By September 2nd, read Sections 14.1, 14.3, and 14.4. On your own, do the following problems:

• 14.1: 3, 7, 9, 13, 31-34, 37, 47-66.
• 14.3: odds 11-23, 35, 39, 43, 45, 49, 55, 59, 65, 69.
• 14.4: 1, 5, 7.

Problem Set 1, Due Tuesday, September 2: Write up and turn in at most three of the exercises in Exercise List A.
 

This week, the lectures will be about (1) max-min methods for functions of two real variables (2) the multidimensional chain rule, and (3) the Implicit and Inverse Function Theorems.

This week, read Sections 14.5, 14.6, and 16.1. On your own, do these problems:

• 14.5: 1, 3, 7, 9, 13, 15, 19, 25, 37, 41, 43, 45, and 59. Some of these will be done in class, but you may wish to try them on your own first.
• 14.6: 1, 3, 7, and 11.
• 16.1: 1, 3, 29, 30, 31, and 32.

Problem Set 2, Due Thursday, September 11: Write up and turn in up to three of Exercises B1, B2, and B3 in Exercise List B. Turn in each Exercise that you do as a separate paper.

Lecture on Tuesday: Introduction to the ideas in the three exercises for Thursday. On Thursday: D'Alembert's approach to the one-dimensional wave equation; directional derivatives; the gradient, giving the direction of fastest increase, and normal to a level set; basic min-max methods for functions of more than one variable.

This week, read Sections 14.6 (again!), 14.7, and 14.8. On your own, do these problems:

• 14.6: 23, 33, and 43.
• 14.7: 1, 3, 7, 15, 27, 29, 31, 39, and 43.
• 14.8: 1, 3, 11, and 13.

Here's solution set for Problem Set 2.
 

The lecture on Tuesday will deal with the method of Lagrange multipliers. The one on Thursday will deal with (1) vector fields, and how to tell when one is conservative, and how to find a potential when one exists; and (2) integration over plane regions.

Problem Set 3, Due Thursday, September 18: Write up and turn in up to three of Exercises B4, B5, and B6 in Exercise List B. Turn in each Exercise that you do as a separate paper. Solutions here. (--available after the due date)

This week, read Sections 15.1, 15.2, 15.3. On your own, do these problems:

• 15.1: 11, 13.
• 15.2: 1, 3, 9, 13, 25.
• 15.3: 1, 5, 9, 19, 33, 35, 39.

Test 1 will be on September 25. It will be OK to use a scientific calculator, but no graphing calculators or computers. Here is an outline of preparation for the test.

• Study Section 14.1 and the problems that were assigned in that Section. Be able to look at an expression for a function of two variables and visualize its graph.
• Be able to "take" partial derivatives of a given function; to determine whether a function is harmonic or not; and to find the gradient of a function.
• Be able to apply the multidimensional chain rule to take first- and second-order partials of composite functions, as for example when changing from Cartesian to polar coordinates in the plane.
• Given a function, a point, and a direction, be able to find the directional derivative.
• Be able to use basic min-max methods (finding critical points and classifying them using the Hessian); and to use the method of Lagrange multipliers; and to use those methods in combination.
• Be able to set up and evaluate iterated integrals, as in Chapter 15.
• Given the two first-order partial derivatives of a function f(x,y), be able to identify the function.
• Review all the assignments made above in Chapters 14, 15, and 16.

Problem Set 4, due September 30th: Write up and turn in up to two of the problems B8 and B9. Make your presentations neat and thorough. Solutions to B8, B9.

On Test 1, there were one A (43-50), 6 Bs (33-42), 5 Cs (23-32), and one D (23-22). Solutions for Test 1.

This week, read Sections 16.1, 16.2, 16.3, and 16.4 (note especially Example 5). On your own, do these problems:

• 16.2: 1, 5, 13, 17, 19, 31, 37, and if your physics is up to it, 44. The book's intention is for you to do the problems in Section 16.2 by evaluating pullbacks. Indeed, you should know how to use pullbacks, but you should feel free also to use the Theorem on page 1059 whenever that's practical.
• 16.3: 1, 3, 5, 13, 15, 21, 23, 27, and 33.

Problem Set 5, due October 9: Write up and turn in solutions for any or all of these problems: B7, B10, C4, C5, C15, C16, C17. Some of these are from the C-list of problems.

Solution to B7.

Recall that in the October 7 lecture, I got a result of minus theta which should have been plus theta. Here is a one-page note that I hope will clear up the problem.
 

