Essential Math

Essential Math
L01 Essential Mathematics for PHY3063
Steven Detweiler and Yoonseok Lee
November 17, 2015
The lack of mathematical sophistication is a leading cause of difficulty for students in
Classical Mechanics and other upper level physics courses.
An official pre-requisite of
PHY3063 is PHY2048, PHY2049, and the math requirements include MAC 2311, 2312 and
2313 (Vector Calculus), and MAP2302. These math courses together cover derivatives and
integrals of trig and log functions, series and sequences, analytic geometry, vectors and
partial derivatives, multiple integrals, and differential equations. We will casually be using
math from all of these subjects. None of these should be completely unfamiliar to you.
Fluency in these math skills is a necessary but not sufficient condition for your
success in all upper level physics courses.
The following discussions and questions are grouped by subject and in approximate order
of difficulty—easiest first. These are representative of the level of mathematics which is
expected in this course. You should be very comfortable and fluent with mathematics at
this level, at least through section F. Section D, on differential equations, is probably
more difficult for you but important. The answers to the questions are not always given. If
you do not know if your answer is correct, then you are not comfortable with mathematics
at this level. The importance of Section G cannot be overemphasized. You will use
Taylor expansion over and over again as far as you are dealing with physics. If
you understand the Taylor series in section G, then you are likely to find section H, on
calculators, interesting and amusing. Don’t be surprised if my discussions seem confusing
at first: To understand math and physics often requires multiple, multiple readings and
drills while working out algebraic details with paper and pencil in hand. Never ever try
to go through math or physics problems with your eyes. You may think you understand
the problem or the subject. But when a similar problem is given in an exam, you will
feel that you have seen this before but cannot solve it. This is why I often hear from
many students ”I studied very hard (with my eyes) but I do not perform well
in exams.” Try to build a habit to write your equations or perform calculations in order
as you write a lengthy program. You make less mistakes and can debug your mistakes
1
easier! Finally, Section I involves an ordinary differential equation that has an interesting
application to radioactivity.
A.
Algebra
Q1. Solve for x:
f (x) = ax2 + bx + c = 0.
For what value of x is f (x) a maximum or a minimum?
Q2. Make a sketch of the function y(x) where y = mx + b and where m and b are constants.
What are the meanings of the constants m and b in terms of your sketch?
Q3. Factorize
(a2 + 4ab + 4b2 )
B.
and
(a2 − 9b2 )
Vector Algebra
~ = 1ı̂ + 2̂ + 3k̂ and B
~ = 4ı̂ + 5̂ + 6k̂.
Q4. Let A
~
What is |A|?
~ · B?
~
What is A
~ × B?
~
What is A
~ and B?
~
What is the cosine of the angle between A
Do you know
ı̂ ̂ k̂ ~×B
~ = 1 2 3 ?
A
4 5 6
2
~·C
~ = 10 and the angle between A
~ and C
~ is 30◦ , then what is the magnitude of C
~
Q5. If A
~ = |C|?
~
if |A|
C.
Calculus
Q6. If x0 , v0 and a are constants and
1
x(t) = x0 + v0 t + at2
2
then what is dx/dt? What is d2 x/dt2 ? If a < 0, does the function x(t) curve up or down?
If x is negative when t = 0 and x is positive when t is very large: then for precisely which
values of t is x positive?
Q7. Evaluate the derivative
d
A cos(ωt + φ)
dt
where A, ω and φ are constant.
Q8. If f (x) =
x
,
cos x
what is
df (x)
?
dx
Q9. If f (x) = tan(ax2 + b), what is
df (x)
?
dx
Q10. Plot y = 13 x3 − 2x2 + 3x + 2 without using a graphing calculator. By just looking at
the functional form can you tell how many extrma at most exist in this curve?
Q11. What is the value of x where the following function peaks (a is a positive constant)?
f (x) = p
1
(a2
− x2 ) 2 + 4
Can you plot the above function?
Q12. Evaluate the following integrals
Z
π
sin θ dθ,
0
3
.
Z
where k is a constant, and
k
dx
x2
x
Z
1
k
dx.
x
Is sin x an odd or even function? How about cos x, sin2 x, x + 5x5 ?
If function f (x) is even, then f (x) = f (−x). If f (x) is odd, then f (x) = −f (−x). Do you
see then why
Z
+a
f (x)dx = 0
−a
for any odd function f (x)?
D.
Differential equations
These next two problems might be difficult or possibly unfamiliar to you, but take a careful
look at them because these are very important in classical mechanics.
Q13. Find two different functions which satisfy the differential equation
d2 f (x)
− λ2 f (x) = 0.
