Informal lecture notes for Nov. 29
1
1.1
Complex differentiation, continued
Complex differentiation rules
Simple functions like constant functions and f (z) = z can be differentiated just using the definition, as we saw. However, just as in the rea
Practice Questions for the 1st Midterm
The following consists of practice questions to help you prepare for the midterm. They are similar in style and difficulty to what you can expect to see on the actual midterm. The actual midterm will be shorter,
Calculus IV: Practice
1
Practice Midterm
Name:
Problem 1 2 3 4 5 Total:
Max
20 20 20 20 20
100
Scores
Instructions
Time for test: 75 minutes.
Write clearly in pen or pencil, and make your final answer easy to find.
To receive any partial credit you mus
Sample questions for complex integration
1. Let C be the curve z(t) = t2 + it3 for 0 t 1, and let f (z) = eiz . Compute 2. Let C be the line segment from i to 1, and let f (z) = z z. Compute
C
f (z) dz.
C
f (z) dz.
3. Let C be the curve consis
Solutions to sample questions for complex integration
1. Let C be the curve z(t) = t2 + it3 for 0 t 1, and let f (z) = eiz . Compute f (z) dz. C We see right away that f (z) is holomorphic (since it's the composition of functions we know to holomor
Solutions to the Practice Final
Note: You may have noticed that none of the questions based on Chapter 16 had hints about how to approach them. Some of the questions on the actual final will have such hints, but in order to allow you to practice deci
Informal lecture notes for Dec. 4
Now that we have a solid background in the basics of complex differentiation, it's time to move on to complex integration.
1
Contour integrals
Consider a curve C in the complex plane parametrized by z(t) for a z
Practice Questions for the 2nd Midterm
The following consists of practice questions to help you prepare for the midterm. They are similar in style and difficulty to what you can expect to see on the actual midterm. The actual midterm will be shorter,
Homework #8
1. (Appendix G: #39) Find the cube roots of i and sketch them in the complex plane. 2. Find functions u(x, y) and v(x, y) from R2 to R such that z + (z)2 = u(x, y) + iv(x, y).
3. Find all possible values of log
e 2
+
e i 2
.
4. Co
Practice Questions for the Final
The following consists of practice questions to help you prepare for the final. They are similar in style and difficulty to what you can expect to see on the actual final. Note that just because a topic does not appea
Solutions to the 2nd Practice Midterm
1. Let C be the curve in the plane given by the graph of y = x2 from the point (1, 1) to the point (2, 4). (a) Integrate f (x, y) = 2y/x along C (with respect to arc length). Since C is a graph, we can parametriz
Solutions for the second midterm
1. Let C be the helix given by x(t) = cos t, y(t) = sin t, z(t) = t for 0 t 2.
(a) Integrate f (x, y, z) = x + 3z 2 along C with respect to arc length. We have been given a parametrization; using it, we have r (t)
Informal lecture notes for Nov. 27
I'll assume you're familiar with the review of complex numbers and their algebra as contained in Appendix G of Stewart's book, so we'll pick up where that leaves off.
1
Elementary complex functions
In one-variabl
Solutions to the 1st Midterm
1. Consider the following integral
1 0 1-2x -1 0 1-y 2
f (x, y, z) dz dy dx. (a) Rewrite this integral so that the inner-most integral is with respect to z, the middle integral is with respect to x, and the outer integra
Solutions to the 1st Practice Midterm
1. Evaluate the following integrals. (a)
1 0 0
2
1-x2
ex
2
+y 2
dy dx
We can't just proceed by taking the anti-derivative of ey , because it doesn't have an antiderivative given by elementary functions. I
THE CALCULUS OF VARIATIONS
PAUL SIEGEL
1. Introduction
It is well-known that the shortest path between two points in the plane is a straight line. This statement
seems so obvious that it is tempting not to give it a second thought, but understanding why i