 0 = 2y
4.2.1
1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
From the slope field of y 0 = 2y we can see that y = 0 is one of the solution curves.
It is a constant or equilibrium solution.
4.2.2
Equilibrium solutions
An equilibrium solution is a constant solution y(t) = c to the ODE
dy
= f (t, y).
dt
The graph of an equilibrium solution is a horizontal line. Such a line has a slope of
zero, i.e., y 0 = 0. This can only happen if f (t, y) = 0 has a solution y = c for some
real constant c.
150
4. ORDINARY DIFFERENTIAL EQUATIONS
Example: Find the equilibrium solutions of y 0 (t) = −3(y − 1).
!"#%$
3
2
1
0
−1
−2
−3
−3
−2
−1
0
1
2
3
4.2. SLOPE FIELDS AND EQUILIBRIUM SOLUTIONS
151
Example: Find the equilibrium solutions of y 0 (t) = 2t + 1.
"!
2
1.5
1
0.5
0
−0.5
−1
−1.5
−2
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
152
4. ORDINARY DIFFERENTIAL EQUATIONS
Example: Find the equilibrium solutions of y 0 = y(1 − y).
!"#
2
1.5
1
0.5
0
−0.5
−1
−1.5
−2
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
4.2. SLOPE FIELDS AND EQUILIBRIUM SOLUTIONS
4.2.3
153
Stability of equilibrium solutions
A pencil sitting balanced vertically is in an equilibrium state. But make one small
perturbation and it will topple over. This is an unstable equilibrium. On the other
hand, a pendulum hanging vertically is also in an equilibrium state. But if you
perturb it slightly, it will eventually (with friction) return to its equilibrium. This
is a stable equilibrium.
From the slope field, you can decide if an equilibrium solution is stable or not by
looking at whether solution curves will tend toward the equilibrium solution or away
from it as time increases.
Formally, an equilibrium solution y(t) = y0 to the differential equation y 0 = f (t, y)
is stable if the initial value problem:
dy
= f (t, y),
dt
y(0) = y0 ± has a solution y(t) which satisfies
lim y(t) = y0 .
t→∞
In other words, if you start sufficiently close to a stable equilibrium solution, then
you will approach that equilibrium solution.
154
4. ORDINARY DIFFERENTIAL EQUATIONS
Example: Here is the slope field for y 0 = sin(4y).
!#"$%
1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
For this ODE, which equilibrium solutions are stable and which are unstable?
4.2. SLOPE FIELDS AND EQUILIBRIUM SOLUTIONS
4.2.4
155
Existence and uniqueness of solutions
Consider the IVP
dy
= f (t, y),
y(t0 ) = y0 .
dt
Then, if f (t, y) is “smooth” in some rectangle about (t0 , y0 ), there exists a unique
solution y = y1 (t) at least in some small neighbourhood of (t0 , y0 ).
y
y0
t0
t
It is an important point that equilibrium solutions cannot be crossed by other
solution curves. In fact, no solutions curves can cross each other because this would
mean that in some point(s) y 0 (t) has more than one value. Equilibrium solutions
therefore partition the solution space.
4.2.5
Long-term behaviour
Exercise: True or False? Assume the ODE
dy
= −3(y − 1)
dt
has initial condition y(t0 ) = c with c < 1. Then y(t) < 1 for all t > t0 .
156
4. ORDINARY DIFFERENTIAL EQUATIONS
Example: Consider the ODE y 0 = y 2 . It has one equilibrium solution y = 0.
For y > 0, y 0 > 0 so that y(t) is increasing and hence any solution with y > 0 tends
away from y = 0.
For y < 0, y 0 > 0 so that y(t) is increasing and hence any solution with y < 0
approaches y = 0.
Therefore y = 0 is neither stable nor unstable.
4.2.6
Main points
• You should understand how to generate and interpret slope fields.
• You should understand what is meant by an equilibrium solution and how to
find one.
• You should be able to determine the stability of an equilibrium solution.
• You should understand that uniqueness of solutions to an IVP implies that
solutions cannot cross.
4.3. EULER’S METHOD FOR SOLVING ODES NUMERICALLY
4.3
157
Euler’s Method for Solving ODEs Numerically
Stewart, Section 9.2 (Section 10.2) covers Euler’s method. The method gives a
simple approximate solution to an ODE and is closely related to the notion of a
slope field.
4.3.1
Euler’s method using tangent lines
What is the equation of the straight line with slope m that passes through the point
(t, y) = (a, b)?
Euler’s method uses tangent lines as approximations to solution curves. The tangent
line to a solution curve of y 0 = f (t, y) at (t0 , y0 ) is
y = y0 + f (t0 , y0 )(t − t0 ).
This approximates the curve when t is close to t0 .
Now imagine a family of solution curves to the differential equation. We can calculate
an approximate value for y at some later time by taking lots of small steps in time.
At each step we will use the tangent line to the solution through our current point.
This is Euler’s method.
Using ∆t as the step size for our algorithm, let
t1 = t0 + ∆t,
t2 = t1 + ∆t,
t3 = t2 + ∆t, . . . , tn = tn−1 + ∆t.
To find approximate y values at these times, use the following equations in sequence:
y1 = y0 + f (t0 , y0 )∆t
y2 = y1 + f (t1 , y1 )∆t
y3 = y2 + f (t2 , y2 )∆t,
and so on.
158
4. ORDINARY DIFFERENTIAL EQUATIONS
Example: Use Euler’s method with ∆t = 0.2, to find an approximate solution to
y(0.6) for the IVP
dy
= 2t,
y(0) = 0.
dt
Compare your answer with the actual value.
4.3. EULER’S METHOD FOR SOLVING ODES NUMERICALLY
159
The method is only accurate if you make ∆t small, which means a large number
of steps is usually required. For this you might want to use Matlab. Define a t
vector and an initial y value, then use the for command to iterate. In the example
∆t = 0.05 and 12 steps have been chosen to take t to 0.6.
t=(0:0.05:0.6);
y(1)=0;
for i=1:12
y(i+1)=y(i)+2*t(i)*0.05;
end
y(13)
plot (t,y,’-’)
Using Matlab we find y(0.6) = 0.33 which is much much better than the previous
estimate value for y(0.6).
Note that we start with y(1) = 0, whereas in the theoretical work we write y(0) = 0.
Matlab does not allow you to put an index of zero. Keep this in mind in case you
ever get the Matlab error message:
???
Index into matrix is negative or zero.
"!$#&%(')!
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
0
0.1
0.2
!
0.3
0.4
0.5
0.6
160
4.3.2
4. ORDINARY DIFFERENTIAL EQUATIONS
Euler’s method using Matlab; an example
Consider the IVP
dy
= sin(xy),
y(0) = 1.
dx
Matlab code to estimate y(2), using Euler’s method with step size ∆x = 0.01 is:
x=(0:0.01:2);
y(1)=1;
for i=1:200
y(i+1)=y(i)+sin(x(i)*y(i))*0.01;
end
y(201)
plot (x,y,’-’)
Using Matlab we find y(2) = 1.8243.
"!#
$%&
2
1.9
1.8
1.7
1.6
1.5
1.4
1.3
1.2
1.1
1
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
4.3. EULER’S METHOD FOR SOLVING ODES NUMERICALLY
4.3.3
161
Demonstrating errors in Euler’s method
Use Euler’s method to approximate the solution curve of the initial value problem
y 0 = y(1 − y),
y(−2) = 2.
Use the slope field below as a guide to draw in the approximate solution curve.
What goes wrong in this case?
!"#
2
1.5
1
0.5
0
−0.5
−1
−1.5
−2
−2
4.3.4
−1.5
−1
−0.5
0
0.5
1
1.5
2
Main points
• You should understand how slope fields lead to Euler’s method.
• You should know how to approximate a solution to an IVP using Euler’s
method by hand.
• You should know how to implement Euler’s method in Matlab.
• You should understand that errors can be a problem in the approximate solution, especially if a large step size is used.
162
4. ORDINARY DIFFERENTIAL EQUATIONS
4.4
Separable ODEs
Separable first-order ODEs are one of several classes of ODE we will study in
MATH1052. It is very important that you become skilled at identifying and solving
this type of ODE. See also Stewart, Section 9.3 (Section 10.3).
4.4.1
Definition
A first-order ODE is called separable if it can be written in the form
dy
= f (x)g(y).
dx
Exercise: Which of the following ODEs are separable?
•
dy
= y(1 − y),
dt
•
dy
= ex+y ,
dx
2
• y 0 = e(x+y) ,
• ẏ =
•
ty + y
,
t2
dy
= ty + y 2 .
dt
4.4. SEPARABLE ODES
4.4.2
163
Solving separable ODEs
1 dy
= f (x).
g(y) dx
Z
2. Integrate both sides with respect to x:
1. Rewrite the equation as
1 dy
dx =
g(y) dx
Z
f (x)dx.
3. Note the integral Zon the leftZ is a “substitution”, so that we can replace the
dy
= f (x)dx.
last equation by
g(y)
4. If one is lucky (or very smart, or both) one or both of the integrals can actually
be evaluated.
5. If one is even more lucky (or even smarter) one can then finally explicitly
express y as a function of x.
Important remark: The final two steps will not always be possible. Whenever
you are asked to solve a separable ODE or IVP in this course you are supposed to
always “go as far as possible”. For good reasons we assume you are infinitely smart;
not being able to compute simple integrals is never a valid excuse.
For the IVP y 0 (x) = f (x)g(y), y(x0 ) = y0 , we again might have the problem that
we can’t evaluate one of both of the integrals needed to solve the separable ODE.
In this case, we can still write down an actual solution, be it a rather implicit one.
Indeed, the solution to the IVP is given by
Z y
Z x
dt
=
f (t)dt.
y0 g(t)
x0
164
Example: Solve the ODE
4. ORDINARY DIFFERENTIAL EQUATIONS
dy
x
= .
dx
y
The solution we found for the above example contains an arbitrary constant so is
the general solution. In the case of an IVP this constant will be fixed to find one,
particular solution.
dy
x
Example: Solve the IVP
= , y(0) = 3.
dx
y
4.4. SEPARABLE ODES
Example: Solve the IVP
165
dy
sin x
=
,
dx
y
y(0) = 1.
166
4.4.3
4. ORDINARY DIFFERENTIAL EQUATIONS
Singular solutions
In step 1 of our recipe for solving separable ODEs we rewrote y 0 (x) = f (x)g(y) as
1 dy
= f (x).
g(y) dx
Since we can’t divide by zero, this means that our method is only valid provided
g(y) 6= 0.
If there is an a such that g(a) = 0 then the ODE will have the equilibrium solution
y(x) = a. This is easy to check:
Such a solution is also known as a singular solution because it won’t generally arise
from our earlier recipe. Whenever you are asked to solve a separable ODE you will
have to check for singular/equilibrium solutions.
dy
= y(1 − y),
Example: Solve the IVP
dt
logistic equation.
y(0) = y0 . This corresponds to the
4.4. SEPARABLE ODES
167
The logistic equation has many uses in population dynamics, physical science and
economics.
168
4. ORDINARY DIFFERENTIAL EQUATIONS
Example: Solve
4.4.4
dy
= y2.
dx
Main points
• You should be able to identify a first-order separable ODE.
• You should know how to solve a separable ODE.
• You should understand that equilibrium and singular solutions are equivalent
and must be checked for when solving a separable ODE.
4.5. APPLICATIONS: LAW OF COOLING, POPULATION GROWTH
4.5
169
Applications: Law of Cooling, Population Growth
Here we look at using some simple ODEs to model cooling and population dynamics.
See Stewart Section 9.4 (Section 10.4).
4.5.1
Newton’s law of cooling
Newton’s law of cooling states that the rate at which a “body” cools is proportional
to the temperature difference between the body and its surrounding medium.
If T is the temperature of the body and Tm the temperature of the surrounding
medium then, according to Newton,
dT
= −k(T − Tm ),
dt
k > 0.
Here the constant k is chosen such that if T > Tm , T 0 (t) is negative, describing
cooling.
Example: Solve this ODE with the intial condition T (0) = T0 .
170
4. ORDINARY DIFFERENTIAL EQUATIONS
In real-life problems the cooling constant k is usually not known and you will have
to infer it from measurements.
Example: CSI Victoria Carl W.’s body was discovered in his cell at Barwon
Prison, Victoria on the 19th of April 2010. The coroner’s report stated that:
• C.W.’s body discovered at 11am on 19 April 2010.
• Body temperature at 11am: 34.8◦ C.
• Body temperature at 12.30pm: 34.1◦ C.
• Temperature in C.W.’s cell: 21.1◦ C.
Use the above information to determine the time of Carl’s death, assuming his body
temperature at the time of death was 37◦ C.
4.5. APPLICATIONS: LAW OF COOLING, POPULATION GROWTH
171
172
4.5.2
4. ORDINARY DIFFERENTIAL EQUATIONS
Malthus: “. . . doomed to misery and vice”
Without inhibiting factors the rate at which the human population grows is proportional to its existing population; if P is the population at time t then
dP
= rP,
dt
where r is the growth rate (per human per year, say).
The world population is currently growing at a rate of approximately r = 0.0125. If
