MWF 1:40-2:35 ES 147
Instructor: Professor Edward Thomas ES 132F
Best way to reach me is by email: firstname.lastname@example.org
Office hours: MWF 10:25-10:45, 12:50-1:30, 2:45-3:15 and by appointment
A link to the class webpage will be found at : http://www.albany.edu/~et392/
Text: Mathematical Methods in the Physical Sciences by Mary Boas. You need to have this text on the first day of class, not at some indeterminate later date. If you don’t have the text, I will take this as a sign of LACK OF PREPARATION.
Same ground rules as usual:
> First...there will be NO CELL PHONES, LAPTOPS or any other type of electronic devices in use during class. Please take care of business and TURN THEM OFF before you enter the classroom.
The syllabus for this course was developed in conjunction with the Physics Department. The emphasis is on those aspects of Mathematics that are can be readily applied to problems in Physics and Engineering. This second semester, we will concentrate on some of the classical Partial Differential Equations of Physics and then take up Complex Analysis and its applications to Applied Mathematics.
There will be assignments every day. These will be corrected and returned to you at the next class meeting.
Assignments will count as one third of the final grade, the other two thirds coming from the mid-term and final exams.
1) Page 620 # 2 (a)
2) On Friday, as we went about solving the steady state heat equation, we came to a fork in the road. Suppose we opted to choose the separation constant to be positive : c = k2. Show that this would lead to the trivial solution. (As a matter of fact, show that when you figure out what X(x) and Y(y) are, the right and left boundary conditions imply that X(x) must vanish identically.)
3) a) Using the first 3 terms of the Fourier series developed in class, estimate T(5, y) for y = .1, .2 and .5 ( don’t forget the 400/Pi out in front)
b) For future reference, figure out the general form of cos( nPi/2) ( you don’t need a single formula here, just the three different values depending on what n is)
4) a) Continuing with the last homework problem, make better estimates by averaging the 3-term partial sums ( which you already computed) and the 4-term partial sums, as discussed in class.
b) Compute the Fourier sine coefficients…the bn’s…for the function f(x) = 96 when 0<x<L/2, f(x) = 0 when L/2<x<L.
5) a) A curious feature of Fourier series…continuing part (b) above, and using the series we got in class, estimate the value of T (L/2,0) by averaging the 7th and 8th partial sums.
(b) ( The wrong prong) Referring to the heat diffusion equation, what goes wrong if you take a positive separation constant c = k2?Why won’t your solution make physical sense?
(c) Expand f(x) = ( 100/L)x in a Fourier sine series on the interval 0<x<L, as set up in class.
6)Turning to the 1-dimensional Schrodinger equation with V=0, as we did in class, first separate variables and call the separation constant E. Next, find the solution for T(t), not forgetting the properties of the imaginary constant. (The answer is in the book).
Now let k2= 2mE/h2 ( h = normalized Plank’s constant) and show that, if we assume psi(x) =0 when x=0 and L, then psi is a pure sine term and k = n Pi/L.
With these assumptions, this is called the “particle in a box” problem of quantum mechanics.
We’ll pick it up there next time.
7) Page 633 #11. (Don’t bother to plot Psi^2)
8) Explain the reasoning why we select a sine or cosine in the time-dependent part of the eigenfunctions in “plucked” versus “ hammered”.
And, do problem #1 on page 637 Added Tuesday at 1;30…this freakin’ problem took me three pages to write out. In the end, you get a whole bunch of stuff to cancel out. In case I didn’t mention it, the first two terms of the answer are given as ( 4.9) on page 635.Kudos to anyone who actually is able to stay with it all the way.
9) page 590 # 4 Verify this by writing out the first 5 terms of each series …then if you want, you can do the general proof using the sigma notation ( pain in the neck)
10) On page 646, extend the figure to include the next two columns and rows. So all together, there will be sixteen modes of vibration, including the four that are already pictured.
Then, on a lighter note, draw a 3-dimensional picture of the mode n=0, m=2. It should resemble a familiar object from south of the border.
11) page 51 write # 2- 8 in polar form and page 52, do # 1-7
The Midterm Exam will be on Friday, March 14th
12) page 63 # 14, 18-21 and page 66 # 3, 4, 5, 9
13) page 667 # 1-4, 6 and 9
14) Part 1. Take # 1-4, 6, 9, 14 and 15 on page 667 and test each using the CR equations. Which are differentiable and which are not?
Part2. Assume that f(z) is differentiable. Compute what you get for the derivative of f(z) when you let z + delta(z) approach z along a vertical line. Equate this to the expression we got in class ( using a horizontal line) and show that you get two equations and they are none other than the CR equations !!