2
dx
Q14. Find two different functions which satisfy the differential equation
d2 f (x)
+ ω 2 f (x) = 0.
2
dx
E.
Trigonometry and Geometry
Euler’s identity,
eiθ = cos θ + i sin θ,
4
is probably new to you. But it provides a convenient and easy way to derive some of the
basic trig identities such as
ei(α+β) = eiα eiβ
cos(α + β) + i sin(α + β) = (cos α + i sin α) × (cos β + i sin β)
or, after multiplying out the right hand side,
cos(α + β) + i sin(α + β) = cos α cos β − sin α sin β + i(cos α sin β + sin α cos β)
The real and the imaginary parts of this equation give the well known trig identities:
cos(α ± β) = cos α cos β ∓ sin α sin β
and
sin(α ± β) = sin α cos β ± cos α sin β.
or, after multiplying out the right hand side,
cos(α + β) + i sin(α + β) = cos α cos β − sin α sin β + i(cos α sin β + sin α cos β)
At this point let’s check if you are comfortable with complex numbers.
Any complex number z can be represented in one of the following ways:
z = a + ib (a, b ∈ R)
b
(arg(z) = θ = arctan ) (Polar Form)
a
iδ i(θ−δ)
iθ0
= |z|e e
= Ce
(C ∈ C).
= |z|eiθ
And,
0
z ∗ = a − ib = |z|e−iθ = C ∗ e−iθ ,
√
|z| = zz ∗ .
What is
√
i? Can you evaluate this? It seems an odd mathematical expression but if
you use the Euler’s identity, you can easily evaluate this.
√
i=
√
ei(2n+π/2) = (ei(2n+π/2) )1/2 = ±(cos π/4 + i sin π/4).
5
√ √
√
Q15. Use the polar form to show i(1 − i 3)( 3 + i) = 2(1 + i 3).
Q16. A real function x(θ) is given by
x(θ) = C1 eiθ + C2 e−iθ
(C1 , C2 ∈ C,
= B1 cos θ + B2 sin θ
and θ ∈ R)
(B1 , B2 ∈ R).
Show that (i) C2 = C1∗ and (ii) B1 = C1 + C2 , B2 = i(C1 − C2 ).
Q17. Use the Euler identity to show that
sin2 θ + cos2 θ = 1.
Hint: start with eiα e−iα = 1 and then use the Euler Identity.
At this point, it is appropriate to introduce hyperbolic functions.
e±x = cosh x ± sinh x.
Q18. What is the first derivative of tanh x?
Q19. See the figure above. AB = BC = 2, DE = 1, and ∠(DF E) = π/2.
What is ∠(F AD)?
What is AF ?
What is DF ?
What is cos θ (θ = ∠(EDF ))?
Let’s call the crossing point of AD and BE G. What is DG?
F.
Sums
Q20. Evaluate the sum (infinite geometric series)
S(x) =
∞
X
xn
n=0
6
for |x| < 1 .
FIG. 1:
Ans: Note that
S(x) =
∞
X
n
x = 1+
n=0
∞
X
xn
n=1
∞
X
= 1+x
xn
n=0
= 1 + xS(x)
So we have
S = 1 + xS
(1 − x)S = 1
S(x) =
and, finally
1
1−x
(x 6= 1).
By simply substituting x = −a, you have
∞
X
1
S(a) =
= 1 − a + a2 − a3 + · · · =
(−1)n an .
1+a
n=0
Here, if a = 0.1, S(0.1) = 0.909090... Now calculate the sum value using the right hand
side expression by adding only the first 2, 3, and 4 terms. You get 0.9, 0.91, and 0.909,
respectively. These answers are within 1%, 0.1%, and 0.01% of the exact results. In physics,
I can take the answer within 1% in general. The smaller a is, the more accurate your answer
obtained using a truncated sum. See Taylor expansion below.
7
Q21. How about the following summation?
SN (x) =
N
X
xn .
n=0
The summation formula also works for complex numbers:
SN (z) =
N
X
zn
(z 6= 1).
n=0
Using this, you can derive Lagrange’s trigonometric identity.
1 + cos θ + cos 2θ + cos 3θ + · · · + cos nθ =
G.
1 sin [(n + 12 )θ]
+
2
2 sin θ/2
(0 < θ < 2π).
