MATH 550 -- Vector Analysis
Professor Matt Miller (miller@math.sc.edu)
Section 001, MWF 2:30-3:20, LC 405

Note: If this page looks out of date, please Reload or Refresh it. Caution: while dates are mostly correct, only assignments above the green horizontal rule have been updated from 2008 to 2012. In general the changes will be minor, so you can get ahead of the game, but our pace, coverage, and emphases will change subtly.

Text: Vector Calculus by Jerrold E. Marsden and Anthony J. Tromba, published by Freeman, 6th ed.,

Office hours: My new office hours are MW 3:30-5:00, and of course by appointment. I will be in LC 300i.

My teaching style: You will observe that I assign homework BEFORE we talk about a section. That is because I really want you to READ the section, and struggle a bit; then the class and my lecture will make more sense to you, and you will be in a better position to ask good questions. Attempting problems and seeing where you get stuck is much more useful (albeit painful) than churning through problems that you basically already know how to do! In fact, you will find that the text itself is written in this style: many important results are developed in the exercises.

Weeks are indicated on a Tuesday-Monday basis; problems marked with a * will be collected on Thursday of the week of that Monday, unless otherwise indicated.

Class topics and problems
  • Aug. 23-30. Basic ideas from MATH 241. Review vector addition, scalar multiplication, cross product and dot product. Also equations of lines and planes and how to use these equations. We proceed to paths and curves, tangent vectors, partial derivatives, surfaces and gradients.
  • Jan. 15-21. Basic ideas from MATH 250/241, Geometry of maps. Read section 4.4 of the text. Make of the interpretations what you can; we will discuss these in greater depth later on. Examples 10 and 15 are quite important. Prove identities 7, 9*, and 11* on p. 306; you may assume that all functions are defined on a suitable domain in (x,y,z)-space, and are C^2 on that domain. While divergence and gradient are straightforward to define in n-space, curl, involving as it does the cross product, is something of an anomaly, and its generalization to higher dimensions is not discussed until section 8.6. Work on Worksheet 1; we will discuss these problems in class. Read section 6.1 of the text. Do problems 1, 2, 3, 4*, 6, 7*, 8*, 9, 10 (due Thursday, Jan. 24). For review purposes (or maybe this is not review--let me know!) I recommend problem 28 from p. 315, problems 17-22 from p. 314, and problems 3, 13 from page 558.
  • Jan. 22-28. Transformations, linear and otherwise; potential functions. Read section 6.2 of the text. Do problem 28* from p. 315 and problems 1, 2*, 3, 5, 6*, 8* from 6.2 for Tuesday, Feb. 5. If you need to review cylindrical and spherical coordinates see section 1.4 of the text. Work on Worksheet 2. Here are some Homework #1 solutions.
  • Jan. 29-Feb. 4. The Change of Variables Theorem, improper integrals Read section 6.2 of the text all over again! Example 5 is quite important, and the argument is pretty. Do problems 7, 12*, 14, 17, 18, 21, 22*, 23, 24, 29 from 6.2, and problem 2* from Worksheet 2 for Thursday, Feb. 7. Read section 6.4. There is a nice graphic of the "volume element" in spherical coordinates on page 573 (I don't know why it appears so late). Here are Quiz #1 solutions. By the way, don't kill yourself doing long and tedious integrations: use the tables (for example, I think formula 58 comes up somewhere). Some solutions for Homeworks #2 and #3: 6.2 #8, #12, 6.1 #6, 6.2 #29, 22.
  • Feb. 5-11. Change of Variables (conclusion) and improper integrals, path integrals. Do problems 1, 3, 5, 6, 8*, 10*, 16* from section 6.4. Go on to read 7.1 (a warm-up for 7.2, not so terribly important in its own right) and the first half of 7.2 of the text and do problems from 1, 2*, 4, 5, 6*, 7, 8b, 8c, 16 from 7.1. The starred problems are due on Thursday, Feb. 14. In problem 6 of 7.1 you can replace t in the formula in the middle of page 424 with theta, which obviously paramaterizes the curve by x = r cos(theta), y = r sin(theta), only here r also depends on theta. The idea then will be to replace dt by d(theta), and the (x')^2+(y')^2 by stuff involving r, theta, and r'(theta)=dr/d(theta). Notice this gives you a polar coordinate version of ds. See if you can give an intuitive pictorial explanation for this result.
  • Feb. 12-18. Path and line integrals. Quiz #3 on Thursday will cover chapter 6 material and 7.1, and the answers are here. The solution to problem 7.1 #6b is readable if you use the little rotate button on the menu. Worksheet 3 has review problems for the exam. Continue to read 7.2 and do problems 1abd, 2a*, 2c*, 4, 5*, 6*, 8, 9, for Thursday, Feb. 21. Begin thinking about the problems from Worksheet 4 (18.1) 1-4, 12*, 13, 19, 20, 21, 22; this will also be referred to as the yellow handout. Based on your homeworks, I added a smidgen to 6.2 #8, #12.
  • Feb. 19-25. Exam #1 and line integrals (continued). Continue to do problems from section 7.2: 10* (see a more explicit statement in problems 21, 22 on the yellow sheet), 11*, 15, 16* (if one straight line segment is nice, maybe more would be even nicer), 17 from 7.2. Take a look at #19 and #20; these might form the cores of small projects. Also work on Worksheet 4 (18.1) 14*, 15, 16*, 18, 24, 25*. Note that the vector r = (x, y, z). Examples 1 and 6 of 2.6, Example 5 of 4.3, and in general the pictures and examples throughout 4.4 all are likely to be useful for these problems. This homework is due Thurs., Feb. 28.