15) page 673 #45 this problem establishes an intriguing connection between the Cauchy Riemann equations for a function f(z) = u +iv and the physics of the associated vector field F = vi + uj where the red color denotes the standard unit vectors from Calc III.
16) First, prove part 1 of the Beautiful Theorem, namely: If f = u +iv is analytic in a region R, then both u and v are harmonic in R. (Hint: Express the second partials of u with respect to x and y as mixed partials of v using the CR equations….add ‘em up to get 0. do the same thing for v.)
Second, in the example we did in class to illustrate part 2 of the Beautiful Theorem, figure out what the final function f(z) turns out to be.
17) page 676 # 1. Do this using the “ordinary calculus way”, using the antiderivative for f(z) = z and verify that you get the same answer as we did by parametrizing. (b) do problem 3(b). Parametrize the semicircular arc using the polar angle , theta, and parametrize the straight line from -1 to +1 just like we did in class with problem 1.
18 ) Let C be the unit circle centered at the origin and oriented counterclockwise. Calculate the integral of f(z) = 1/z and also f(z) = 1/z2 around C , using the polar angle theta as parameter. (The point of this is to contrast your answers with Cauchy’s Theorem.)
19) Show that co(z) and sin(z) satisfy everybody’s favorite trig identity. Then do problems # 3, 6, 9 on page 69 and # 1-6 on page 74)
20) page 677 # 17, 18, 19 Note: to apply Cauchy’s Integral Formula, the denominator must have the form z-a. Plus, compute the integral of z dz/ (z2 – 2) around the circle of radius 1 centered at z =1
21) page 678 # 22, 23 plus compute the integral around the circle of radius 2 centered at the origin of ez divided by z2 + 6z + 5
22) A handout on Laurent series…copies posted on my door.
23) Find the Laurent series expansion of f(z) =1/z2( 2 – 3z) which is valid in a punctured disk
0< mod(z)< ? and circle the residue.
24) (a) Expand f(z) = 1/ z5( 2-z)(5-z) in the innermost region (centered at z0 = 0) and find the residue. (b) Expand g(z) = 1/(2-z)3(3-z) in the punctured disk at z0=2 and find the residue.
25) Find the integral of dz/z(4-z) around the following circles. Find residues by Laurent series when needed: (1)the circle centered at 0 of radius 1, (2) center 2, radius 1 and (3) center 3 and radius 2. Plus page 686 # 14 and 15( Sometimes in a Laurent series, you need to expand 1/(1+r) rather than 1/(1-r). Well. 1/(1+r) expands as the alternating geometric series 1 – r +r2 – r3 + etc
26) (a) Prove that if f(z) = 1/(az+b)(cz+d) then the residues at the two singularities of f are the negatives of each other. (b) Compute the integral around the unit circle of 1/i times (2z2 + 5z + 2)-1 and (c) compute the integral of eiz (1+z2)-1 around a simple closed curve C that lies in the upper half plane and encloses the complex number i.
27) page 699 # 1, 2, 5 Hint on #1: z2 + 26iz -5 factors to give two singularities ( poles) and one of these lies inside the unit circle.
28) pg 699 #10. Be sure, at the very least, to pay lip service to what happens to the integral along C+R as R approaches infinity.
29) page 699 # 13 and 15 ( if you don’t get the author’s answer on #15, join the club)
30) (a)Water flows in and out of a wedge shaped channel bounded by the rays theta = 0 degrees and theta = 60 degrees. Find the equation of a typical streamline. (b) Suppose a particle starts at the point ( 1,1) and flows along its streamline ultimately heading southeast. Approximately where will it cross the line x = 3? (c) Discuss what other wedge shapes can be readily analyzed this way.
31) Prove that ez and log(z) satisfy the Cauchy-Riemann equations. There was a handout reminding you of the real and imaginary parts of these two functions. In particular, log(z) = ln((x2 + y2 )1/2) + i(arctan (y/x)).
32) Several problems on transforming rectangles into annular regions and vice versa with ez and log(z). Plus page 716 #4
33) A handout project to help in doing problem #9 on page 717. I’ll put some copies of the handout on my door but if you miss this class you’re probably gonna have a difficult time.
h34) Problem 6 on page 717 with some bells and whistles….namely, take the circle to have center at (0, ½ ) and radius ½. Find the temperature in the shaded region and find the equations of the isotherms and the lines of heat flow. Identify the isotherms geometrically.
What is the actual temperature at x = 315 and y = 147 ? and at x=2, y=3? Oh yeah… Anyone who would like to show us how come the real and imaginary parts of an analytic function describe orthogonal trajectories, please let me know.
Final exam times: Reading Day 10AM and 1PM and the next day, Saturday, 10AM and 1 PM.
If those don’t work for you, contact me.