Math 550 (Vector Analysis) Section 001
Spring 2015

Lectures in LeConte 112

MWF 9:40 am - 10:30 am

Professor: Peter J. Nyikos

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

Office Hours: 10:40 - 12:40 MWF; 2:30 - 3:30 MW; 2:30 - 3:00 Friday, or by appointment (or any time I am in). Exceptions posted on door and announced in advance whenever possible.

Special office hours for exam week:
Monday, April 27: 10:40 - 12:40 and 2:00 - 3:45
Tuesday, April 28 (reading day): 9:30 - 12:00 and 1:30-3:30
Wednesday, April 29: 1:00 - 4:00
Thursday, April 30: 9:30 - 12:00 and 1:30-4:00
Friday, May 1: 9:30 -12:00

The final exam is on Wednesday, April 29, 9am -- the morning after reading day! Information on all final exam times can be found outside of "Self Service Carolina" at this good old fashioned Registrar's website.

The final exam covers the following sections: 2.2, 2.3, 4.2, 4.3, 4.4, 6.2, 7.1 through 7.6, 8.1, 8.3, and 8.4. Besides the practice problems, the problems from Quiz 5 and quiz 6; problem 3 from Homework 2: problems 2 and 3 from Homework 3; problems 1c, 2, 3, and 4 from Test 2; and all problems (especially 3 and 4) from Test 3, will all be of help. Answer keys for these problems are outside the door of my office in a manila envelope.

The first hour test for this course was on Monday, February 16. The problems on the test were similar to practice problems below for Chapter 1 and sections 2.2, 2.3, and 2.4, but only number 1 for Section 2.3.

The second hour test was on Wednesday, March 18. It covered the part of Section 2.3 on the derivative in multiple variables (a matrix of partial derivatives) 4.1, 4.2, 4.3, 4.4, 5,2, 5.3, and 5.4. Practice problems were good preparation, as well as the quizzes and easier problems on the homework.

The third hour test was on Monday, April 20, on section 6.2 and sections 7.1 through 7.6. As usual, practice problems and quizzes were of help, and to some extent hand-in homework as well.

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 8.2, highlights of rest of Chapter 8.

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 quiz was on Friday, Jan. 23, on sections 1.1 through 1.3 as far as we got in class on Friday, Jan. 16. The second quiz was on Friday, Jan 28, on Section 1.4 and on matrix multiplication. The third quiz was on Friday, February 6, on level curves. The fourth quiz was on Friday, February 20, on gradients and directional derivatives.

Practice problems, not to be handed in:

1.1 numbers 5, 13, 15
1.2 numbers 7, 13
1.3 numbers 1, 3, 5, 7, 15 (a) (d)
1.4 do more examples from 1 (a) and (b) [the last example in each was shown in class]
1.5 numbers 1, 5, 7

2.1 draw the first three level curves for 5 and 9.
2.2 numbers 5(a), 9(b), and show that the following limit does not exist: limit as (x,y) goes to (0,0) of xy/x^2 + y^2.
2.3 number 1 and 7(a) (b)
2.4 numbers 5, 7, and 15
2.6 numbers 1, 3(b)(c), 5(a)(c), and 13(a)(b).

4.1 numbers 1, 3
4.2 numbers 1, 3, 7
4.3 number 15
4.4 numbers 3, 13, 17

5.1 numbers 1, 2(b)
5.2 numbers 1(a), 3
5.3 numbers 1, 7
5.4 numbers 1 (a)(b)

6.2 number 3, following the method of Example 3, pp. 382-383 for both parts.

7.1 number 3
7.2 numbers 1, 3
7.3 numbers 1, 3 [for the third hour test, just look at quiz 6 and the calculations of Tu and Tv and their cross product in Examples 2, 3, and 4)
7.4 numbers 3, 5, 9 and finish number 1.
7.5 number 3
7.6 number 1

8.1 numbers 1, 3(a) (d), 13, 15 (note: x = a cos t, y = b sin t)
8.3 numbers 3, 7, 15(a) (c) and see class notes on 15(b)
8.4 numbers 1, and 3 using Gauss's theorem.

Homework assignment handed in on Wednesday, February 25:
1. Do 19 on page 37, but with the ship at (1,3) and the rock at (2,1).
2. Find the volume of the parallelepiped determined by the vectors u = -i+j, v = i+2j +k, w = 3i + j. [See boxed formula on page 49.]
3. Find Df(x_0) if x_0 = i + 3j + pi/4 k, and f(x, y, z ) = (x^2yz, e^x - cos z, y - tan z).

Homework assignment handed in on Wednesday, March 2:
1, Find the determinant of the matrix whose four rows are
1   2   1   0
3   0   1   2
2   0   2   1
-1   -1   0   1
2. Sketch the flow line of the vector field   F(x, y) = (-2x, y) that passes through (1,1), labeling at least two other points.
3. Show that the curve c(t) = (e^t, 2 ln t, 1/t), t > 0, is a flow line of F(x, y, z) = (x, 2z, -z^2).

Homework assignment handed in on Wednesday, April 15:
1. with the change of coordinates (see Section 6.2 for theorems on how to handle these) u = y/x, v = xy, evaluate the double integral of exp(xy) over the region D whose associated D* is [1/2, 1]x[1,2]. [recall that exp(t) = e^t]
2. Find Tu, Tv and an equation of the tangent plane at Phi(1, 2) if Phi(u, v) = (2u, u^2+v, v^2).
3. Let F be the vector field xi + yj + zk. Evaluate the double integral over Phi of F . dS if Phi(u,v) = (e^u cos v, e^u sin v, v) with domain [0, ln 2]x[0, pi].

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.

2. Let c(t) = t^2i + tj + t^3k, t in [1, 2], with C the range of c and oriented by it.

[6 points]
a. Find a parametrization r(t) of C that reverses the orientatilon of c(t) and thus gives -C.
Hint: look at page 437.

[12 points]
b. Let F(x, y, z) = 2zi -yj + xk. Evaluate the integral of F . ds over C and -C using the method where the integral of F . dp in general is found by evaluating the integral of F(p(t)) . p'(t) over the interval of interest for p(t). For C use c(t) and for -C use r(t) (obtained in part a.) in place of p(t),

[10 points]
3. Let C be the border of the triangle in Figure 7.5.3, page 479, parametrized counterclockwise. Use Stokes' theorem to find the integral over C of F . ds if F(x, y, z) = 2zi + 8x j + (3x +y)k.