Math 550 (Vector Analysis) Section 001
Fall 2014

Lectures in Gambrell

MWF 12:20 -- 1:10 PM

Professor: Peter J. Nyikos

Prof. Nyikos's Office: LeConte 406.     Phone: 7-5134

The final exam is on Monday, December 8, at 4pm. Information on all final exam times can be found outside of "Self Service Carolina" at this good old fashioned Registrar's website. On the first two chapters, problems very similar to the ones on the first test will be given. Tests 2 and 3 will also be good preparation for the final exam, but also look at quizzes and practice problems for all except chapters 1 and 2 below, especially those for Chapter 7.

Special office hours Monday, Dec. 8: 10:00 - 10:45 am, 2:00 - 3:45 pm; Tuesday, Dec. 9 and Wednesday, Dec. 10: 10:00 - 12:00 and 1:30 - 3:45; Thursday, Dec. 11: 2:00 - 5:30 pm. Office hours for Friday will be posted as they become definite.

Textbook: Vector Calculus, , Fifth Edition, by Marsden and Tromba, (2003) W.H. Freeman and Co.

The course covers the following chapters and sections:
Highlights of Chapter 1
Highlights of Chapter 2, with emphasis on 2.5
Highlights of Section 3.1; details of 3.2 and 3.3; rest of Chapter 3 as time permits
Highlights of Chapter 4, except for detailed treatment of 4.4
Highlights of Chapter 5
Sections 6.1 and 6.2
Highlights of sections 7.1 and 7.2, detailed coverage of the remaining sections of Chapter 7 except 7.7
Sections 8.1 through 8.4, highlights of rest of chapter.

Only simple calculators (in other words, those that may be used in taking SAT tests) are needed for this course, and they will be needed only a small fraction of the time, outside of class. Neither the quizzes, nor the tests, nor the final exam will require their use, although they may save some time on a few problems. Programmable calculators are not permitted for quizzes, tests, or the final exam.

The course grade will be based on quizzes, homework, three tests, a final exam, and attendance. Details on this and on various policies can be found here.

Learning Outcomes: Students will master concepts and solve problems based upon the topics covered in the course, including the following: vector fields, line and path integrals, orientation and parametrization of lines and surfaces, differentiation of vector fields, change of variables in double and triple integrals, Jacobians, oriented surface integrals, theorems of Green, Gauss, and Stokes.

The first test was on Wednesday, September 24. It covered Chapter 1, partial derivatives, directional derivatives, and gradients.
The second test was on Wednesday, October 29. It covered the sections of Chapters 3 and 4 from which practice problems were assigned (see below)
The third test was on Monday, November 24, on Sections 6.2 and 7.1 through 7.4.

The following homework was due Monday, October 13:

1. Find the Jacobian determinant for the map from R^3 to R^3 that takes (rho, theta, phi) to (x, y, z) by the usual equations:
x = rho(sin phi) (cos theta)
y = rho(sin phi) (sin theta)
z = rho(cos phi)

2. Section 2.4, number 20

3. Section 2.5, number 5d

4. page 177, number 32

5. Find f_xx, f_yy, and f_xy for f(x, y) = xy tan y. Do it in general and also at the point (x, y) = (3, pi/3). Here _xx designates a double subscript, for second order partials.

The following homework featured some problems which were only fully solvable using infinite series. The portion that required this is now part of the extra credit problems.

1. Let Phi be the function Phi(u,v) = (2 cos u, 2 sin v, v) with domain [0, 2pi] x [-1, 1]. Use the integral over S (the range of Phi) as in Section 7.4, to calculate the area of S.
2. With Phi and S as before, integrate the function f(x, y, z) = xy on S.
The most common mistake in these problems was not putting absolute value signs around 2 sin u when it was brought out from under the radical. This called for splitting the integral into one part where the absolute value is 2 sin u and another where the absolute value is -2 sin u. It is only after this is solved that the rest of the integration requires infinite series.

Practice problems, not to be handed in:

Section 2.3: number 1
Section 2.4: numbers 9, 11, 13, 15, 17
Section 2.5: numbers 5, 9, 13
Section 2.6: numbers 1, 3bc, 5, 8

Section 3.1 numbers 1, 3, 9, 15
Section 3.2 numbers 1, 3
Section 3.3 numbers 1, 17

Section 4.3 numbers 13, 15
Section 4.4 numbers 1, 3, 13, 15

Section 6.2: numbers 2, 3

Section 7.1: numbers 3, 9
Section 7.2: numbers 1ab, 2b, 15
Section 7.3: numbers 1 and 5, part 1
Section 7.4: numbers 1, 3
Section 7.6: number 1

Extra Credit Problems

From time to time, extra credit problems will be assigned in class. Extra credit problems are to be done strictly on your own, except that I am willing to give you advice. You are not to discuss them with anyone else.

There is no due date for extra credit, but once a fully correct solution is handed back, the problem is no longer eligible for extra credit. This is true of any problem crossed out below

If you can't quite get the solution but have some ideas, hand them in for partial credit. I will keep adding to your score as you improve your work on it.

To get full credit, it is not enough to get the correct answer. You need to get it in such a way that someone who has not seen the problem before can tell that you did, indeed, get the right answer.

1. [worth 8 quiz points] Let C be the border of the triangle in Figure 7.5.3 on page 479, parametrized counterclockwise. Use Stokes' theorem to find the line integral of F = (2z, 8x, 3x+y) on this curve.

2. [worth 8 quiz points] Use Gauss's Divergence Theorem to find the surface integral of the vector field F = (x^3 , y^3 , z^3) outward from the surface of the region enclosed by the upper hemisphere of the unit sphere and the plane z = 0.

3. [worth 10 quiz points] Find a Taylor series expansion for the square root of 1 + 4cos^2 and a formula for the series one gets by integrating it term by term (with + C because it is an indefinite integral.

4. [worth 8 quiz points] Use the first two nonzero terms of a Taylor series for the square root of 1 + 4cos^2 to approximate its definite integral between x = -1 and x = 1.