we take as initial condition the year 2000 with roughly 6 billion people on the planet,
and ignore global warming so that we may take the total area of land on Earth to
be constant at 150 million square kilometres, find in which year each human will
have, on average, one square metre of land left to stand on.
4.5. APPLICATIONS: LAW OF COOLING, POPULATION GROWTH
173
174
4.5.3
4. ORDINARY DIFFERENTIAL EQUATIONS
Verhulst to the rescue; down with the Malthusian law
The problem with the Malhusian population model is that it does not contain a
“damping” factor reflecting issues such as over-population, limited food or water
supply, etc. The logistic model of Verhulst is the simplest model that has such
built-in damping:
dP
P
= rP 1 −
,
dt
θ
where θ is some very large constant, known as the carrying capacity.
Note that when P is small(ish) then P/θ is small and we approximately have
dP
≈ rP
dt
as before. When the population becomes close to the carrying capacity (i.e., P/θ
very close to 1 so that not just 1 − P/θ ≈ 0 but also P (1 − P/θ) ≈ 0) then growth
comes almost to a standstill.
If we introduce new variables y = P/θ and τ = rt then the ODE becomes
dy
= y(1 − y).
dτ
We have already seen that if y(0) = y0 then this has solution
y=
y0 eτ
.
1 − y 0 + y 0 eτ
If P (0) = P0 in our original variables then we have
θP0 ert
P =
.
θ − P0 + P0 ert
4.5. APPLICATIONS: LAW OF COOLING, POPULATION GROWTH
175
Example: Identify the equilibrium (or singular) solutions to the logistic population model, and interpret these solutions in terms of population growth. Include a
diagram.
176
4. ORDINARY DIFFERENTIAL EQUATIONS
Example: The Pacific Halibut Fishery uses a logistic model plus an extra term to
take into account their harvesting, which is at a rate proportional to the existing
population.
If E is the constant of proportionality for harvesting, their model is
dP
P
= rP 1 −
− EP.
dt
θ
What restriction must be put on E so that the population will not die out?
4.5. APPLICATIONS: LAW OF COOLING, POPULATION GROWTH
4.5.4
177
Extra reading: numerically solving ODEs
Matlab has some more sophisticated numerical methods for solving ODEs than Euler’s method. Matlab’s ode45 uses a Runge–Kutta fourth-order integration technique. To use it you need to write two little programs. The first one, say logistic.m,
defines the differential equation
function dy=logistic(t,y)
dy=y*(1-y);
The second program will numerically solve the ODE and plot your solution.
y0=0.1;
tfinal =6;
[t,y]= ode45 (‘logistic’, tfinal, y0);
plot (t,y)
4.5.5
Main points
• You should be able to apply Newton’s law of cooling and solve the resulting
IVP.
• You should know how to model population growth and solve the resulting IVP.
• You should be able to interpret the obtained solutions.
178
4. ORDINARY DIFFERENTIAL EQUATIONS
4.6
Solving Linear First Order Equations
If you cannot solve a differential equation by separation, your next plan of attack
should be to test if it is linear and use the integration factor method described below.
Further exposition and examples are given in Stewart Section 9.5 (Section 10.5).
4.6.1
Definition
A first-order ODE is linear if it can be put in the form
dy
+ p(t)y = q(t)
dt
for some functions p(t) and q(t).
Exercise: Which of the following are linear?
• y0 = x
• y 0 = log x
• y0 = y2
• y0 = y
• y0y = 1
• y 0 = y + 3 sin x
• y 0 = t5 − t2 y
• y 0 = sin y + ex
• (y 0 )2 = y
4.6. SOLVING LINEAR FIRST ORDER EQUATIONS
Recall the product rule for differentiation (written here in reverse order):
f
dg df
d
+
g = (f g).
dt
dt
dt
This formula is key to solving first-order linear ODEs.
Example: Solve
dy 2t + 2
y = t2 + 1.
dt
t +1
179
180
4. ORDINARY DIFFERENTIAL EQUATIONS
The lucky thing (it wasn’t luck really) about the previous example was that after
multiplication by t2 + 1, the equation exactly matched the product rule. This raises
the following natural question.
Question: Can we always multiply a first-order linear ODE by some function I(t),
so that its left-hand side matches the left-hand side of the product rule?
This question can be answered in the affirmative and any function I(t) that does
the trick is known as an integrating factor.
Given the ODE
dy
+ p(t)y = q(t)
dt
we multiply by the yet-to-be-found integrating factor I(t):
dy
+ I(t)p(t)y = I(t)q(t).
(∗)
dt
We think of I as g and y as f . Then the first term on the left is f 0 g and consequently
we want the second term on the left to be f g 0 = yI 0 . Given that this second term is
in fact Ipy, we infer that I must satisfy the ODE
I(t)
I 0 (t) = I(t)p(t).
This is good news since this is a separable ODE which we already know how to
solve:
Z
I(t) = exp
p(t)dt .
Important remark: Any integrating factor will do. Hence when you try to solve
I 0 = Ip, i.e., when you try to compute the above integral (assuming you can explicitly
do so), you do not need to worry about keeping track of constants. For example, if
p(t) = 2t, you would get
Z
2
2
2
I(t) = exp 2 tdt = et +C = eC et = B et ,
where B = exp(C) 6= 0. When you substitute this in (∗) the constant B would
simply be an overall factor which may thus be divided out. It is therefore perfectly
okay to write on your assignment: Because constants of integration do not
matter,
Z
I(t) = exp 2
2
tdt = et .
To confuse your tutor, you might even write: Because constants of integration
do not matter,
Z
I(t) = exp 2
2 +65π 123
tdt = et
.
4.6. SOLVING LINEAR FIRST ORDER EQUATIONS
181
Summary
To solve a first-order linear ODE:
dy
+ py = q.
dt
Z
2. Find an integrating factor I = exp
pdt .
1. Write the equation in the form
3. Multiply the ODE by I and apply the product rule to get
d
(Iy) = Iq.
dt
4. Integrate both sides with respect to t. (This time, don’t forget about constants
of integration!).
182
Example: Solve the ODE
4. ORDINARY DIFFERENTIAL EQUATIONS
dy
= y + t.
dt
4.6. SOLVING LINEAR FIRST ORDER EQUATIONS
Example: Solve the IVP x2
dy
+ xy = 1,
dx
183
y(1) = 2.
184
Example: Solve the ODE
4. ORDINARY DIFFERENTIAL EQUATIONS
dy
− x2 y = x2 .
dx
4.6. SOLVING LINEAR FIRST ORDER EQUATIONS
4.6.2
185
Worked-out example: RL circuit
The ODE describing the current in a circuit with a resistor and an inductor (known
as an RL circuit) turns out to be linear. To obtain this equation consider the change
in voltage around the circuit. The voltage drop due to a resistor is RJ where R is
the resistance (in Ohms) and J is the current (in Amperes).
R
E
L
The voltage drop due to the inductor is L
dJ
where L is the inductance (in Henries).
dt
According to Kirchhoff’s Law the sum of the voltage drops is equal to the supplied
voltage E(t) (in Volts). Hence
L
dJ
+ RJ = E(t).
dt
(∗)
Example: Suppose in an RL circuit a battery supplies a constant voltage of 80V,
the inductance is 2H and the resistance is 10Ω. (i) Find an expression for J(t), and
(ii) determine the the current after 1 second if J(0) = 0.
Solution: We first put (∗) into standard form:
E(t)
dJ RJ
+
=
.
dt
L
L
An integrating factor for this ODE is
Z
I(t) = exp(R/L
dt) = exp(Rt/L).
Multiplying the ODE by this factor and using the product rule gives
e
Rt/L dJ
dt
Rt/L RJ
+e
E(t) Rt/L
=
e
L
L
⇒
d Rt/L E(t) Rt/L
e
J =
e
dt
L
Integrating the previous equation leads to
Z
Z
1
1 −Rt/L
Rt/L
Rt/L
e
J=
E(t)e
dt ⇒ J(t) = e
E(t)eRt/L dt.
L
L
186
4. ORDINARY DIFFERENTIAL EQUATIONS
So far we have treated everything completely generally. Now we use that the battery
supplies constant voltage, i.e., E does not depend on t. This yields
Z
L
E −Rt/L
E
J(t) = e
eRt/L dt ⇒ J(t) = e−Rt/L
eRt/L + C ,
L
L
R
so that we finally obtain
E
+ D e−Rt/L .
R
Since J(0) = 0, the constant D = −E/R:
J(t) =
J(t) =
E
1 − e−Rt/L .
R
Since E = 80, R = 10 and L = 2:
J(t) = 8 1 − e−5t .
We can finally determine J(1) as J(1) = 8(1 − exp(−5)) Amperes.
Remember that we derived a completely general solution to the ODE for any applied
voltage E(t):
Z
1 −Rt/L
J(t) = e
E(t)eRt/L dt.
L
So if we had a non-constant voltage such as E(t) = E0 sin(ωt), we would be able to
find the solution for this case by evaluating the integral.
4.6.3
Main points
• You need to be able to identify a first-order linear ODE.
• You need to be able to solve a first-order linear ODE using an integrating
factor.
4.7. LINEAR SECOND-ORDER DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS
4.7
187
Linear Second-Order Differential Equations with Constant Coefficients
Second-order ODEs are covered in Stewart in Sections 17.1 and 17.2 (Sections 18.1
and 18.2). They often correspond with unforced (homogeneous) oscillators in our
applications, but they arise in many other problems.
4.7.1
Definition
If the unknown function and its derivatives appear linearly in an ODE, the ODE is
said to be linear. Otherwise it is nonlinear.
Exercise: Which of the following ODEs are linear?
• y 00 + y 0 = y log x
• y 0 = y 2 + y 00
• y 00 y 0 = y
• y 0 y = y 00
• y 0 + x3 y 00 = xy + 3 sin x
• y 0 = y 00 t5 − t2 y
• y 00 = sin y + ex
• y 00 + (y 0 )2 = y
A second-order linear ODE is an ODE that can be written in the form
dy
d2 y
+ p(t)
+ q(t)y = r(t).
2
dt
dt
The functions p(t) and q(t) are called the coefficients of the ODE and if r(t) = 0
the ODE is called homogeneous.
188
4.7.2
4. ORDINARY DIFFERENTIAL EQUATIONS
The superposition principle
If y1 (t) and y2 (t) are solutions of
d2 y
dy
+ q(t)y = 0,
+ p(t)
2
dt
dt
then so is
y(t) = c1 y1 (t) + c2 y2 (t).
Proof
4.7. LINEAR SECOND-ORDER DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS
189
Theorem: If y1 (t) and y2 (t) are two linearly independent solutions
(i.e., y2 (t) 6= Cy1 (t)) of the homogeneous second-order linear ODE
d2 y
dy
+ q(t)y = 0
+ p(t)
2
dt
dt
then the general solution is given by
y(t) = c1 y1 (t) + c2 y2 (t).
Important remark: For first-order ODEs we have seen that the general solution
contains one arbitrary constant. The above result suggests that for second-order
ODEs the general solution should contain two arbitrary constants, and this is in
fact the case.
4.7.3
Reduction of order
The previous theorem states that all we need to do in order to solve a homogeneous
linear second-order ODE is to find two linearly independent solutions. Unfortunately, that is where the good news ends: no general method (such as the integrating
factor method for linear first-order ODEs) is known to find two such solutions.
There is however a method to find a second solution once a first solution has been
found. So, if you are clever enough to guess one solution (and a guess it will have
to be, given that we have no general method) you can obtain a second, linearly
independent solution. This goes by the name of reduction of order.
Assume your smart little sister has guessed that y1 (x) is a solution to
dy
d2 y
+ p(x)
+ q(x)y = 0.
2
dx
dx
To find a second solution y2 (x) you set
y2 (x) = u(x)y1 (x).
with u(x) a yet-to-be-determined function. Substituting this in the ODE gives
u00 y1 + 2u0 y10 + uy100 + p(u0 y1 + uy10 ) + quy1 = 0.
This may also be written as
u00 y1 + u0 (y1 p + 2y10 ) + u(y100 + py10 + qy1 ) = 0.
190
4. ORDINARY DIFFERENTIAL EQUATIONS
But y1 is a solution to the ODE so the third term actually vanishes! Thus
y10 00
0
u +u p+2
= 0.
y1
This is a first-order linear ODE in v = u0 which we known how to solve using an
integrating factor.
Example: One solution to the ODE x2 y 00 + 2xy 0 − 6y = 0 is y(x) = x2 . Use
reduction of order to find the general solution.
4.7. LINEAR SECOND-ORDER DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS
191
192
4.7.4
4. ORDINARY DIFFERENTIAL EQUATIONS
Homogeneous linear second-order ODEs with constant coefficients
We next discuss how to solve a homogeneous linear second-order ODE with constant
coefficients:
ay 00 + by 0 + cy = 0,
a 6= 0.
The trick is to try to find a solution of the form y = eλt . Substituting this into the
ODE gives
aλ2 eλt + bλeλt + ceλt = 0.
Since eλt 6= 0 we may divide both sides of the above equation by this term to be left
with the quadratic equation
aλ2 + bλ + c = 0.
This equation is called the characteristic equation of the ODE.
Important remark: When you are solving ODEs of this type on your assignments
and exams, you don’t have to derive the characteristic equation. It is fine to simply
start from the characteristic equation and solve the ODE from there.