  • February 29 -- HAPPY BIRTHDAY TO ME!! You can skip doing homework this night. I'm creeping up to you guys in age.
  • Feb. 26-March 3. Line integrals (conclusion), parameterization of surfaces. Read sections 7.3, 7.4 and 7.5 and do problems 1, 2, 4*, 5, 7, 9, 10*, 11, 13, 14, 17 from 7.3; 1, 4*, 6, 10*, 14, 17, 18 from 7.4; and 1, 3, 6*, 7, 8 from 7.5 for Thursday, March 6. You may want to refer back to sections 2.6 and 3.5 for discussions of surfaces that are graphs or level surfaces of some scalar function. Here are some solutions for problems from 7.2. There will be a quiz on March 6.
  • March 4-17. Surface integrals of scalar and vector fields, Spring Break Read section 7.6 and do problems 1, 2, 3, 4*, 5, 6*, 9, 10* (use outward pointing normals), 16, and S*. Problem S* is to extend problem 6 by computing the line integral of the vector field F itself along the curve C that is the boundary of S that lies in the plane z = 0, oriented counterclockwise. Then compute the surface integral of curl(F) on the piece of the xy-plane that is inside C, using k as the unit normal vector. These problems are due Thursday, 20 March. Here are some solutions to the problmes of 7.3-7.5, and Quiz #4 solutions.
  • March 18-24. Surface integrals (conclusion), Green's Theorem. Read 8.1. Do problems 15 and 18 from 7.6, then problems 1, 2, 7, 10*, 12*, 14, 15, 17*, and 20 from 8.1. These problems are due Thursday, 27 March. Here are a couple of solutions from the manual.
  • March 25-31. Stokes's Theorem. Read 8.2, but skip the stuff on other coordinate systems for now. Do problems 8, 9, 12b*c*, 16, 18 from the review exercises for chapter 7 on pages 514-515 to prepare for the exam. Then do problems 1, 2, 3, 5*, 6*, 9, 10 (hint: do almost no work!), 14, 19*, 25 from 8.2. Wherever possible, consider using Stokes' Theorem twice as described in problem 15 and discussed in class. These problems will be due Tuesday, April 8.
  • April 1-7. Conservative vector fields. Read 8.3 after the exam to prep for next week. Do problems 1--4, 6*, 8, 9, 10*, 12*, 13 from 8.3, due Thursday April 10. Also do the three problems on simply connected domains (handout). There is also a bonus problem, which you may turn in on April 17.
  • April 8-14. Exam #2, Conservative fields (conclusion), Gauss's Divergence Theorem. Read section 8.4. Do problems 1, 2, 3, 5, 8*, 11* (hint: use the "product rule" for div(fF)--see (7) and Example 15 on p.306) and 16* for Thursday, April 17. Also read the material on the "flowbox" interpretation of divergence by me and by someone else, as well as pages 307-310 in your text. Figure 4.4.11 is also relevant. Think about these questions. Get started on your projects!
  • April 15-21. Gauss's Divergence Theorem (cont.), physical and coordinate-free interpretations of divergence and curl. Here are the solutions to Quiz #7. Read the blue handout, which will guide you towards computation of formulas for divergence and curl in the three main coordinate systems (that is, Cartesian, cylindrical, and spherical). You will find a complete and formal chain rule approach, but also a more geoemetric and somewhat intuitive and informal approach in the exercises #21-25. The equivalent material in our text may is to be found in Example 4 in section 8.2, on pages 542-543, and on pages 572-573. I find it fun to give these heuristic derivations (motivations might be a better word) for these results based on the ideas of infinitesimal circulation per unit area and flux per unit volume; to me these convey more information than the mechanical workings of the chain rule.
  • April 22-28. Div, grad, curl in alternate coordinate systems, project presentations, applications, teaching evaluations. See the homework just now posted for last week! Do problem 26 on the blue handout; but give the full description of div(F), curl(F) dot k, and whether or not F is conservative for each diagram. I will not ask you to compute any of these quantities in cylindrical or spherical coordinates, but you should be aware of their meanings in a coordinate-free setting. We won't likely have project presentations in class, for lack of time, but I will ask each individual or group to meet with me for 15-20 minutes to go over the written project report, just to confirm that you are on top of what you have written (so don't copy fancy stuff that you don't understand, since you can be sure that I'll focus on just that. By the way, the Honors College would like for you to complete their online survey; the departmental ones will be on paper on Thursday. Here is the solution to the bonus problem.
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    • Project possibilities: I will continue to modify this list as ideas come to me. The deadline for submission of the written report is April 24 (if you need an extension discuss this with me before the due date); we won't have time for many, if any, class presentations; instead I will schedule a meeting with you to discuss the written report and to make sure that you really do understand what you have written.

