Math 20E. Vector Calculus - Winter 2013


It is important that you have a look at the material before the lectures since it will help you to follow the lectures, see the schedule where you also find lecture notes. Please ask questions in lectures, since if you don't quite understand something others may not understand either, and the explanations will help everyone understand better and keep the lectures at a pace you can follow. The best way to learn math is by doing examples so try to do all the homework problems and more similar problems. Let us know if you have any complaints and suggestions for improvements.


It can be helpful with different perspectives: Multivariable calculus texts give an easy introduction. Davis, Snider, Vector analysis used to be the text. Colley, Vector calculus is similar to the text. Shey, Div, grad, curl and all that is a popular introduction from a physics perspective. Kettler Multivariable calculus, applications and theory is free. Online Lectures in Multivarible Calculus Further study: 150A Differential geometry of curves & surfaces, B Vector calculus on manifolds, differential forms Partial Differential Equations 110A,B, Physics classes. Analysis-proofs 140, 142 or Hubbard, Hubbard, Vector calculus, linear algebra, and differential forms -the Honors Calculus 31 text.


Bring identification to exams. No calculators, books or notes are allowed in exams, but you will be allowed one hand-written 'cheat sheet'. The exams cover material in homeworks due before the exams, and corresponding lectures and reading. Questions about the grading should be brought up with your TA.

Calculation of Grade

Your grade will be calculated as the better of the two options as follows:

Option 1: homework (10%), 2 midterms (20% each) and the final (50%).

Option 2: homework (10%), the better of the 2 midterms (20%) and the final (70%)

For both options we will drop your worst homework score for the homework component.

Syllabus, Schedule and summary of lectures.

Below you find a tentative syllabus for the class. We may possibly go a little faster or slower in the course; in the beginning we will likely be going faster.
It is important that you have a look at the material before it is covered in the lectures. See below for a link to a similar table, where you can also find lecture notes by another professor. While my lectures will not be exactly the same as for that other course, the notes may still be helpful.
wk  date  Monday  Wednesday  Friday
  1  1/7  1.1-3 Overview, vector operations  1.3 Cross product, determinants,  1.5 Matrices-linear maps   2.1 Graphs-surfaces, (2.2 Limit)
  2  1/14  2.3 Derivative matrix,
 2.4 curves, 2.5 chainrule
 2.5 Chain rule, 2.6 Gradient  3.2 Taylor's theorem-vector func.
  3  1/21  Holiday  4.1-2 Curves, arc length  4.3-4 Vector fields-flow lines
  4  1/28  4.4 Divergence and curl  5.1-3 Double-, iterated-integrals  Exam
  5  2/4  5.4 Changing order of intergration  5.5 Triple integrals  6.1 Change of variables,geometry
of maps in plane(3.5 Inv Func Th)
 6.2 Change variables in integrals
  6  2/11  7.1 Path-, 7.2 Line-integrals  7.2-3 Parametrized surface  7.4 Area of surface
  7  2/18  Holiday  7.5 Integrals over surfaces  7.6 Integrals over surfaces
  8  2/25  7.6 Integrals over surfaces  8.1 Green's thereom  Exam
  9  3/4  8.2 Stokes' theorem  8.3 Conservative fields  8.4 Gauss' theorem
10  3/11  8.5 Electromagnetism, Fluids  8.6 Differential forms  Review
11  3/18  Final 3-6pm  No class  No class
The schedule was taken from Professor Lindblad's course page. If you go to this page, you will find a similar syllabus (with different dates); if you click on the appropriate chapter, you will find Lindblad's LECTURE NOTES; they now refer to a different book, but the numbering of the chapters is almost the same, and so are the notes. I am not planning to exactly follow them, but they may still be useful for our lectures as well.

Homework Assignments

All homeworks should be handed in, but we only have resources to grade 3-4 problems per set and some sets might be returned ungraded. The lowest homework score will be discarded. Attempt to solve all problems yourself to learn it. Homeworks are usually supposed to be turned in on Wednesdays. Mailboxes will be available on the 6th floor of APM. Please turn in your homework on the given date by 5pm.

for 1/16: Sec. 1.1: 16, Sec. 1.2: 8, 14, 22, Sec. 1.3: 6, 7, 12, 16a, Sec. 1.5: 4, 18, Sec. 2.1: 6, 11 (level sets only), 21, 28,

for 1/23: Sec. 2.3: 6a, 8cd, 12c, 18, 20, Sec. 2.4: 8, 18, Sec. 2.5: 4, 5c, 15, 8, 12, Sec. 2.6:3c,4a, 5c, 6a,9,12,

for 1/30: Sec. 3.2: 2,6, Sec. 4.2: 2,8,10, Sec 4.3: 5,6,9, 11, 14, Sec. 4.4: 4, 8, 19, 26. Material until including this assignment will be relevant for the midterm.

Midterm I: The first midterm takes place on February 1 in class. The material will go until Section 4.4. Exam problems will be similar to homework problems or problems in the practice exam posted below. You are allowed one hand-written cheat-sheet (you can use both sides), but no further notes, books or calculators.

Practice test for Midterm I

Solution sketches for practice test

for 2/6: Sec. 5.1: 1a, 10, Sec. 5.2: 2d, 4, Sec. 5.3: 2b, 8, Sec. 5.4: 2b, 8, Sec. 5.5: 4, 11,

for 2/13: Sec. 6.1: 2, 3, 4, 10 Sec 6.2: 1,4a,,8,9,19, 25, 29, Sec 7.1: 2a,4a, Sec 7.2: 1c, 2a, 3,

for 2/20: Sec 7.2: 12,15,18, Sec 7.3: 2, 5, 12, 14, Sec 7.4: 1, 4, 6, 11,

for 2/27: Sec 7.5: 2, 4, 6, 8, 10, Sec 7.6: 2(look at example 4 in book),3, 7, 10,

Midterm II: We will have our second midterm this coming Friday. The material will go until the current homework assignment, i.e until including Section 7.6. The same rules apply as for the first midterm: one cheat sheet, no calculators or books. Below is an old midterm to practice. You could also look at some more examples in Section 7.6, and more homework problems in that section (The problems 5 and 8 in that section are somewhat time consuming - I will not give such lengthy problems on the midterm).

Practice test for Midterm II

for 3/6: Sec. 8.1: 1, 3d, 4(see Theorem 4), 9, 12, 13,

for 3/13: Sec 8.2: 3, 6, 10, 11, Sec. 8.3: 4, 7, 13ac, Sec. 8.4: 2, 3, 5a, 8

Final: The final will be cumulative, going to including the last homework assignment, with an emphasis on the material after the second midterm. For this you will also have to be familiar with all kinds of integrals which we have studied starting from Section 5. Material before midterm I will only be relevant if it is needed in connection with problems from later sections. I have posted a practice final below which should give you some idea. The usual rules apply for the final: one cheat sheet (both sides OK), no calculators, no books.

Office hours for final: Friday 3-4 (right after class), M 1-2

Practice final

Unfortunately, there is no official solution set available for the practice exam. But you are welcome to ask questions about the practice exam problems in my or the TAs' office hours.

------------------------ --------------------------------------------------- please ignore below line------------------------------------------------------------------------------------------------------------------------