Various scenarios can now arise, which we discuss in more detail below.
Case 1: Two distinct real roots
If b2 − 4ac > 0 the characteristic equation aλ2 + bλ + c = 0 has two distinct real
roots, say λ1 and λ2 .
In this case the general solution to the ODE is
y = c1 eλ1 t + c2 eλ2 t .
Question: What will this solution look like for very large t?
4.7. LINEAR SECOND-ORDER DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS
Example: Solve the ODE y 00 − 3y 0 + 2y = 0.
193
194
4. ORDINARY DIFFERENTIAL EQUATIONS
Case 2: A single root of multiplicity 2
If b2 − 4ac = 0 the characteristic equation aλ2 + bλ + c = 0 has one root, say µ, of
multiplicity 2. (Of course µ = −b/(2a) but it is more convenient to work with µ).
This is somewhat of a problem because now we only have single solution to the
ODE: y = eµt .
Fortunately, we know that reduction of order will give us a a second linearly independent solution, so that we try y = u(t)eµt . By the product rule
y 0 = u0 eµt + µueµt
y 00 = u00 eµt + 2µu0 eµt + µ2 ueµt .
Substituting this into ay 00 + by 0 + cy = 0 gives
h
i
aµ2 + bµ + c u + 2aµ + b u0 + au00 eµt = 0.
Since µ is a solution to the characteristic equation, aµ2 + bµ + c = 0. Moreover,
since µ = −b/2a, 2aµ + b = 0. Hence we are left with just
u00 = 0.
This is easily solved as
u(t) = (a + bt).
Using this with a = 0 and b = 1 we now have two linearly independent solutions:
y1 (t) = eµt
and y2 (t) = teµt ,
where µ = −b/2a.
Important remark: In actual problems you do not have to repeat the above
derivation. It is enough for you to claim that if the characteristic equation has
a single root of multiplicity 2 then the above are a pair of linearly independent
solutions and
y(t) = (c1 + c2 t)eµt
is the general solution.
Question: What will this solution look like for very large t?
4.7. LINEAR SECOND-ORDER DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS
Example: Solve the IVP y 00 + 6y 0 + 9y = 0,
y(0) = 2,
y 0 (0) = 0.
195
196
4. ORDINARY DIFFERENTIAL EQUATIONS
Case 3: Complex roots
If b2 − 4ac < 0 the characteristic equation aλ2 + bλ + c = 0 has a pair of complex
conjugate roots, say α ± iβ.
This leads to the general solution
i βt
− i βt
y(t) = c1 e + c2 e
eαt .
For those who want the solution to be manifestly real we note that
e± i βt = cos(βt) ± i sin(βt),
so that
y(t) = (c1 + c2 ) cos(βt) + (c1 − c2 )i sin(βt) eαt .
Introducing new constants b1 and b2 by b1 = c1 + c2 and b2 = (c1 − c2 )i we thus get
y(t) = b1 cos(βt) + b2 sin(βt) eαt .
Example: Solve y 00 + 9y = 0.
4.7. LINEAR SECOND-ORDER DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS
Example: Solve the IVP y 00 − 4y 0 + 5y = 0,
y(0) = 1,
y 0 (0) = 0.
197
198
4. ORDINARY DIFFERENTIAL EQUATIONS
Challenge problem: Solve the ODE y 0000 + y 000 − y 00 + y 0 − 2y = 0.
4.7. LINEAR SECOND-ORDER DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS
4.7.5
199
Extra reading: numerical solutions
Numerical solutions of second-order ODEs usually require one second-order equation
to be split into two first-order equations. Then the two coupled ODEs are solved
together. For example, the equation
z 00 + z 0 − z = cos(x),
z(0) = 1,
z 0 (0) = 2
can be converted into:
z0 = y
y 0 = −y + z + cos(x)
with initial conditions z(0) = 1, y(0) = 2. Then an Euler method can be used on
these equations. For instance, the code to implement Euler’s method would look
like:
deltax=0.1;
x(1)=0;
z(1)=1;
y(1)=2;
for i = 1:100
x(i+1) = x(i) + deltax;
z(i+1) = z(i) + deltax*y(i);
y(i+1) = y(i) + deltax*(-y(i) + z(i) + cos(x(i)));
end
plot(x,z)
4.7.6
Main points
• You should be able to identify a linear second-order ODE.
• You should understand the superposition principle and the general solution
theorem for homogeneous linear second-order ODEs.
• You should know how to solve homogeneous linear second-order ODEs with
constant coefficients.
200
4.8
4. ORDINARY DIFFERENTIAL EQUATIONS
Simple Harmonic Motion and Damped Oscillations
This section deals with the modelling of oscillators by second-order ODEs. This
material is covered in Stewart Section 17.3 (Section 18.3).
4.8.1
The undamped spring
A spring of (natural) length L is stretched a distance s by a weight with mass m.
In equilibrium we take the position of the weight (viewed as a point-particle) to be
x = 0, with the positive x-axis pointing down.
If we pull down the weight and then release it, it will start to oscillate.
If we assume the weight-spring-system moves free of any resistance (no air-resistance
and no internal friction in the spring) we say the spring is undamped.
To derive an ODE for the undamped spring we must combine Newton’s second law
and Hooke’s law.
According to Hooke’s linear spring law the restoring force T exerted by the spring
(“trying to unstretch”) is proportional to the distance d it is stretched:
T = −kd.
Here k > 0 is measured in N m−1 and is called the spring constant. Since k > 0,
the tension in the spring acts to restore the spring to its natural length. The larger
the spring constant, the more tightly-coiled the spring.
Remark: As we all know from experience, Hooke’s law only works if we do not
stretch a spring too much.
4.8. SIMPLE HARMONIC MOTION AND DAMPED OSCILLATIONS
201
fixed support
L
T
natural
length
L
s
s
T
+tve
mg
x(t)
mg
According to Hooke’s law the total force F is given by
F = −k(s + x) +
| {z }
restoring force
mg
|{z}
,
gravitational force
with x = x(t) the displacement out of equilibrium and g (≈ 9.8 m/s2 ) the acceleration on the Earth’s surface.
Before we pulled down the weight the system was in equilibrium, so that
ks = mg.
We are thus left with
F = −kx.
(∗)
According to Newton’s second law of motion the total force equals mass times acceleration:
F = mx00 .
(∗∗)
Equating (∗) and (∗∗) leads to the equation of motion
mx00 + kx = 0
(which holds throughout the Universe, not just on Earth).
The equation of motion only depends on the ratio of k and m, and it is customary
to introduce
k
ω2 = ,
m
where ω > 0 is known as the angular frequency.
Then the equation of motion for the undamped spring is thus given by
x00 + ω 2 x = 0.
202
4. ORDINARY DIFFERENTIAL EQUATIONS
This is a homogeneous second-order linear ODE with constant coefficients.
The corresponding characteristic equation is given by
λ2 + ω 2 = 0
with purely imaginary roots
λ1 = iω
and λ2 = −iω.
Two linearly independent solutions are thus
e i ωt
and
e− i ωt
or
cos(ωt) and
sin(ωt).
and the general solution is
x(t) = c1 cos(ωt) + c2 sin(ωt).
Although we could stop here it is standard to rewrite the above solution in the
alternative form
x(t) = A cos(ωt − φ),
where the two constants c1 and c2 have been traded for two new constants A > 0
and φ ∈ (−π, π], known as the amplitude and phase shift.
4.8. SIMPLE HARMONIC MOTION AND DAMPED OSCILLATIONS
Example: Find the relation between c1 , c2 and A, φ.
203
204
4. ORDINARY DIFFERENTIAL EQUATIONS
Example: A spring with mass 9kg and spring constant 4N/m is pulled down 1m
and given an initial upward kick of −0.5 m/s. Solve for position as a function of
time and explicitly determine the amplitude of the oscillations.
4.8. SIMPLE HARMONIC MOTION AND DAMPED OSCILLATIONS
4.8.2
205
The damped spring
If you try the “spring-weight” system at home it is unlikely to display true simple
harmonic motion. If that were possible you could build a perpetuum mobile.
To make the model more realistic we need to take into account the effect of damping.
This is caused by air-resistance and mechanical friction of the spring.
The most common model of damping assumes that it corresponds to a force Fd ,
proportional to the velocity of the moving object, and opposite to the direction of
movement. Hence
Fd = −βx0 ,
where β > 0 is the damping constant measured in N s m−1 .
The total force F for the damped spring is thus given by
F =
−βx0
| {z }
damping force
−k(s + x) +
| {z }
restoring force
mg
|{z}
.
gravitational force
As before,
ks = mg,
so that
F = −βx0 − kx.
Thanks to Newton’s second law this yields the equation of motion
mx00 + βx0 + kx = 0.
This equation only depends on the ratios of k/m and β/m, and it is customary to
not only set
k
ω2 =
>0
m
but also
β
2p =
> 0.
m
Then the equation of motion for the damped spring is given by
x00 + 2p x0 + ω 2 x = 0.
This is a homogeneous second-order linear ODE with constant coefficients.
206
4. ORDINARY DIFFERENTIAL EQUATIONS
The corresponding characteristic equation is given by
λ2 + 2pλ + ω 2 = 0,
which has discriminant
D = 4(p2 − ω 2 ).
There are three different cases to consider.
• D < 0 (i.e., p < ω).
This is the underdamped or weakly-damped spring.
• D = 0 (i.e., p = ω).
This is the critically-damped spring.
• D > 0 (i.e., p > ω).
This is the overdamped or strongly-damped spring.
4.8.3
Underdamping
For p < ω the characteristic equation
λ2 + 2pλ + ω 2 = 0
has two complex conjugate roots
p
λ1 = −p + i ω 2 − p2
and λ2 = −p − i
p
ω 2 − p2 .
Two linearly independent solutions are thus
e−pt cos(ωp t) and
e−pt sin(ωp t),
where
ωp =
p
p
ω 2 − p2 = ω 1 − (p/ω)2 .
The general solution is
x(t) = e−pt c1 cos(ωp t) + c2 sin(ωp t) .
4.8. SIMPLE HARMONIC MOTION AND DAMPED OSCILLATIONS
207
This figure shows an example of underdamped motion:
"!$#%'&)(+*-,.0/1!$'&)(2*4356/178
1
0.8
0.6
0.4
B
0.2
?A>@
0
−0.2
−0.4
−0.6
−0.8
−1
0
10
20
9;30: #=<
40
50
60
It is best to write the general solution in terms of a single trigonometric function:
x(t) = Ae−pt cos(ωp t − φ) = A(t) cos(ωp t − φ).
Here A(t) is a time-dependent amplitude (as shown in the above figure with the
dashed lines):
A(t) = Ae−pt
and ωp is the frequency of the underdamped spring.
Since
ωp p
= 1 − (p/ω)2 < 1
ω
the damping of the spring leads to a frequency red-shift and an increase in the period
of the spring.
When p approaches ω the period tends to infinity so that we may expect that
the critically-damped spring does not display oscillatory motion. This is the case
considered in the next section.
208
4. ORDINARY DIFFERENTIAL EQUATIONS
Example: A mass of 1kg is attached to a spring hanging under gravity with
damping constant 0.2 and spring constant 4. Find the position of the mass after
time t if it is pulled down 1m from the equilibrium position and released without
kick.
4.8. SIMPLE HARMONIC MOTION AND DAMPED OSCILLATIONS
4.8.4
209
Critical damping
For p = ω the characteristic equation
λ2 + 2pλ + ω 2 = 0
has one root of multiplicity two: λ = −p. Two linearly independent solutions are
thus
e−pt and te−pt
and the general solution is
x(t) = e−pt (c1 + c2 t).
Unlike the underdamped spring it is straightforward to express c1 and c2 in terms
of the initial conditions x(0) and x0 (0):
c1 = x(0) and c2 = x(0)p + x0 (0).
Therefore
x(t) = e−pt (x(0) + x(0)pt + x0 (0)t).
The linear polynomial x(0) + x(0)pt + x0 (0)t vanishes for
t=−
x(0)
.
x(0)p + x0 (0)
This corresponds to a positive time t if x0 (0) < −x(0)p.
210
4. ORDINARY DIFFERENTIAL EQUATIONS
Depending on the initial conditions, the critically-damped spring behaves as shown
in this figure:
!"##$"%#&'($)*$+!#$#$ ,.-0/21 436587:9;-=<>[email protected]=<>DE
FHGJIKL8MONQPSRUTWV
FHGJIKL8MOKQPSRUTWV
FHGJIKL8M*XYNZP[RUTWV
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
−0.2
0
1
2
3
4
5
6
7
8
9
10
• The top-most curve corresponds to x(0) and x0 (0) both positive, so that the
weight is not just pulled down a distance x(0) but is also given a downward
kick on release.
• The middle curve corresponds to x0 (0) = 0 so that we only pull down the
weight and then release it.
• The third curve corresponds to x0 (0) < 0 so that upon release the weight is
kicked upward. Since the spring overshoots the equilibrium we must also have
that x0 (0) < −x(0)p.
Example: A mass of 1kg is attached to a spring hanging under gravity, with spring
constant 4. What value of the damping constant implies critical damping?
4.8. SIMPLE HARMONIC MOTION AND DAMPED OSCILLATIONS
4.8.5
211
Overdamping
For p > ω the characteristic equation
λ2 + 2pλ + ω 2 = 0
has two distinct real roots
λ1 = −p −
p
λ2 = −p +
p
p2 − ω 2 =: −β
p2 − ω 2 =: −α
with α > β > 0.
Two linearly independent solutions are thus
e−αt
and
e−βt
and the general solution is
x(t) = c1 e−αt + c2 e−βt .
For generic initial conditions x(0) and x0 (0), the constants c1 and c2 are both nonzero.
Assuming this, we have that for sufficiently large t