Problem Set 6, due October 16: Write up and turn in solutions for any or all of these problems: B11, B12, C1, C2, C3, C4, C5, C6, C7, C8, C9, C10, C15, C16, C17. These links are operational: Solutions to C1-3. ... Solutions to C4-5. ... Solutions to C15-18. ... These will be operational later: Solutions to C11. ... Solutions to C12-14.

Read Section 15.7, Triple Integrals. On your own, do these problems:

• 15.7: 1, 3, 11, 15, 17, 23, 29, 31, 41.

Read Section 15.4, Double Integrals in Polar Coordinates. On your own, do these problems:

• 15.4: odds 1-9, 25, 23 25, 27, 29.

Test 2 will be on Thursday, October 23rd. Scientific calculators only will be allowed. An outline:

• You may be asked to state (completely and correctly) and to prove Green's Theorem for a rectangle. You may give the proof done in class or a more general treatment like the one in the book.
• Be prepared for a Lagrange multipliers problem.
• Be able to compute, using a pullback, an integral along a path. The path may be an arbitrary parametrized path, but be familiar with the cases of line segments and graphs of functions.
• Be prepared to apply Green's Theorem. For practice: in 16.4, do 1, 3, 5, 7, 11, 13, and 19.
• Be able to compute double integrals in Cartesian and in polar coordinates.
• Be prepared to evaluate triple integrals.
• Study everything else indicated by the lectures, reading, and problems assigned since the last test.

Integrals over surfaces. An introduction to the exterior algebra of differential forms and the operator d. The Generalized Fundamental Theorem of Calculus. In the next few weeks we will cover these notes on differential forms.

Notes on Test 2.

Problem Set due Thursday, October 30: You may write up and turn in either or both of the problems C11 and C12. Solution to C11.

Read carefully, as a preview, Section 16.10, which occupies just one page, page 1118. By the end of the semester, you should understand these formulas throroughly. Then read Sections 16.5 (Curl and Divergence) and 16.6 (Parametric Surfaces and their Areas). On your own, do these problems:

• 16.5: 1, 3, 5, 7, 9, 10, 11, 12, 13, 17, 19, 21, and 22.
• 16.6: odds 1-9; 11-17, 19, and 21. Some of these problems will be difficult at first and become easier later.

Read Section 16.7 (Surface Integrals) and pay special attention to the Examples. (I suggest giving last priority to Example 1; try to see the case of integrals over a graph of a function f as a special instance of integrals over a parametrized surface.) On your own, do these problems:

• 16.7: 5, 7, 11, 13, 15. Setting up the integral is the key thing. You should be able then to evaluate them by hand, but feel free to use Mathematica instead.
• Do also these problems in 16.7, on surface integrals of vector fields: 19, 21, 23, 27 (this one's really easy to do by hand!), 35, 39, 41.

For the Problem Set due Thursday, November 6, you may do any or all of these problems: C13, C14, D6, D7, D8. Here's the D-list of problems.

Solutions to C12-C14.

Solutions to D1-D2.

Solutions to D6-D8.
 

Review Section 16.5 and the problems assigned there. Use these questions for study:

• Given a vector field F, how is curl F defined and computed?
• What is the significance of the condition curl F = 0 ?
• Given a vector field F, when can you find a potential f for F?
• Given a vector field F, why is the divergence of its curl always 0? (The proof is easy.)
• Given a scalar field f, why is the curl of its gradient always 0?
• What is the divergence of the gradient of a vector field?
• What can you say about the curl of the divergence of a vector field?

For the Problem Set due Thursday, November 13, you may do any or all of these problems: D1, D2, D3, D4, D5, D6, D7, D8. That's right; you can take a second shot at some of those assigned for last week.
 

Test 3, on the material of Weeks 10-13, will be Thursday, November 20. As you prepare for this test and for the Final Exam (Dec. 8), you may find it useful to look at this collection of tests I've given in previous years, some with answers and some not. Ignore the Sturm-Liouville problems that appear on the 4038 test.
 

Read Sections 16.8 and 16.9. Problems to do on your own:

• 16.8: 1, 3, 5, 7, 15, 19.
• 16.9: 1, 2, 3, 5, 7, 19, 23, 25.
• Try the True-False Quiz on page 1119.

The final problem set is due December 4. You may write up solutions to any or all of the eight problems found on List E.

The final exam will be comprehensive. For material from the earlier part of the course, see the advice found above under Week 5 and Week 9. Here's a study outline. It's in Hypertext Markup Language, for which I apologize, but it's readable.