Taylor expansions of a function
Any differentiable function f (x) may be approximated in the neighborhood of a point x0 by
the Taylor expansion
f (x) = f (x0 ) + (x − x0 )
df
1
d2 f
1
dn f
+ (x − x0 )2 2
+ · · · + (x − x0 )n n
+ · · · (1)
dx x=x0 2
dx x=x0
n!
dx x=x0
For example, consider f (x) = 1/(1 − x), expanded about x0 = 0. Then
f (x) = 1/(1 − x),
df
= [1/(1 − x)2 ]x=0 = 1
dx x=0
d2 f
= 2[1/(1 − x)3 ]x=0 = 2
dx2 x=0
d3 f
= 6[1/(1 − x)4 ]x=0 = 6
3
dx x=0
dn f
= n![1/(1 − x)n+1 ]x=0 = n!
dxn x=0
The Taylor expansion for 1/(1 − x) with x0 = 0 is now
1
1
1
1
= 1 + x + x2 × 2 + x3 × 6 + · · · + xn × n! + · · ·
1−x
2
6
n!
8
And this is easily seen to be
∞
X
1
=
xn ,
1 − x n=0
the same as our example for doing sums above!
Taylor expansions of this sort are extremely useful in physics. You will have to use Taylor
expansion over and over again in physics. Trust me! For example in special relativity
when we are interested to see how close special relativity is to Newtonian physics for small
speeds v, we usually make the assumption that v/c 1. Then we make Taylor expansions
of the relevant formulae, and include only the terms proportional to v/c or maybe also v 2 /c2 .
Q22. Try this! You do not have to understand physics here. Just follow the mathematical
procedure. The displacement z of a particle of rest mass mo , resulting from a constant force
mo g along the z-axis is
z=
gt 1
c2
{[1 + ( )2 ] 2 − 1},
g
c
including relativistic effect. Find the displacement z as a power series in time t. Compare
with the well-known classical result,
1
z = gt2 .
2
Here, g is the gravitational acceleration and c is the speed of light.
Hint: You should realize gt/c << 1 and behave as a small parameter as in the formulae
above. In the complete classical limit where the speed of light is considered infinite, you
will recover the classical result. You know that you cannot just put c = ∞ in the above
expression, which will give you meaningless z = 0.
Common Taylor expansions give approximations such as
1
= 1 + + O(2 ),
1−
(1 + )n = 1 + n + O(2 ),
9
√
1
1 − = 1 − + O(2 ),
2
1
1
√
= 1 + + O(2 ),
2
1−
1
1
1
e = 1 + + 2 + 3 + 4 + O(5 ),
2
6
24
1
1
1
ei = 1 + i − 2 − i 3 + 4 + O(5 ),
2
6
24
1
1
ln(1 + ) = − 2 + 3 + O(4 ),
2
3
1 2 1 3
ln(1 − ) = − − − + O(4 ).
2
3
The O(n ) term here is standard mathematical notation to mean a function which is less
than some constant times n in the limit that → 0. In other words for small , O(n ) is
no bigger than something times n .
We can use the Euler identity ei = cos + i sin to easily pick off the purely real terms
from this last expansion which give the expansion of cos for a small angle , and the purely
imaginary terms, which give the expansion of sin for small :
and
1
1
cos = 1 − 2 + 4 + O(6 )
2
24
1 3
sin = − + O(5 ).
6
Since cos is an even function, you can only have terms of a even power such as 0 = const,
2,4,.. . Similarly for sin only odd power terms exist.
Q23. Expand tan x =
sin x
cos x
in powers of x to O(x2 ).
Ans: You can do the expansion directly by taking the first and second derivative of tan x.
However, the following method is much easier when you have a fractional function in general.
First do Taylor expansion of the denominator and numerator to the desired order, separately:
tan x ≈
x
1− 12 x2
(The numerator has the next term in x3 . So no need to put.) Then do the
geometric expansion of the denominator: tan x ≈ x × (1 + 12 x2 ) ≈ x + O(x3 ).
Q24. Expand
cos x
1−x2
to O(x2 ).
10
Ans: The coefficient of the x2 term is − 32 .
Q25. The relativistic energy of a particle of mass m moving with a speed u is
mc2
E(u) = q
.
2
1 − uc2
In the non-relativistic limit, E(u) = mc2 + 21 mv 2 . Show this.
Let’s go back to Eq.(1). If a function f (x) has a extremum (maximum or minimum) at
x = xo , then
df
= 0.
dx x=x0
This means that around x = xo , the shape of the function is parabolic bending upward
(minimum) if
d2 f
dx2 x=x0
> 0 or bending downward (maximum) if
d2 f
dx2 x=x0
< 0. An analytic
function can be approximated by a quadratic function near its minimum or maximum!
Q26. Show that for x << 1, tanh x ≈ x, and tanh x → ±1 as x → ±∞. Then, sketch the
curve of y = tanh x on the graph.