x(t)critial damping ∼ c2 te−pt
and
x(t)overdamped ∼ c2 e−βt .
√
Since β = p − . . . < p we see that the overdamped spring takes longer to return
to equilibrium than the critically damped spring.
The overdamped spring describes a door-closer that is too tight. It may take a long
time for the door to actually close because the strong damping restricts (nearly) all
movement.
A tradeswoman will try to adjust a door-closer such that the damping will be as
close as possible to critical damping. As always, Maths Rules!
212
4. ORDINARY DIFFERENTIAL EQUATIONS
Example: A mass of 1kg is attached to a spring hanging under gravity with
damping constant 5 and spring constant 4. Find the position of the mass after time
t if it is pulled down 1m from the equilibrium position and released without kick.
4.8. SIMPLE HARMONIC MOTION AND DAMPED OSCILLATIONS
4.8.6
213
Summary of motion of a spring:
The equation of motion of a mass m attached to a damped spring is described by
the ODE
x00 + 2p x0 + ω 2 x = 0,
where
k
β
> 0,
2p =
>0
m
m
and k is the spring constant and β the damping constant.
ω2 =
The roots of the characteristic equation λ2 + 2pλ + ω 2 = 0 determine the type of
motion:
• p = 0; No damping — pure oscillatory or harmonic motion.
The roots are ±iω and the general solution is given by
x(t) = c1 cos(ωt) + c2 sin(ωt) = A cos(ωt − φ).
• p < ω; Underdamped motion. p
The roots are −p ± iωp , ωp = ω 1 − (p/ω)2 and the general solution is given
by
−pt
x(t) = e
c1 cos(ωp t) + c2 sin(ωp t) = Ae−pt cos(ωp t − φ).
• p = ω; Critically damped motion.
The single root (of multiplicity 2) is given by −p and the general solution is
given by
x(t) = e−pt (c1 + c2 t).
• p > ω; Overdamped motion.
p
The two roots (denoted −α and −β) are −p ± p2 − ω 2 and the general
solution is given by
x(t) = c1 e−αt + c2 e−βt .
214
4.8.7
4. ORDINARY DIFFERENTIAL EQUATIONS
The pendulum
Some of you may have seen those beautiful old pendulum clocks. In such a clock a
pendulum of mass m is suspended from a pivot by a long, thin metal rod.
Now suppose that the pendulum moves in an arc of a circle with radius r. If the
pendulum makes an angle (say in the anti-clockwise direction) θ with its equilibrium position, then it has travelled a distance rθ from this position. According to
Newton’s second law of motion
m
d2
(rθ) = F,
dt2
where F is the force in the direction of motion. Hence
rm
d2 θ
= −mg sin θ.
dt2
This is a nonlinear ODE which displays very interesting nonlinear behaviour. If we
however assume that θ is very small (the pendulum does not swing very far out,
unlike your little sister on the swing perhaps), then we have sin θ ' θ. Making this
approximation we get
d2 θ g
+ θ = 0.
dt2
r
2
Setting g/r = ω as before we obtain the same ODE describing simple harmonic
motion.
Question: What is the period of the undamped pendulum?
Remark: It is interesting to note that the ODE describing the undamped pendulum
does not depend on the mass of the pendulum.
4.8. SIMPLE HARMONIC MOTION AND DAMPED OSCILLATIONS
215
As before one can make the model more realistic by assuming damping proportional
to velocity.
Example: If the damping constant is γ show that this leads to the ODE
θ00 + 2p θ0 + ω 2 θ = 0,
where
ω2 =
4.8.8
g
> 0,
r
2p =
γ
> 0.
m
Main points
• You should know how to derive and solve the equations of motion for a damped
oscillator given just the mass, damping and spring constants.
• You should also be able to classify the motion into under, over or critically
damped, and know what these notions mean.
216
4.9
4.9.1
4. ORDINARY DIFFERENTIAL EQUATIONS
Inhomogeneous Linear Second-Order ODEs
Inhomogeneous linear second-order ODEs with constant coefficients
Inhomogeneous second-order ODEs, also known as ODEs with forcing, are covered
in Stewart in Section 17.2 (Section 18.2). They are very common in applications
and often they require a bit of ingenuity to solve.
We will only consider inhomogeneous linear second-order ODEs with constant coefficients, i.e., equations of the form
ay 00 + by 0 + cy = r(x),
where r is a continuous function.
First we outline the general strategy of solving such equations.
1. First set r(x) = 0 and solve the corresponding homogeneous ODE. Its solution,
yH (x) = c1 y1 (x) + c2 y2 (x), is known as the complementary function.
2. Find a solution, yP (x) to the inhomogeneous equation. Any solution will do
and goes by the name of a particular solution.
3. The general solution to the inhomogeneous equation is then given by
y(x) = yH (x) + yP (x).
Clearly the remaining question is: How does one find a particular solution? Below
we will describe two methods. The first is again reduction of order and is the more
cumbersome, albeit most general method. We will illustrate it in the following
example.
4.9. INHOMOGENEOUS LINEAR SECOND-ORDER ODES
217
Reduction of order partially-worked-out example
Solve the ODE y 00 + y 0 − 2y = x2 − 2x + 3.
• First we need to find the complementary function. The characteristic equation
for the homogeneous ODE is
λ2 + λ − 2 = 0.
This has solutions λ = 1 and λ = −2, meaning that the complementary
function is given by
yH (x) = c1 ex + c2 e−2x .
• Next, set either y(x) = ex u(x) or y(x) = e−2x u(x). We will take the first
option but the second option works in the same way.
Since y = uex ,
y 0 = ex (u0 + u) and y 00 = ex (u00 + 2u0 + u).
Substituting this into the ODE and dividing by ex gives
u00 + 3u0 = e−x (x2 − 2x + 3).
This is a linear first-order ODE for u0 .
• We can use the integrating factor I = e3x to solve this linear first-order ODE.
The solution turns out to be
1
u(x) = (−2x2 + 2x − 7)e−x + c1 + c2 e−3x .
4
• Finally, we use y(x) = ex u(x) to find the general solution to the inhomogeneous
equation as
1
y(x) = (−2x2 + 2x − 7) + c1 ex + c2 e−2x .
4
This is y(x) = yP (x) + yH (x) with yP (x) = 14 (−2x2 + 2x − 7).
• By direct substitution you can check that the above yP (x) is indeed a particular
solution of the ODE.
Important remark: The previous method produced the general solution for the
inhomogeneous equation given the complementary solution. It did not produce just
a particular solution, which is all that is needed once the complementary solution is
known. Below we will show that in many simple examples, you can save yourself a
lot of work by only going for a particular solution.
218
4. ORDINARY DIFFERENTIAL EQUATIONS
Simpler method
We will consider the following cases for the inhomogeneous term r(x):
• r(x) is a polynomial.
• r(x) is a polynomial times an exponential.
• r(x) is a combination of the above.
Case 1: r(x) is a polynomial.
The method is best illustrated through an example, and again we take
y 00 + y 0 − 2y = x2 − 2x + 3.
Because the polynomial on the right is of degree 2 we differentiate the ODE 2 times
to get
y 000 + y 00 − 2y 0 = 2x − 2
y 0000 + y 000 − 2y 00 = 2
Remembering that we are only after a particular solution, not the general solution,
we can take yP00 = −1 as solution of the last equation since it correctly gives
0 + 0 − 2(−1) = 2.
If yP00 = −1 then yP0 = −x + C. Substituting this into
y 000 + y 00 − 2y 0 = 2x − 2
gives
0 + (−1) − 2(−x + C) = 2x − 2
so that C = 1/2, i.e., yP0 = −x + 1/2, or yP = (−x2 + x + D)/2. Substituting this
in the original ODE gives
(−1) + (−x + 1/2) − (−x2 + x + D) = x2 − 2x + 3,
so that D = −7/2 and
1
yP = (−2x2 + 2x − 7).
4
4.9. INHOMOGENEOUS LINEAR SECOND-ORDER ODES
219
Important remark: In the above computations we twice had to fix a constant.
Because “constants are constants” we could have ignored all x-stuff and streamlined
the computations as follows:
. . . then yP0 = −x + C. Substituting this into
y 000 + y 00 − 2y 0 = 2x − 2
and ignoring any x’s gives
0 + (−1) − 2(C) = −2
so that C = 1/2, i.e., yP0 = −x + 1/2, or yP = (−x2 + x + D)/2. Substituting this
in the original ODE again ignoring any x’s gives
(−1) + (1/2) − (D) = 3,
so that D = −7/2 and
1
yP = (−2x2 + 2x − 7).
4
Example: Find a particular solution of 3y 00 − 2y 0 + 4y = 2x3 + 1.
220
4. ORDINARY DIFFERENTIAL EQUATIONS
Case 2: r(x) is a polynomial times an exponential
Example: Find a particular solution to y 00 − y = 3xe2x .
Partially worked-out solution: In this case r(x) = 3xe2x is not a polynomial so
no amount of differentiating will reduce it to a constant.
We thus have to get rid of the exponential first.
The trick is to put y(x) = u(x)e2x with u(x) an unknown function.
Exercise: Show that u must satisfy the ODE u00 + 4u0 + 3u = 3x.
We do not need to find the general solution of the ODE satisfied by u, but again
only need a particular solution.
Exercise: Find a particular solution uP .
Exercise: Use uP to find a particular solution yP .
4.9. INHOMOGENEOUS LINEAR SECOND-ORDER ODES
221
Important remark: The above method also works in the case that r(x) is a
polynomial times a simple trigonometric function. For this case, you use the fact
that e i x = cos x + i sin x to write your trigonometric function as either the real or
imaginary part of an exponential. This allows you to use the above method, and
then you need to take either the real or imaginary part (depending on if you had a
cos or a sin term) of your particular solution at the end of the calculation.
Case 3: r(x) is a sum of the previous two cases.
The above two cases are enough to also deal with linear combinations such as
r(x) = x2 + 2x + 1 + e2x (x + 1)
or
r(x) = x3 + xex + e3x .
Theorem: If yP1 is a particular solution of ay 00 + by 0 + cy = r1 (x) and
yP2 is a particular solution of ay 00 + by 0 + cy = r2 (x), then yP = yP1 + yP2
is a particular solution of ay 00 + by 0 + cy = r1 (x) + r2 (x).
222
4. ORDINARY DIFFERENTIAL EQUATIONS
Example: Find a particular solution of y 00 − 2y 0 + y = x2 + 1 + e−2x .
4.9. INHOMOGENEOUS LINEAR SECOND-ORDER ODES
223
Example: Finding a general solution
Find the general solution to y 00 + 4y 0 + 5y = 5x + 1.
4.9.2
Main points
• You should be able to find a particular solution for the cases we have talked
about.
• You should know how to find the general solution as the sum of a particular
solution and the complementary function.
224
5
5. PARAMETRISATION OF CURVES AND LINE INTEGRALS
Parametrisation of Curves and Line Integrals
Stewart covers parametrisation in Chapters 10 and (some of) 12 (Chapters 11 and
(some of) 13). There are some features of Matlab, especially plotting, that will work
with parametric forms.
5.1
Parametrisation of Curves
Imagine some curve in the xy-plane, for example, x2 + 2y 2 = 1 or 3x + y = 2. One
way to think of the curve is as the trajectory of a moving particle, such that at time
t the particle is at the point (x(t), y(t)) on the curve. In other words, we describe
the curve by
(
x = f (t)
y = g(t).
Such equations for the curve are called parametric equations, and the auxiliary
variable t, is called a parameter. The functions f and g are said to provide a
parametrisation of the curve.
y
t2 > t1
(f (t2 ), g(t2 ))
(f (t1 ), g(t1 ))
x
Before we look at some examples, it should be clear from our mental picture of a
moving particle that a parametrisation of a curve is not unique. For example,
(
x = F (t)
y = G(t),
where, say F (t) = f (3 − t) and G(t) = g(3 − t), is another parametrisation of the
same curve; we simply have changed the velocity and initial (i.e., t = 0) position of
the imaginary particle.
5.1. PARAMETRISATION OF CURVES
5.1.1
225
Sketching parametric curves
Example: Sketch the curve parametrised as
(
x = t2 − 2t
y = t + 1.
Give an equation for the curve in terms of just the Cartesian coordinates x and y
and give two alternative parametrisations of the curve.
226
5. PARAMETRISATION OF CURVES AND LINE INTEGRALS
Sometimes one has reasons to restrict t to a finite interval. In general, the curve
with parametric equations
(
x = f (t)
a≤t≤b
y = g(t)
starts at the point (f (a), g(a)) and finishes at the point (f (b), g(b)). One has to be
careful though, since even a finite time-interval can still give rise to an infinitely-long
curve.
(
x = t2 − 2t
Example: Sketch the curve
0 ≤ t ≤ 4.
y =t+1
5.1. PARAMETRISATION OF CURVES
227
Example: Sketch the Archimedean spiral given by
(
x = t cos t
y = t sin t
and verify that it may also be described by x + y = arctan2 (y/x).
2
If you like spirals you could

