Math Self-Assessment

Math Self-Assessment
PHY 3221
Mathematics Self-Assessment
Steven Detweiler
Fall 2012
modified by Yoonseok Lee
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
PHY3321 is PHY2048, PHY2049, and the math requirements include MAC 2311, 2312 and
2313 (Vector Calculus). These math courses together cover derivatives and integrals of trig
and log functions, series and sequences, analytic geometry, vectors and partial derivatives
and multiple integrals. We will casually be using math from all of these subjects. None of
these should be completely unfamiliar to you.
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 C on calculus. 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 that your answer is correct then you are not comfortable
with mathematics at this level. Sections E, F, G, H and I are more advanced than is
necessary as a prerequisite for this course. But, these ideas will often be discussed in class.
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 while
working out algebraic details with paper and pencil in hand. 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
Vector Algebra
~ = 1ı̂ + 2̂ + 3k̂ and B
~ = 4ı̂ + 5̂ + 6k̂.
Q4. Let A
~
What is |A|?
~ · B?
~
What is A
1
(a2 − 9b2 )
~ × 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
~ ·C
~ = 10 and the angle between A
~ and C
~ is 30◦ , then what is the magnitude of C?
~
Q5. 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. Evaluate the following integrals
Z
π
sin θ dθ,
0
k
dx
x2
Z
where k is a constant, and
Z
1
x
k
dx.
x
2
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.
Q11. Find two different functions which satisfy the differential equation
d2 f (x)
− λ2 f (x) = 0.
2
dx
Q12. Find two different functions which satisfy the differential equation
d2 f (x)
+ ω 2 f (x) = 0.
dx2
E.
Trigonometry and Geometry
Euler’s identity,
eiθ = cos θ + i sin θ,
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.
Q13. 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.
3
FIG. 1: See Q15.
At this point, it is appropriate to introduce hyperbolic functions.
e±x = cosh x ± sinh x.
Q14. What is the first derivative of tanh x?
Q15. 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
Q16. Evaluate the sum
S(x) =
∞
X
xn
∞
X
xn = 1 +
n=0
for |x| < 1 .
Ans: Note that
S(x) =
n=0
∞
X
xn
n=1
∞
X
= 1+x
xn
n=0
= 1 + xS(x)
So we have
and, finally
S = 1 + xS
(1 − x)S = 1
1
.
S(x) =
1−x
4
Q17. How about the following summation?
SN (x) =
N
X
xn .
n=0
G.
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 )
1
1
d2 f
dn f
df
+ · · · + (x − x0 )n n
+ · · · (1)
+ (x − x0 )2 2
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
dx3 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!
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 .
5
Q18. 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
gt 1
c2
z = {[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 graviational acceleration and c is the speed of light.
Hint: You should realize gt/c << 1 and behave as a small parameter as ǫ 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 + ǫ + O(ǫ2 ),
1−ǫ
(1 + ǫ)n = 1 + nǫ + O(ǫ2 ),
√
1
1 − ǫ = 1 − ǫ + O(ǫ2 ),
2
1
1
√
= 1 + ǫ + O(ǫ2 ),
2
1−ǫ
1
1
1
eǫ = 1 + ǫ + ǫ2 + ǫ3 + ǫ4 + O(ǫ5 ),
2
6
24
1
1
1
eiǫ = 1 + iǫ − ǫ2 − i ǫ3 + ǫ4 + O(ǫ5 ),
2
6
24
1
1
ln(1 + ǫ) = ǫ − ǫ2 + ǫ3 + O(ǫ4 ),
2
3
1 2 1 3
ln(1 − ǫ) = −ǫ − ǫ − ǫ + O(ǫ4 ).
2
3
The O(ǫn ) term here is standard mathematical notation to mean a function which is less
than some constant times ǫn in the limit that ǫ → 0. In other words for small ǫ, O(ǫn ) is
no bigger than something times ǫn .
We can use the Euler identity eiǫ = cos ǫ + i sin ǫ to easily pick off the purely real terms
from this last expansion which give the expansion of cos ǫ for a small angle ǫ, and the purely
imaginary terms, which give the expansion of sin ǫ for small ǫ:
and
1
1
cos ǫ = 1 − ǫ2 + ǫ4 + O(ǫ6 )
2
24
1 3
sin ǫ = ǫ − ǫ + O(ǫ5 ).
6
6
√
Q19. Expand 1 − 2tz + t2 in powers of t assuming that it is small. What is the coefficient
of the linear term, t in this
√ expansion?
Ans: You should reach 1 − 2tz + t2 ≈ 1 + zt + 21 (3z 2 − 1)t2 .
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
2
2
(minimum) if ddxf2 x=x0 > 0 or bending downward (maximum) if ddxf2 x=x0 < 0. An analytic
function can be approximated by a quadratic function near its minimum or maximum!
Q20. 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
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,
7
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
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
8
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
1 t/t1/2
2
.
Now divide out the N0 , and take the natural logarithm of both sides
1
t
t
ln
. (We are using ln(AB ) = B ln(A) and ln e = 1)
− =
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.
9
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