H.
Calculators
When solving a physics problem, think with your brain not with your calculator! Before
touching your calculator, check to see that your algebraic answer has the correct units and
that it has the expected behavior for various limits. It is nearly impossible to check the
correctness of an answer once you touch your calculator. You might find it amusing that the
number 10100 has been given the name googol, and 10googol is called googolplex—and these
names were coined well before the internet was invented. But, note the difference in spelling
between googol and the name of the internet search engine. As an aside: The internet was
invented by physicists who wanted to exchange easily experimental data between the United
States and Europe.
Here are a couple examples which are relevant to one of the homework problems for this
course. Let f (n) = n2 , where n is an integer. First evaluate f (102 ) − f (102 − 1) on your
11
calculator. You should get 199. Now try to evaluate f (10100 ) − f (10100 − 1). Your calculator
will choke on this problem, but your brain can easily find the answer to 100 significant digits.
Note that
f (n) − f (n − 1) = n2 − (n − 1)2 = n2 − (n2 − 2n + 1) = 2n − 1.
With n = 10100 , it is easy to see that f (n) − f (n − 1) = 2 × 10100 − 1 ≈ 2 × 10100 with 100
significant digits.
Here is a second, more challenging, problem. Let f (n) = n−2 where n is an integer. Evaluate
f (10100 − 1) − f (10100 ). Your calculator will also choke on this problem, but again you can
easily find the answer to about 100 significant digits. Use the Taylor expansion
f (n + δn) = f (n) + δn
df
2
df
+ . . . ⇒ f (n + δn) − f (n) = δn
+ . . . = −δn 3 + . . .
dn
dn
n
With n = 10100 and δn = −1, we easily have f (10100 − 1) − f (10100 ) = 2 × 10−300 + . . .,
where the . . . represents terms which are comparable to 1/n4 = 10−400 or smaller. Thus, the
answer is correct for the first 100 digits.
For a final example which reveals the limitations of your calculator, evaluate
1−
√
1 − 3 × 10−30
The answer is not zero. Analytically, find an approximation to the answer. In this context,
the word “analytically” means that you should use algebra and calculus to find the answer.
And you shouldn’t touch a calculator or computer.
Hint: use a Taylor expansion.
I.
Radioactivity and a simple differential equation
The radioactive nucleus
14
C spontaneously decays into
14
N a β − and a ν̄e . That is to say,
carbon–14 decays into nitrogen–14, a beta particle (also known as an electron), and an antielectron-neutrino, which is generically described as just a neutrino. If you start with a glass
full of
14
C today, then in 5730 years you will only have half a glass of
14
C. After a total of
11460 years only a quarter of a glass will remain. And so forth. We say that the half-life
12
of
14
C is 5730 years. In general for any radioactive particle, if we start at t = 0 with N0
particles, then after a time t the number remaining is
N (t) = N0
1 t/t1/2
2
where t1/2 is the half-life.
Radioactive decay gives one example of a number N (t) whose rate-of-change in time dN (t)/dt
is proportional to the number N (t) itself. In other words,
dN (t)
∝ N (t).
dt
For definiteness assume that
dN (t)
= −λN (t),
dt
where λ is a constant. We solve this differential equation by rewriting it as
dN (t)
= −λ dt
N
and integrating both sides
Z
dN (t)
= −
N
Z
λ dt
ln N = −λt + constant
or
N (t) = No e−λt
where ln(N0 ) = constant is a constant of integration determined by the initial conditions.
The last line follows by taking the logarithm of both side of the previous equation. With
radioactivity, we often define the “e-folding time” τ ≡ 1/λ, which also happens to be the
“mean-lifetime” of the particle, so that
N (t) = N0 e−t/τ .
τ is called the e-folding time because the number of particles decreases by a factor of e after
a time τ . It is easy to see the relationship between t1/2 and τ by starting with
N0 e−t/τ = N0
13
1 t/t1/2
2
.
Now divide out the N0 , and take the natural logarithm of both sides
−
1
t
t
=
. (We are using ln(AB ) = B ln(A) and ln e = 1)
ln
τ
t1/2
2
Finally, cancel the t, invert each side of the equation, and use the fact that ln(1/2) = − ln 2.
The result is
t1/2 = τ ln 2.
Note that
τ (14 C) = 5370yr/ ln 2 = 7750 yr
is the e-folding time of
14
C. The mean-lifetime (e-folding time) of a muon is about 2µs. So,
the half-life of a muon is about t1/2 (muon) = ln 2 × 2 µs ≈ 1.4 µs.
14
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