cos t

x =
t
t > 0;
sin
t

y =
t
also try to sketch

cos t

x = √
t
t > 0;
sin t

y = √
t
0≤t≤
π
2
2
(
x = e−t cos t
y = e−t sin t
t ∈ R.
For the true connoisseurs there even is the amazing Cornu spiral; who said maths
couldn’t be fun?

Z t


cos(s2 )ds
x =
0Z
t > 0.
t

2

y = −
sin(s )ds,
0
228
5.1.2
5. PARAMETRISATION OF CURVES AND LINE INTEGRALS
Parametrising circles and ellipses
Example: Parametrise the circle of radius r centered at (a, b).
5.1. PARAMETRISATION OF CURVES
229
Example: Parametrise the circle with radius 2 centered at (2, 3), such that the
imaginary particle tracing this circle is travelling in clockwise direction and at time
t = 0 is at (0, 3).
y
(x − 2)2 + (y − 3)2 = 4
3
2
x
230
5. PARAMETRISATION OF CURVES AND LINE INTEGRALS
Example: Parametrise the ellipse
x2 y 2
+ 2 = 1.
a2
b
A much more fancy parametrisation may be given using Jacobi’s elliptic functions
sn, cn and dn. You will hopefully learn about these functions in a more advanced
mathematics course.
5.1. PARAMETRISATION OF CURVES
5.1.3
231
Parametric curves in R3
In general, a curve in three dimensions can be parametrised in the form



x = f (t)
y = g(t)


z = h(t).
A helix is a three-dimensional curve that has shape of a wound spring, or a piece of
string wrapped around a cylinder.
One possible parametrisation of the helix is



x = r cos(πt)
y = r sin(πt)


z = t
for t ∈ R.
Example: Sketch the helix for t ∈ [−2, 2].
232
5.1.4
5. PARAMETRISATION OF CURVES AND LINE INTEGRALS
Piecewise linear paths
Example: Sketch the curve parametrised by
5.1.5
x(t) = 1,
y(t) = t,
0≤t≤1
x(t) = 2 − t,
y(t) = t,
1<t≤2
x(t) = 2 − t,
y(t) = 2,
2 < t ≤ 3.
Plotting parametrised equations in Matlab
The ezplot function handles parametric plots. To plot an ellipse, the command is
simply
ezplot(’2*cos(t)’,’1/3*sin(t)’)
To plot a spiral, simply use
ezplot(’exp(-t/100)*cos(t)’,’exp(-t/100)*sin(t)’)
Executing this command Matlab uses a default range for the t values. To get a
longer spiral, 0 ≤ t ≤ 100, run
ezplot(’exp(-t/100)*cos(t)’,’exp(-t/100)*sin(t)’,[0,100])
5.1. PARAMETRISATION OF CURVES
5.1.6
233
Main points
• You should understand what a parametric representation of a curve is.
• You should be able to parametrise circles and ellipses.
• You should be able to sketch curves from their parametric representation. This
includes being able to sketch parametric representations of parabolas, circles,
ellipses, spirals, helices and piecewise linear paths.
• You should be able to plot parametric forms in Matlab.
234
5.2
5. PARAMETRISATION OF CURVES AND LINE INTEGRALS
Position Vectors, Velocity and Acceleration
Stewart covers the following material in Section 13.4 (Section 14.4). Parametric
equations are used to describe position, velocity and acceleration.
5.2.1
Position vector
The position vector
r(t) = x(t)i + y(t)j
traces out the path given by the parametrisation (x(t), y(t)).
y
t2 > t1
(x(t1 ), y(t1 ))
(x(t2 ), y(t2 ))
r(t1 )
r(t2 )
x
Similarly in three dimensions, the position vector
r(t) = x(t)i + y(t)j + z(t)k
traces out the path given by the parametrisation (x(t), y(t), z(t)).
Example: Find the position vector of a car moving along the helical path
x = cos 5t,
y = sin 5t,
z = t.
5.2. POSITION VECTORS, VELOCITY AND ACCELERATION
5.2.2
235
Parametrising straight lines
We know from chapter 1 of this workbook that the position vector of any point on
a straight line parallel to the vector v and passing through the point r 0 is given by
r(t) = r 0 + tv.
Example: Find the position vector of any point on the line passing through the
1
point (0, 1, 3) and parallel to the vector 2i − 3j + k.
2
236
5. PARAMETRISATION OF CURVES AND LINE INTEGRALS
Say we want to parametrise the straight line between the points (a, b) and (c, d). We
also require that (a, b) corresponds to parameter value t = t0 , and (c, d) corresponds
to t = t1 , with t1 = t0 + ∆t and ∆t > 0.
Let r 0 be the position vector for (a, b) and r 1 the position vector for (c, d). The
vector
v = r1 − r0
is a direction vector for the line that goes through (a, b) and (c, d).
(c, d)
(a, b)
r 1 = r(t1 )
r 0 = r(t0 )
v
0
We know from chapter 1 that the position vector for points on the line through r 0
and r 1 is
r = r 0 + λv = (1 − λ)r 0 + λr 1 ,
λ ∈ R.
So all we need to do to parametrise the portion of the line from r 0 to r 1 is choose
a relationship between λ and t that gives λ = 0 when t = t0 and λ = 1 when t = t1 .
The linear relation between λ and t that does this is
λ=
t − t0
.
∆t
A parametrisation for the line segment is therefore
r(t) = r 0 +
t − t0
v,
∆t
t0 ≤ t ≤ t1 .
5.2. POSITION VECTORS, VELOCITY AND ACCELERATION
237
Example: Parametrise the path given in the following diagram, so that path C1
is given for 0 ≤ t ≤ 1 and the path C2 is given for 1 ≤ t ≤ 3.
y
−1
x
C2
−1
C1
−2
238
5.2.3
5. PARAMETRISATION OF CURVES AND LINE INTEGRALS
Velocity vector
The vector tangent to the path of motion, with magnitude equal to the speed is the
velocity vector. If r(t) is the position vector, then
∆r = r(t + ∆t) − r(t)
is approximately tangent to the curve traced out by r(t). The approximation gets
better as ∆t → 0.
r(t)
0
∆r
r(t + ∆t)
The velocity vector v is given by
r(t + ∆t) − r(t)
∆t→0
∆t
∆r
= lim
∆t→0 ∆t
dr
=
.
dt
v(t) = lim
In component-form, if r(t) = x(t)i + y(t)j + z(t)k, then the velocity is
v(t) =
dx
dy
dz
dr
=
i+
j+
k.
dt
dt
dt
dt
Speed is the magnitude of velocity:
s
speed = kv(t)k =
dx
dt
2
+
dy
dt
2
+
dz
dt
2
.
Note that if an object moves with constant velocity, this means that it has constant
speed and direction, i.e., it travels in a straight line.
5.2.4
Acceleration
The acceleration vector is defined as
a(t) =
dv
d2 x
d2 y
d2 z
= 2 i + 2 j + 2 k.
dt
dt
dt
dt
5.2. POSITION VECTORS, VELOCITY AND ACCELERATION
239
Example: The velocity of an object is v = 2i + 3j + 4k and at t = 0 the object
passes through (−2, 1, 0). Find the position vector.
Example: Find the velocity vector of a car moving along the helical path (e.g.,
the Indooroopilly Shopping Centre carpark)
x = cos 5t,
y = sin 5t,
z = t.
240
5.2.5
5. PARAMETRISATION OF CURVES AND LINE INTEGRALS
Quiz
Can you match each position vector with one of the curves? Then locate the points
corresponding to t = 0.
(a) r(t) = (−t, 3/2 − t)
(b) r(t) = (1, t − 1)
1
(c) r(t) = (− sin t, cos t)
2
(d) r(t) = 2(cos 3t, sin 3t).
y
y
1
2
1
x
1
2
x
−1
y
y
2
3
2
2
x
3
2
x
5.2. POSITION VECTORS, VELOCITY AND ACCELERATION
5.2.6
241
Extra reading: stone on a string
Imagine you spin a stone on a string above your head so that it moves in a circle
of radius 1m at a height of 2m above the ground at a constant speed with period π
seconds. If the string breaks, find the position, velocity and acceleration vectors of
the stone after the string breaks.
Before the string breaks
r(t) = cos 2ti + sin 2tj + 2k
v(t) = −2 sin 2ti + 2 cos 2tj,
so that r(0) = i + 2k and v(0) = 2j.
After the string breaks the stone moves under gravity
ma(t) = −mgk
⇒
a(t) = −gk
⇒
dv
= −gk.
dt
Hence
v = −gtk + v(0) = 2j − gtk.
Since
dr
= v this in turn implies that
dt
1
r(t) = 2tj − gt2 k + r(0)
2
so that
5.2.7
1 2
r(t) = i + 2tj + 2 − gt k.
2
Main points
• You should understand the relationship between a position vector and the
parametric equations of a curve.
• You should understand the relationship between position, velocity and acceleration vectors.
• You should be able to parametrise lines and piecewise linear paths.
242
5.3
5. PARAMETRISATION OF CURVES AND LINE INTEGRALS
Line Integrals and Work Done
The content of this section is covered in Stewart over a few sections: arc length in
Sections 8.1 and 10.4 (Sections 9.1 and 11.4) and general line integrals in 16.2 (17.2).
5.3.1
Length of a curve
Say we have a curve in R3 parametrised by
x = x(t), y = y(t), z = z(t), a ≤ t ≤ b.
We aim to calculate the length of the curve. This is commonly referred to as the
arc length.
We can work out how to do this by first approximating the curve as a series of
straight lines. We can draw a picture of what we are doing for a curve in R2 , but
we can do the same thing for a curve in R3 .
5.3. LINE INTEGRALS AND WORK DONE
243
In this case, with ∆si representing the length of a small segment of the curve corresponding to a small change ∆t in the parameter from ti to ti + ∆t, and given
r(t) = (x(t), y(t)) we have the approximation
dr ∆si ≈ ∆t.
dt t=ti
The same formula applies in R3 and then the length of the curve is approximately
the sum of the lengths of each line segment:
244
5. PARAMETRISATION OF CURVES AND LINE INTEGRALS
This approximation improves for smaller (and hence more) line segments. In the
limit when the number of line segments tends to infinity, we obtain an exact result,
giving the arc length formula
s
Z b 2 2 2
dx
dy
dz
+
+
dt.
arc length =
dt
dt
dt
a
This formula is often written more succinctly using the notion of a line integral. If
the function s(t) gives the length of the curve C between r(a) and r(t) — so that
the arc length we are after is nothing but s(b) — then
s 2 2
2
ds
dx
dy
dz
=
+
+
.
dt
dt
dt
dt
Then the length of the arc between r(a) and r(b) is given by the line integral
Z
ds.
arc length =
C
In general, if f (x, y) is defined on a smooth curve C in R2 then the line integral of
f along C is
Z
f (x, y)ds.
(1)
C
If C is parametrised by r(t), a ≤ t ≤ b and f (x, y) > 0, then this line integral gives
the area of a ribbon whose base is the curve C and height above the point (x, y) is
f (x, y).
z
y
r(a)
C
o
f (x, y)
x
r(b)
Line integrals are a generalisation of one-dimensional definite integrals.
5.3. LINE INTEGRALS AND WORK DONE
Example: Find the length of the helix
245



x = cos 10t
y = sin 10t


z = t
0 ≤ t ≤ π.
246
5. PARAMETRISATION OF CURVES AND LINE INTEGRALS
Example: Use the arc length formula to show that the length of a semicircular arc
of radius 2 is 2π.
5.3. LINE INTEGRALS AND WORK DONE
247
Example: Find the length of the spiral

x(t) = e−t/10 cos t
for t ≥ 0.
y(t) = e−t/10 sin t
Solution:
We compute
r(t) = e−t/10 (cos t i + sin t j)
1
1
−t/10
−t/10
v(t) = e
− cos t − sin t i + e
− sin t + cos t j
10
10
r
1
1√
kv(t)k =
101e−t/10 .
+ 1 e−t/5 =
100
10
Hence
Z
∞
arc length =
0
1√
kv(t)k dt =
101
10
The improper integral can be computed as
Z ∞
Z
−t/10
e
dt = lim
A→∞
0
Z
∞
e−t/10 dt.
0
A
e−t/10 dt
0
A
= −10 lim e−t/10 0
A→∞
= −10 lim
A→∞
e−A/10 − 1
= 10.
Therefore the arc length of the spiral is
√
101.
Example: Find the length of the parabola y = x2 for x between 0 and 2.
Solution: First we parametrise the parabola as x = t, y = t2 for 0 ≤ t ≤ 2. Hence
r(t) = (t, t2 )
⇒ kv(t)k =
p
1 + (2t)2
⇒
⇒
v(t) = (1, 2t)
Z 2p
arc length =
1 + (2t)2 dt.
0
(If you are really clever you can perhaps show that this gives
√
17 + arcsinh(4)/4.)
248
5.3.2
5. PARAMETRISATION OF CURVES AND LINE INTEGRALS
Work done by a constant force
In one dimension, the work done by a constant force F in moving an object along a
straight line of length d is W = F d.
In two or three dimensions, the work done by a constant force in moving a particle
−→
along a straight line from P to Q is W = F · P Q.
5.3. LINE INTEGRALS AND WORK DONE
5.3.3
249
Work done over a curve
We now consider the more general case of the work done by a force field
F (x, y, z) = F1 (x, y, z)i + F2 (x, y, z)j + F3 (x, y, z)k
which moves an object along a curve C.
First we give an approximation: divide C into n arcs, so that the ith arc has length
∆si .
We approximate ∆si as a straight line (for ∆si small enough), and F as constant
over ∆si by evaluating F at a specified point Pi on the arc. Let T (Pi ) be a unit
tangent vector to the curve C at the point Pi . We then approximate the direction
over the arc as the vector T (Pi )∆si .
Thus the work done over the ith arc is approximately
Wi ≈ F (Pi ) · T (Pi )∆si .
Summing up over all arcs gives an approximation to the total work done:
X
X
W ≈
Wi ≈
F (Pi ) · T (Pi )∆si .
Taking the limit ∆si → 0 gives
Z
F · T ds.
W =
C
This last integral, which is an integral over the curve C, is again known as a line
integral.
250
5. PARAMETRISATION OF CURVES AND LINE INTEGRALS
To evaluate the line integral, we use a parametrisation of the curve C. Let C be
parametrised by
r(t) = (x(t), y(t), z(t)),
a ≤ t ≤ b.
We let ∆t represent a small change in t corresponding to ∆si . Let Pi correspond to
parameter value ti .
As we saw previously, as ∆si → 0,
∆si ≈ kr 0 (ti )k∆t,
and we can write the unit tangent vector to the curve at the point Pi in terms of
the parametrisation as
r 0 (ti )
T (Pi ) = 0
.
kr (ti )k
Hence the work done over the ith arc is approximately
Wi ≈ F (r(ti )) · r 0 (ti )∆t.
5.3. LINE INTEGRALS AND WORK DONE
251
Summing up over all arcs gives an approximation to the total work done:
X
X
W ≈
Wi ≈
F (r(ti )) · r 0 (ti )∆t.
Finally, taking the limit as ∆t → 0, we have
Z b
F (r(t)) · r 0 (t)dt.
W =
a
It is common to express these integrals as
Z
F · dr.
C
This is merely a notational convenience. This notation means
Z
Z
F · dr =
work done =
C
where C is parametrised by r(t), a ≤ t ≤ b.
a
b
F (r(t)) · r 0 (t)dt,
252
5. PARAMETRISATION OF CURVES AND LINE INTEGRALS
R
Example: Evaluate C F · dr, where F = (xy, yz, zx) and C is parametrised as