    Exams
  • FIRST EXAM: Tuesday, February 19 (class #11). The exam will be drawn from the class material through February 12 (that is, 4.4, 6.1, 6.2, 6.4, 7.1, a smidgen of 7.2), including the material on computation of divergence, curl, vector identitites, test for gradient vector fields, and computation of potential functions. Review the three quizzes and the first two worksheets. You will have these exams back in time for the "drop deadline" of Monday, February 25 (though I sincerely hope no one will need to consider this!). Here are solutions and a correction with supplement.
  • SECOND EXAM: Thursday, April 8 (class #23). The exam will be drawn from the class material through March 27 (that is, 7.2-7.6, 8.1-8.2, circulation, but not cylindrical coordinates), and worksheets along the way. Note that some problems of chapter 7 become much easier once you know Green's Theorem and Stokes's Theorem (and others will be come easier after learning Gauss's Theorem). See the main course page for solutions to Exam #2 from Spring, 2006. Here are some more solutions to review problems from Chapter 7.
  • FINAL EXAM: Thursday, May 1 (9:00-12:00). There will be two parts, as announced on the syllabus. Part B (60 points) will cover from the end of section 8.2 to the end of the course (conservative and non-conservative vector fields, the Divergence Theorem, flowboxes, infinitesimal geometric interpretation of div, grad, curl and all that), while Part A (120 points) will recap the part of the course that was previously covered by the first two exams. If you have already practiced the quizzes and exams from this semester, see my old ones from 2006 and 2002, including some solutions, as well as those of some of my colleagues for the current and past semesters (it looks like Professor Lu did the Honors 550 last spring, for example).

    • Last modified: April 23, 2008