 x=t
0 ≤ t ≤ 1.
y = t2

3
z=t
5.3. LINE INTEGRALS AND WORK DONE
253
Example: Find the work done by moving a mass of 2 kg along an arc of the circle
x2 + z 2 = h2 from (−h, 0, 0) to (0, 0, h). Does the direction we choose matter?
254
5. PARAMETRISATION OF CURVES AND LINE INTEGRALS
5.3.4
Main points
• You should know how to use a line integral to calculate the arc length of a
curve.
• You should know how to use a line integral to calculate the work done by a
force field in moving a particle along a parametrised path.
5.4. GRADIENT FIELDS
5.4
255
Gradient Fields
Gradient fields are a special type of force field which are conservative, and they give
a relatively easy way to determine if the field is conservative. Stewart covers this in
Sections 16.1 and 16.3 (Sections 17.1 and 17.3).
5.4.1
Z
Conservative fields
B
F · dr is independent of the path taken, then F is called a conservative field.
If
A
Z
B
F · dr will give the same result
That is, if F is conservative then the line integral
A
for any path C that you choose between A and B.
B
C2
C1
A
Example: Gravity is a conservative field because
Z B
Z B
F · dr = −mg
dz = −mg(z(B) − z(A)),
A
A
and it does not matter how you get from A to B. Any constant field must be
conservative by the same reasoning.
256
5. PARAMETRISATION OF CURVES AND LINE INTEGRALS
Example: Take the field F = xi + yj, which you can picture as a set of vectors
F (x, y) at each point (x, y).
y
x
Since
F · dr = (xi + yj) · (dxi + dyj) = xdx + y dy,
we get
Z
B
Z
B
F · dr =
A
xdx + y dy
A
1
=
2
Z
B
A
iB
1h 2
2
d(x + y ) = x + y
.
2
A
2
2
Once again F is a conservative field. But it was chosen in rather a special way. F
is a gradient field, i.e., F = ∇f for some f . In this case f (x, y) = (x2 + y 2 )/2 since
2
x
y2
F =∇
+
= (x, y).
2
2
5.4. GRADIENT FIELDS
257
Z
B
F · dr is
Theorem: Any gradient field F is a conservative and hence
A
path-independent.
To show this, let F = ∇f =
∂f
∂f
i+
j. Then using the chain rule,
∂x
∂y
F (r(t)) ·
The work done is given by
Z
dr
∂f dx ∂f dy
=
+
dt
∂x dt
∂y dt
df
.
=
dt
B
Z
b
F (r(t)) ·
F · dr =
a
A
Z
b
=
a
Z
dr
dt
dt
df
dt
dt
B
=
df
A
= f (B) − f (A).
Important remark: The above means that finding the work done by a conservative field really comes down to finding a function f such that F = ∇f . This
function f is called a potential function for the force field F . If you can find a
potential function f such that ∇f = F then
Z
B
F · dr = f (B) − f (A).
A
Important remark: This equation is by far the simplest way to evaluate the work
RB
done by a conservative field. When asked to evaluate A F · dr for a conservative
field you are expected to use the above formula and not a parametrisation of the
path.
258
5. PARAMETRISATION OF CURVES AND LINE INTEGRALS
Example: Find the work done by the electric field
E=
xi + yj + zk
x2 + y 2 + z 2
with potential
1
log(x2 + y 2 + z 2 )
2
along any path from (1, 0, 0) to (1, 1, 1).
f=
5.4. GRADIENT FIELDS
259
Example: Evaluate the work done by the force F from (0, 0) to (1, 2), where
F = ∇f
and f (x, y) = xy.
Example: Show that F (x, y) = (x + y, x) is a gradient field.
260
5. PARAMETRISATION OF CURVES AND LINE INTEGRALS
Example: Show that F (x, y) =
y
(x + y)
i + j is not a gradient field.
2
2
It turns out that you don’t have to look for a potential function to check whether a
force field F = (F1 , F2 ) is a gradient field. If we assume that F is a gradient field
with potential function f then
∂f ∂f
F = (F1 , F2 ) =
,
.
∂x ∂y
∂ 2f
∂ 2f
We know that
=
when the second order partial derivatives are contin∂x ∂y
∂y ∂x
uous, which means that the components of our force field F must satisfy
∂F1
∂F2
=
.
∂y
∂x
In fact, the components of a vector field in R2 satisfy this condition if and only if F
is a gradient field. This means that we can simply check this condition to determine
whether a force field is a gradient field.
You will look at this result in more detail in MATH2000.
5.4. GRADIENT FIELDS
261
Example: Determine whether the force field F = (y 2 − 2, 2xy) is conservative.
Evaluate the work done by F in moving a particle along the path x = 2t, y = t2 ,
0 ≤ t ≤ 2.
262
5. PARAMETRISATION OF CURVES AND LINE INTEGRALS
5.4.2
Main points
• You should understand the relation between a gradient field and its potential
function.
• You should understand that conservative fields and gradient fields are equivalent and that the work done by these fields in moving a particle between two
points is independent of the path taken.
• You should know that the potential function will give you an easy method to
calculate line integrals for gradient fields.
• You should know how to check whether a force field is a gradient field.
• You should know how to find a potential for a conservative force field.
5.5. WORK DONE OR PATH INTEGRALS FOR NON-CONSERVATIVE FIELDS
5.5
263
Work Done or Path Integrals for Non-conservative Fields
This material is covered in Stewart, Section 16.2 (Section 17.2).
Line integrals for forces that are not conservative depend on the chosen path, which
makes them harder to evaluate. To work them out we need to parametrise the path,
by say t, and convert the integral to one over t. We have already seen how to do
this in section 5.3.
Let F (x, y) = F1 (x, y)i + F2 (x, y)j be a non-conservative field and let x(t) and y(t)
provide a parametrisation of the path along which we wish to integrate. Then
Z
Z b
dr
dt.
F · dr =
F (r(t)) ·
dt
C
a
Or, in component form,
F · dr = F1 (x, y)dx + F2 (x, y)dy
and
Z b
Z
F · dr =
C
a
dx
dy
F1 (x(t), y(t))
+ F2 (x(t), y(t))
dt
dt
dt.
There are some important properties of line integrals that can be useful when evaluating the work done by a non-conservative field:
264
5. PARAMETRISATION OF CURVES AND LINE INTEGRALS
Z
F · dr, where F (x, y) =
Example: Evaluate
C
as shown in the figure below.
y
1
C2
C1
1
x
x+y
2
i+
y
j and the path is
2
5.5. WORK DONE OR PATH INTEGRALS FOR NON-CONSERVATIVE FIELDS
265
Z
F · dr, were F = (x + y)i + yj and C is the arc of the
Example: Evaluate
C
ellipse
x2
+ y 2 = 1 from (0, 1) to (2, 0) as shown in the figure below.
4
y
1
2
x
266
5. PARAMETRISATION OF CURVES AND LINE INTEGRALS
5.5.1
Main points
• You should know how to recognise a non-conservative field.
• You should know how to evaluate the work done by a non-conservative field
by performing a parametrisation of the path.
267
6
MATLAB Guide
6.1
6.1.1
Introduction
Getting started
• Log in using your library username and password. Ask your tutor if you have
a problem.
• Open MATLAB just like any other Windows program.
Start→Programs→MATLAB→MATLAB R2008b
• Save all files to your student directory on H:/ or your own USB flash drive.
Files saved onto individual computers or S:/ will be deleted at the end of
semester.
As MATH1051 is a companion for MATH1052, it is generally assumed that you
will still be in the process of developing a basic knowledge of MATLAB. Such basic knowledge includes navigating to the correct directory, executing functions and
defining variables. Many of these fundamentals will be re-introduced in this document, but some simple operations, etc. may be used without explanation. If you
need further guidance then consult MATLAB Help or the MATH1051 notes.
6.1.2
Getting help
There are a number of resources available should you need assistance when using
MATLAB. The most easily accessible is MATLAB’s own built-in Help system.
• Clicking the blue question mark button in the toolbar will open the Help
window, as will typing doc in the command window. You can search for the
topic in question or navigate using the panel on the left.
• If you know the name of the function or operation you need help with, typing
help <function name> in the command window will display a short description of the function and how to use it. Typing doc <function name> will
open the Help window and navigate to the page dedicated to that function or
operation.
If you have trouble finding what you need with MATLAB’s Help system, there is
a great deal of documentation and useful articles available online. The Mathworks
268
6. MATLAB GUIDE
website (http://www.mathworks.com/products/matlab/) has a lot of helpful descriptions and solutions which go beyond what is available in the MATLAB Help,
as well as m-files written and uploaded by users. A Google search often turns up
results as well.
6.2
The Basics
The MATLAB Command Window can be used like an advanced calculator. Mathematical operations like addition, subtraction, multiplication, division and exponentiation are executed just as with a calculator, using +, −, ∗, / and ˆ respectively.
MATLAB also has many more complicated functions built in, like sin, cos, tan,
log and exp, as well as constants like π and i. Consult the MATH1051 notes or
MATLAB Help for more details on these or other functions.
>> 1+6*2
ans =
13
>> log(10)
ans =
2.3026
>> exp(cos(pi))
ans =
0.3679
Variables are defined using an equals sign, and can then be used just like regular
numbers.
>> x = 7
x =
7
>> 2^x
ans =
128
Vectors and matrices are defined in MATLAB using square brackets. Spaces or
commas separate columns and semicolons separate rows. MATLAB has many inbuilt
operations for use with matrices, such as addition and multiplication, and more
complex functions like eig and rref. Note that matrix indices start at 1, not 0 (so
the top-left cell in a matrix is in the (1,1) position).
>> x = [1 2; 5 6]
x =
6.3. M-FILES
269
1
2
5
6
>> x^2
ans =
11
14
35
46
>> [2 3]*x
ans =
17
22
>> x(1,2)
ans =
2
Use the command clear <variable name> to remove a variable definition from
MATLAB’s memory, and the command clear all to remove all variables at once.
Variables should usually be cleared in between MATLAB sessions to avoid using
incorrect values.
6.3
M-files
M-files are programs written in MATLAB’s own programming language. They are
essentially a series of commands which are executed successively when the M-file is
run. There are two main types of M-files in MATLAB:
• M-file scripts are sequences of commands which could normally be entered into
the command window. They are usually intended to be executed a number
of times, so time is saved by putting the commands into the M-file. The
operations inside a M-file script have access to all the variables, etc. which
have been saved in the command window.
• M-file functions begin with
function [output variables] = function name(input variables)
and operate like ordinary functions such as sin or log. They can take input
variables, perform operations on them, and produce output variables. The
operations inside an M-file function do not have access to the variables saved
in the command window – they can only access the input variables.
270
6. MATLAB GUIDE
Anything followed by a % in a M-file is a comment, and will not affect the execution
of the file. You should comment all your M-files to help yourself or others understand
how the program works.
For example, put the following commands into a new M-file, and save it as example script.m.
x = [1 2 3 4 5];
y = 2*x’;
z = x*y
Now run this script by typing the command example script in the command window:
>> example script
z =
110
Note that only the value of z was ouput, because the definitions of x and y were
followed by semicolons (suppressing their output).
Put the following into an M-file and save it as example function.m. It will evaluate the polynomial 4x3 + 2x2 − 10x − 7.
function output = example function(x)
% Evaluates the 3rd order polynomial at x
output = 4*x^3 + 2*x^2 - 10*x - 7;
Execute the function as you would any regular function in MATLAB:
>> example function(4)
ans =
241
Note that the name of the function (the word following the = sign at the top of the
M-file) must match the name of the M-file itself.
6.3.1
If statements
An if statement in MATLAB is a special type of command which is executed only
if a certain condition is satisfied. The general structure of an if statement is always
if 1st condition
1st commands
elseif 2nd condition
6.3. M-FILES
271
2nd commands
else
3rd commands
end
The conditions above should be boolean statements; i.e. they should take the value
0 or 1. MATLAB has many of these built in; for example, x == y returns 1 if x = y
and 0 otherwise, and x <= y returns 1 if x ≤ y and 0 otherwise. If 1st condition
takes the value 1, then 1st commands will be executed, and the if statement will
be exited. Otherwise, the next condition will be checked, and so on. If none of the
conditions take the value 1, then the commands following else will be executed.
You can have as many elseif lines as you want (or none at all), and you do not
need to have an else line (in which case MATLAB will do nothing if none of the
conditions hold).
For example, the following function will return the absolute value of a real number,
as well as stating whether it is positive, negative or zero.
function output = abs val check(x)
% Outputs the absolute value of x and states whether x
% is positive, negative or zero
if x > 0
output = x;
display(’positive’)
elseif x < 0
output = -x;
display(’negative’)
else
output = 0;
display(’zero’)
end
The function can now be executed in the command window in the usual way.
>> abs val check(5)
positive
ans =
5
6.3.2
For and while loops
Loops allow MATLAB to perform repetitive operations automatically. There are
two main types of loops:
272
6. MATLAB GUIDE
• For loops instruct MATLAB to perform a fixed number of operations.
• While loops instruct MATLAB to repeatedly perform operations for as long
as a certain condition holds.
The basic structure of a for loop is
for index = vector
commands
end
For each value in the vector, the variable index is assigned that value, and commands
are executed.
For example, the following function will calculate the sum of the first 10 integer
powers of x.
function output = sum powers(x)
% Evaluates the sum of the first 10 powers of x
temp = 0;
for j = 1:10
temp = temp + x^j;
end
output = temp;
Evaluate the function in the usual way:
>> sum powers(2)
ans =
2046
While loops look a bit like if statements – their basic structure is
while condition
commands
end
On the first iteration of the loop, MATLAB checks whether condition holds. If it
does, then commands are executed, and the process is repeated. If on any iteration
condition does not hold, then MATLAB exits the loop.
For example, the following function performs the same operation as the one above.
function output = sum powers2(x)
% Evaluates the sum of the first 10 powers of x
temp = 0;
6.4. VISUALISING FUNCTIONS
273
j = 1;
while j <= 10
temp = temp + x^j;
j = j + 1;
end
output = temp;
Here we are increasing the value of j by 1 in each iteration, and we finish the loop
after j = 10.
There is a danger with while loops: if the condition always holds, then the loop
will keep iterating forever. When writing while loops, make sure that there will
always be some point at which the condition will fail. If you happen to find yourself
running an infinite loop, use the command Ctrl+c to terminate it.
6.4
Visualising Functions
There are many situations, especially when dealing with multivariate calculus, when
a function or a set of data is difficult to visualise. There are a number of ways to
plot and view functions and data in MATLAB.
6.4.1
Ezplot
The ezplot command allows you to plot functions of one variable. You can control
the domain and range if you wish, or let MATLAB assign the default domain of
[−2π, 2π]. For example, to plot y = x sin(x) over the domain [−π, π] and range
[−10, 10], use the command
>> ezplot(’x*sin(x)’,[-pi,pi,-10,10])
Remember to insert the quotation marks ‘ ’ around the function. To plot multiple
functions at once, use the hold on command before you start plotting. You can
change the domain and range of the figure without re-plotting by using the axis
command; for example,
>> axis([0,pi,-10,10])
You can edit the plot or provide a title and axis labels by using the Property
Editor in the View menu.
The ezplot command can also be used to plot points of the form f (x, y) = 0. For
example, to plot the points which satisfy sin(x) + cos(y) = 0 in the domain and
range [−2π, 2π], use the command
274
6. MATLAB GUIDE
>> ezplot(’sin(x)+cos(y)’,[-2*pi,2*pi,-2*pi,2*pi])
If you wish to open a new figure window to plot functions on a new axis, use the
figure command.
The ezplot command, despite only being able to plot functions of one variable,
can help in visualising functions of two or more variables. This is because it can
be used to plot cross sections. For example, to plot cross sections of a function
z = f (x, y), use hold on and simultaneously plot the functions z = f (x, −2), z =
f (x, −1), . . . , z = f (x, 2). Using different colours, line types and/or labels for each
plot may make it easier to visualise the cross sections. Consider also plotting cross
sections where x instead of y is held constant (but not on the same axis!).
For example, to plot cross sections of the function z = f (x, y) = sin(x) cos(y) for
−2π ≤ x ≤ 2π and −2 ≤ y ≤ 2, you could use the commands
>>
>>
>>
>>
>>
>>
hold on
ezplot(’sin(x)*cos(-2)’)
ezplot(’sin(x)*cos(-1)’)
ezplot(’sin(x)*cos(0)’)
ezplot(’sin(x)*cos(1)’)
ezplot(’sin(x)*cos(2)’)
6.4.2
Ezcontour
Cross sections may help in visualising a multivariate function, and can often be
done easily by hand. However there are many cases where they only produce very
confusing or uninformative pictures. Contour diagrams are sometimes a better way
of visualising multivariate functions. The ezcontour command is the easiest way to
produce contour diagrams for functions. It is used in much the same way as ezplot.
The default x and y domains are both [−2π, 2π], but can be controlled in the same
way as for ezplot.
>> ezcontour(’sin(x)*cos(y)’)
>> ezcontour(’sin(x)*cos(y)’,[-10 10 -10 10])
The lines in the contour diagram represent points where z takes the same value.
The colours of the lines represent the value of z on that line. Use the command
colorbar or go to Insert → Colorbar to insert a colourbar, which will show the
values of z corresponding to the colours.
The ezcontourf command operates in the same way as ezcontour, but colours in
the gaps between the contours.
6.4. VISUALISING FUNCTIONS
275
On the MATH1052 webpage is a M-file called MATH1052quiver.m. This script
looks for a M-file function called quivfunc.m (which you must define yourself), and
produces a contour diagram for the function, as well as an arrow plot. The arrow
plot consists of many small arrows which point in the direction of the gradient at
their location (ie. up the slope of the function at that point). The quivfunc function
must accept inputs x and y.
For example, you could define quivfunc as
function output = quivfunc(x,y)
output = 2*x^2 - y^2;
Running MATH1052quiver will then produce a contour diagram and arrow plot:
>> MATH1052quiver
!#"$
1
0.8
0.6
0.4
"
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
You can alter the x and y axis ranges in the first few lines of the MATH1052quiver
M-file:
xmin
xmax
ymin
ymax
=
=
=
=
-1;
1;
-1;
1;
276
6. MATLAB GUIDE
6.4.3
Ezsurf and ezmesh
In addition to cross sections and contour diagrams, MATLAB can display multivariate functions using surface plots. There are two main MATLAB functions
for plotting surfaces: ezsurf and ezmesh. Their use is virtually the same as for
ezcontour: the function is specified in exactly the same way, as is the domain(s).
The ezsurf command plots a coloured surface, while ezmesh plots a coloured mesh.
In both cases the colour at any point represents the value of the function at that
point; a colourbar can be inserted to aid in viewing the plot.
For example, try using
>> ezsurf(’sin(x)*cos(y)’)
>> ezmesh(’sin(x)*cos(y),[-10 10 -10 10])
Note that Figure Windows have a button called Rotate 3D. Clicking this button
allows you to use the cursor to rotate a plot in three dimensions. Doing so may help
you visualise the function more clearly.
6.4.4
Functions of three variables
Since functions of three variables require four dimensions in order to be plotted
properly, they are often hard to visualise effectively. One helpful method essentially
uses the four-dimensional analog of cross sections. To visualise a function w =
f (x, y, z), we can make a number of surface plots where z (or x or y) is held constant.
x−y
For example, consider the function w = f (x, y, z) = x2 +y
2 +z 2 sin(x − z) for −5 ≤
x ≤ 5, −5 ≤ y ≤ 5 and 0 ≤ z ≤ 2. We can hold z constant at a number of values
between 0 and 2, and produce surface plots for each resulting two-variable function:
f (x, y, 0), f (x, y, 0.5), f (x, y, 1), f (x, y, 1.5) and f (x, y, 2). Obviously we don’t want
all these plots to be on the same axis, so use the figure command before each plot
to create a new Figure Window.
>>
>>
>>
>>
figure
ezsurf(’(x-y)/(x^2+y^2+0^2)*sin(x-0)’)
figure
ezsurf(’(x-y)/(x^2+y^2+0.5^2)*sin(x-0.5)’)
and so on.
6.5
Visualising Data
The commands described in the previous section are useful for functions which can
be easily described using a number of variables. However, there will be times when
6.5. VISUALISING DATA
277
you wish to visualise data which cannot be described using a simple function, or for
which such a function is unknown. MATLAB has several commands for plotting
and viewing such data.
6.5.1
Plot and plot3
MATLAB’s plot command is used for plotting two-dimensional data points. To plot
the points (x1 , y1 ), . . . , (xn , yn ), use the command plot(x,y) where x = (x1 , . . . , xn )
and y = (y1 , . . . , yn ).
For example, to plot the points (1, 2), (4, 2), (3, 1), (7, 4), (2, −3), use
>> x = [1,4,3,7,2];
>> y = [2,2,1,4,-3];
>> plot(x,y)
The plot command can take an optional third argument, consisting of a string
of letters or symbols which tells MATLAB how to plot the data. For example,
plot(x,y,’ro’) will plot the points as red circles, and plot(x,y,’g--’) will plot
the points connected by a green dotted line. See MATLAB’s help files for the full
list of plotting options.
Three-dimensional data can also be plotted in MATLAB: simply use plot3 instead
of plot, and use vectors x, y and z. The same optional arguments are available. For
example, to plot the points (1, 2, 3), (2, 4, 2), (4, 3, −1), (3, 2, 5) as black stars, use the
commands
>>
>>
>>
>>
x = [1,2,4,3];
y = [2,4,3,2];
z = [3,2,-1,5];
plot3(x,y,z,’k*’)
6.5.2
Contour, surf and mesh
Just as plot is the point-wise analog of ezplot, the other plotting commands described in the previous section have point-wise versions. They are contour, surf
and mesh. These work slightly differently to plot3; instead of taking three vectors
as arguments, they take two vectors x, y and a matrix Z. These arguments describe
a table of values for the data: x and y respectively give labels for the columns and
rows of Z, producing a set of (x, y, z) points. For example, consider the following
table:
278
6. MATLAB GUIDE
-1
0
1
2
-1
0
1
2
-2
-1
0
1
-1
0
1
2
0
1
2
3
1
2
3
4
Here, x and y are both [−1, 0, 1, 2], and Z is the 4 × 4 matrix inside the table. Using surf(x,y,Z) would instruct MATLAB to create a surface using all
the (x, y, z) points, where x and y come from the column and row labels respectively, and z comes from the values inside the table. So the points plotted would
be (−1, −1, −2), (0, −1, −1), (1, −1, 0), . . . , (2, 2, 4). If x is of length n and y is of
length m, then Z must be a m × n matrix.
So to produce a surface plot of the above table, one would use the commands
>>
>>
>>
>>
x = [-1, 0, 1, 2];
y = x;
Z = [-2, -1, 0, 1; -1, 0, 1, 2; 0, 1, 2, 3; 1, 2, 3, 4];
surf(x,y,Z)
As with ezsurf, a colourbar can be inserted to assist in viewing the plot.
The surf command can in fact be used with only the argument Z; in this case,
MATLAB assumes that x and y are the vectors [1, 2, . . . , n] and [1, 2, . . . , m], where
Z is a m × n matrix.
The mesh and contour commands function in exactly the same way as surf. As
one would expect, mesh produces a mesh plot of the data, and contour produces a
contour diagram. A coloured-in version of contour, called contourf, also exists.
6.6
Animations
The final method of presenting multivariate data is with animations. This essentially involves plotting a sequence of cross sections (or the three-dimensional analog
thereof), and using each plot as a single frame. Animations are particularly useful
if one of the independent variables is time.
Basically the method for producing an animation is:
• Write a loop which will vary t (or another independent variable) over some
range;
• For each value of t, produce a plot;
6.6. ANIMATIONS
279
• Use the command M(j) = getframe at each plot to store each image into the
‘vector’ M ;
• Use the command movie(M,n) to play the movie n times.
For example, the following will produce an animation of the function f (x, t) =
cos(x) cos(t) for 0 ≤ x ≤ 6.3 and 0 ≤ t ≤ 20. We cannot use ezplot here because
MATLAB will be unable to correctly plot the function (it has two variables); we
will instead use plot, with the points close enough together as to make the curve
sufficiently smooth. The outermost loop (with index j) will cycle through the t
values, and the inner loop (index k) will cycle through x values.
% MATLAB program to produce animation for cos(x)cos(t)
for j=1:101
t=(j-1)/5;
for k=1:64
x(k)=(k-1)/10;
z(k)=cos(x(k))*cos(t);
end
plot(x,z);
axis([0,6.3,-2,2]);
M(j)=getframe;
end
movie(M,5)
Note that we could not start the j and k loops at 0, because both of those indices
were used to index vectors: j was used to index M and k was used to index x and z.
MATLAB always starts matrix indices at 1, not 0; so we had to start those indices
at 1 and then compensate by setting t = (j − 1)/5 and x(k) = (k − 1)/10.
The process is much the same with three-dimensional ‘cross sections’. The only
major difference is that there will now be three nested for loops, instead of two:
one each to cycle through t, x and y.
For example, the following will produce an animation for f (x, y, t) =
cos(x) cos(y) cos(t), with 0 ≤ x ≤ 6.3, 0 ≤ y ≤ 6.3 and 0 ≤ t ≤ 20.
% MATLAB program to produce animation for cos(x)cos(y)cos(t)
for j=1:101
t=(j-1)/5;
for k=1:64
x(k)=(k-1)/10;
for m=1:64
280
6. MATLAB GUIDE
y(m)=(m-1)/10;
Z(m,k)=cos(x(k))*cos(y(m))*cos(t);
end
end
surf(x,y,Z);
axis([0,6.3,0,6.3,-1,1]);
M(j)=getframe;
end
movie(M,5)
6.7
6.7.1
Optimisation
Using fminsearch
An important objective of much mathematical analysis is to locate global maxima
and minima of multivariate functions. MATLAB contains a powerful inbuilt command called fminsearch, which is very effective at finding maxima and minima.
The application of fminsearch is as follows:
[xval,fval] = fminsearch(’functionname’,x0)
where:
• functionname is the name of the M-file function you wish to minimise; it must
take a single vector as input.
• x0 is the initial guess for the location of the minimum; it must be a vector of
the same dimensions as the input of functionname.
• xval is the the location of the minimum.
• fval is the value of the function at the minimum.
For example, consider the function f (x, y) = (x − 1)2 + y 2 + 3. We first need to
write the M-file function:
function output = myfun(input)
% input is a vector: we will separate its elements
% for simplicity
x = input(1);
y = input(2);
output = (x-1)^2 + y^2 + 3;
6.7. OPTIMISATION
281
Then if our initial guess for the (x, y) coordinate of the minimum is (1, 2), we would
use the command
>> [xval,fval] = fminsearch(’myfun’,[1,2])
xval =
1.0000
0.0000
fval =
3.0000
So the (x, y) coordinate of the minimum is (1, 0) and the value of the function at
that point is 3.
Note that you may have to try a number of different initial guesses before you find
the global minimum.
To maximise a function, you need to find the minimum of the negative of the function.
6.7.2
Lines of best fit
When given a set of multi-dimensional data, we often wish to fit it to some kind of
function. This can help us understand the relationships between different properties
and make predictions for future data. A line of best fit is the simplest way to do
so. While MATLAB has its own built-in functions for doing this (for example,
look up polyfit), we will demonstrate here how to use fminsearch to fit a line to
two-dimensional data.
Suppose we have a collection of two-dimensional points (x1 , y1 ), (x2 , y2 ), . . . , (xn , yn )
and we wish to find a line of best fit of the form f (x) = y = mx + c (where x and y
are the independent and dependent variables respectively). The most accurate such
line is the one which minimises the errors between the actual data points and the
corresponding points on the line; that is, it must minimise the errors f (xi ) − yi =
mxi + c − yi . This does not quite make sense: minimising the errors could mean
letting them all go to −∞. We actually wish to minimise the absolute value of the
errors. This can be done, but it turns out to be easier to minimise the squares of
the errors, which are also always positive.
This is not quite specific enough – we may have many data points, so how can we
effectively minimise the square of each error? In fact, we must minimise the sum of
the squares of the errors:
n
X
i=1
(mxi + c − yi )2
282
6. MATLAB GUIDE
From here, the rest is easy. We need to write a function which takes as input m and
c (in a single vector), and outputs the above sum. This function can then be fed
into fminsearch, which will find the appropriate values for m and c which minimise
this sum of squares.
For example, say our experimental data consists of the (x, y) points
(20, 3), (17, 5), (21, 2), (19, 4), (16, 5), (23, 2)
Then our function may look like the following:
function output = best fit(input)
% Finds the m and c values for a line of best fit
m = input(1);
c = input(2);
x = [20 17 21 19 16 23];
y = [3 5 2 4 5 2];
temp = 0;
for i = 1:length(x)
temp = temp + (m*x(i) + c - y(i))^2;
end
output = temp;
and to find the m and c values for the line of best fit, we apply fminsearch in the
usual way.
>> [coeff,fval] = fminsearch(’best fit’,[0,1])
coeffs =
-0.5100
13.3600
fval =
0.83
So the line of best fit for the above example is f (x) = y = −0.51x + 13.36. Plotting
the data points and this line on the same axis demonstrates that this is indeed a
very good line of best fit.
This method works just as well for higher-dimensional examples. For example, a
plane of best fit would have the equation f (x, y) = z = mx + qy + c, where x and y
are the independent variables and z is the dependent variable. The M-file function
would then take as input m, q and c, and output the sum of the squares of the errors:
n
X
i=1
(mxi + qyi + c − zi )2
6.8. DIFFERENTIAL EQUATIONS
6.8
6.8.1
283
Differential Equations
Slope fields
A first-order differential equation dy/dx = f (x, y) determines the gradient of any
solution curve y(x) at any (x, y) point. A slope field is essentially a visual representation of this differential equation: it uses small lines to indicate the gradient at
(x, y) points on a cartesian plane. Given an initial condition, you can use a slope
field to visually ‘solve’ a differential equation, by following the direction of the field
from the given point.
The M-files field.m and func.m can be downloaded from the MATH1052 website.
The field.m file is a MATLAB function that produces a slope field for dy/dx =
f (x, y), and func.m is used to describe the function f (x, y).
For example, to produce a slope field for the differential equation dy/dx = y − 2x2 ,
change func.m to
function ydash = func(x,y)
ydash = y - 2*x^2;
and then run field.m with appropriate input arguments:
>> field(-1,1,20)
The first argument is the lower bound for the horizontal and vertical axes, the second
argument is the upper bound, and the third argument is the number of intervals
between the field lines. Note that this function will always produce a plot on a
square. You should get the following output.
!#"
1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
284
6. MATLAB GUIDE
A possible solution to the differential equation is shown below.
"!#%$'&()
1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Taking the initial condition y(0) = 0 and following the field lines to the left and to
the right yields a curve like the one shown.
6.8. DIFFERENTIAL EQUATIONS
6.8.2
285
Euler’s method
Euler’s method is a numerical method for solving differential equations. The algorithm essentially just uses the slope field ‘solution’ described above:
• Start at a point (x0 , y0 ), described by an initial condition;
• Determine the value of the gradient f (x0 , y0 ) at that point;
• Follow in the direction of the gradient for a short distance, to the point (x1 , y1 );
• Determine the gradient f (x1 , y1 ) at this new point;
• Follow in the direction of the gradient for a short distance, to the point (x2 , y2 );
and so on.
This algorithm is very easy to code in MATLAB. For the differential equation
dy/dx = f (x, y) with initial condition y(x0 ) = y0 , you need to define the function f (x, y) (as we did earlier using func.m), and then write a for loop which
performs the calculation
yi+1 = yi + f (xi , yi )∆x
where ∆x is a small (constant) value which will be the distance along the x-axis
between successive points. The yi points will need to be saved in a vector, as you
will need to store them all for the final solution. The xi points should also be saved
in a vector; you can do this before starting the loop with the linspace command,
or with the ‘:’ (colon) vector notation.
You should produce two sequences of points: the xi points, which will all be a
constant distance ∆x apart, and the yi points, which were calculated in each iteration
of the for loop. Plotting the y vector against the x vector will illustrate your
numerical solution.
You should experiment with changing ∆x and/or the number of points used. Euler’s
method tends to become inaccurate after several thousand steps; you should remember that this is really only an approximate solution to the differential equation. If
possible, find the analytic solution and plot it on the same axis to determine if and
where the numerical solution becomes inaccurate.
6.8.3
Runge–Kutta method
Euler’s method is referred to as a first-order approximation for a differential equation, because the approximation essentially uses a first-order Taylor expansion.
286
6. MATLAB GUIDE
Higher-order methods can yield far more accurate solutions. One such is the Runge–
Kutta method, which is technically halfway between a fourth and fifth order method.
Consequently, the inbuilt MATLAB program for implementing the method is called
ODE45.
Consider a differential equation problem of the form dy/dt = f (t, y), with f (t0 ) = y0
for t0 ≤ t ≤ T . The MATLAB syntax for solving this problem using ODE45 is
ode45(’func’,[t0,T],y0)
where func evaluates f (t, y).
For example, consider the differential equation described previously (with x changed
to t): dy/dt = y − 2t2 , y(0) = 0. We will restrict the time interval to [0, 5]. We can
use the same func.m (you can change x to t if you wish). We now only need to
execute the ODE45 command:
>> ode45(’func’,[0,5],0)
!#"$&%(')+*
0
−100
−200
−300
−400
−500
−600
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Alternatively, we could use the command
>> [t out,y out] = ode45(’func’,[0,5],0);
In this second case, t out will be a vector consisting of the t coordinates of the points
used to calculate and plot the function, and y out will be the y coordinates. The
plot will not be automatically produced. Using the command
>> plot(t out,y out)
will produce similar output to the plot above.
6.8. DIFFERENTIAL EQUATIONS
6.8.4
287
Systems of differential equations
You have so far seen how to use MATLAB to numerically solve single first-order
ODEs. MATLAB’s inbuilt ODE solvers can also solve higher-order differential equations, like
d2 y
dy
(0) = 0
+ ey = 0,
y(0) = 1,
2
dt
dt
and coupled systems of ODEs, like
dx
= f (x, y) = = 2x − 3y,
dt
du
= g(x, y) = = x + 2y,
dt
x(0) = 2
y(0) = 0.
We will start with the system of equations above. We can use ODE45 in more or less
the same way, with only a couple of differences:
• ODE45 stipulates that func must take two input variables: t and another.
However, we have three here: t, x and y. We must write func so that it takes
inputs (t, u), where u = [x, y].
• The function func will have two outputs: f (x, y) and g(x, y). ODE45 requires
that these outputs come in the form of a column vector.
We define our new function as follows:
function output = system func(t,u)
x = u(1);
y = u(2);
f = 2*x - 3*y;
g = x + 2*y;
output = [f;g];
We can now use ODE45 in the same way as before (letting the time interval be [0, 2]):
>> [t out,u out] = ode45(’system func’,[0,2],[2,0]);
The vector t out has the same meaning as before. The first column of u out corresponds to the x values, and the second column corresponds to the y values. Plotting
t out against each of these columns will show how x and y change with time. Plotting the columns of u out against each other will plot x against y (this is called a
phase plot).
288
6. MATLAB GUIDE
>> plot(t out,u out(:,1))
>> plot(t out,u out(:,2))
>> plot(u out(:,1),u out(:,2))
Higher order ODEs can be solved with ODE45 by first converting them into systems
of ODEs. Consider the second-order ODE mentioned previously:
y 00 + ey = 0,
y(0) = 1, y 0 (0) = 0
We introduce a ‘new’ variable x = y 0 (so x0 = y 00 ). This leads to the system of
(first-order) ODEs:
dx
= f (x, y) = −ey
dt
dy
= g(x, y) = x
dt
x(0) = 0
y(0) = 1.
We can now write a function to evaluate these:
function output = secondorder func(t,u)
x = u(1);
y = u(2);
f = -exp(y);
g = x;
output = [f;g];
The command to execute ODE45 is essentially the same as before (note that our
initial value is a vector, because we have two variables):
>> [t out,u out] = ode45(’secondorder func’,[0,5],[0,1]);
Since x was an extra variable inserted to assist with solving, we only actually care
about y, ie. the second column of u out. So to plot y against t, use
>> plot(t out,u out(:,2))
6.9
Parametric Functions
One way of presenting a relationship between variables x and y is to write y as a
function of x, ie. y = f (x). You have seen that MATLAB is well equipped to plot
functions in this form.
Not all relationships between variables can be expressed directly using functions,
however. For example, the unit circle on the cartesian plane cannot be described
6.9. PARAMETRIC FUNCTIONS
289
with a simple function y = f (x). It can be plotted in MATLAB by plotting multiple
functions with hold on:
>>
>>
>>
>>
figure
hold on
ezplot(’sqrt(1 - x^2)’,[-1,1])
ezplot(’-sqrt(1 - x^2)’,[-1,1])
However this is needlessly complicated, and may not be possible for more complex
relationships. Parametric functions are a much easier way to describe relationships
such as these, and MATLAB can plot them just as easily.
Parametric functions relate the variables (eg. x and y) through an additional variable, t, called the parameter. For example, the unit circle can be described with the
parametric functions x(t) = cos(t) and y(t) = sin(t). In this particular case, t can
be interpreted as the angle around the circle going anticlockwise from the x axis.
The syntax for plotting a two-dimensional parametric relationship is
ezplot(’xfunc’,’yfunc’,[a,b])
where xfunc and yfunc are the parametric representations of x and y respectively,
and the parameter t varies over the interval [a, b]. So to plot the unit circle, we can
use the command
>> ezplot(’cos(t)’,’sin(t)’,[0,2*pi])
The default interval for t is [0, 2π].
MATLAB has a very similar inbuilt function for plotting three-dimensional parametric relationships, called ezplot3. Unsurprisingly, its syntax is
ezplot3(’xfunc’,’yfunc’,’zfunc’,[a,b])
For example, to plot a helix (a spring-shaped object) of radius 1 and height 20, you
could use the command
>> ezplot3(’cos(t)’,’sin(t)’,’t/4’,[0,80])
The ezplot3 command has an additional option for showing the progression of the
(x, y, z) coordinate as t changes. Use the syntax
ezplot3(’xfunc’,’yfunc’,’zfunc’,[a,b],’animate’)
When using ezplot3, remember that you can rotate the figure, as well as change
the axis ranges with the axis command. Also consider using the commands axis
square and/or axis equal.
290
6. MATLAB GUIDE
6.10
Common commands and operations
6.10.1
Common numerical operations and constants
The following table lists the MATLAB commands for a number of commonly used
functions which operate on numbers, as well as some constants. Use MATLAB’s
help command for more information.
Operation
x+y
x−y
xy
x/y
xn
√
x
log(x)
log10 (x)
x × 10n
MATLAB
Code
x+y
x-y
x*y
x/y
x^n
sqrt(x)
log(x)
log10(x)
xen
Operation
ex
|x|
sin(x)
cos(x)
tan(x)
arcsin(x)
arccos(x)
arctan(x)
n!
MATLAB
Code
exp(x)
abs(x)
sin(x)
cos(x)
tan(x)
asin(x)
acos(x)
atan(x)
factorial(n)
Constant
π
√
−1
e
∞
−∞
MATLAB
Code
pi
i
exp(1)
Inf
-Inf
6.10. COMMON COMMANDS AND OPERATIONS
6.10.2
291
Vector and matrix operations
The following table lists the MATLAB commands for a number of commonly used
operations which can be performed on vectors and/or matrices.
Operation
Entering a row vector: a = 1 3 4
 
4
Entering a column vector: b = 0
7
2 4
Entering a matrix: A =
0 1
Second component of vector b:
Entry (1, 2) of a matrix A:
Transpose: c = aT
Adding vectors or matrices: b + c
Dot product: b · c
Cross product: b × c
Norm: kak
Multiplying matrices: AB
Matrix determinant: det(A)
Inverse: A−1
Vector dimension:
Matrix dimensions:
Minimum value in vector b:
Maximum value in vector b:
6.10.3
MATLAB Code
a=[1,3,4]
b=[4;0;7]
A=[2,4;0,1]
b(2)
A(1,2)
c=a’
b+c
dot(b,c)
cross(b,c)
norm(a)
A*B
det(A)
inv(A)
length(a)
size(A)
min(b)
max(b)
Special matrices
MATLAB has a number of special matrices built-in, and these can be produced with
certain commands, given in the table below.
Matrix
n × n identity matrix
m × n matrix of ones
m × n matrix of zeros
MATLAB Code
eye(n)
ones(m,n)
zeros(m,n)
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