## Math 20E: Vector Calculus, Fall 2003

under construction!

Office hours: M4-5, W3-4, F11-12

Office: APM 5256, tel. (858) 534-2734

Teaching assistants: Graham Hazel (ghazel@math.ucsd.edu) and Daniel Felix (dfelix@math.ucsd.edu)

Computation of grade: The grade is computed from your scores in the final (40%), 2 midterms (25% each), and homework (10%). Passing grade for final required for passing the course!

Dates of exams:

Midterms: 10/22 and 11/17 (new date !)

### Texts

• H. Davis and A. Snider, Vector Analysis, 7/e wm. C. Brown Publishers 1995.
Alternative texts that use linear transformations to explain derivatives and surface area element are
Marsden & Tromba, Vector Calculus Freeman 1996,   Colley Vector Calculus Prentice Hall 2000
and Barr Vector Calculus Prentice Hall 1997.   Many multivariable calculus texts contain an easy
introduction. A brief text from a physics perspective is Shey Div, grad, curl and all that Norton 1998
For lots of problems see Spiegel Schaums's Outline Series: Vector Analysis McGraw-Hill 1992
For further study of the geometry of curves and surfaces take Math 150A and for partial
differential equations Math 132A or physics classes in fluid mechanics and electromagnetism.
For the deeper analysis and proofs take Math 140, 142 or read e.g.:
Hubbard & Hubbard Vector Calculus, Linear Algebra, and Differential Forms. Prentice Hall 2002.

### Preliminary syllabus

Week 1: 1.1-1.6, 1.8-1.13 : Scalar and vector product, equations of lines and planes, determinants.
Week 2: 2.1-2.2, 3.1-3.2 : Curves, velocity, tangents and arclength. Level surfaces and gradients.
Week 3: 3.3-3.5, 3.7 : Flow lines, divergence and curl. Taylors formula.
Week 4: 4.1-4.4,(4.5): Line integrals, simply connected domains, conservative and irrotational fields
Week 5: 15.1, 15.6 Double and triple integrals, change of variables, cylindrical and spherical coord.
Week 6: 4.6-4.8: Surface area, surface integrals and flux.
Week 7: 4.9: Introduction to Divergence Theorem and Stokes' Theorem.
Week 8: 4.9, 5.1 : Divergence Theorem.
Week 9: 5.4-5.5 , 5.2, (5.3): Greens and Stokes Theorems, Laplace Equation.
Week 10: (5.7)-5.8: Orthogonal transformations. Electromagnetism or Differential forms or Review.

Sections 15.1, 15.6 refers to the handout. Sections 1.1-1.13, 2.1-2.2 and 3.1-3.2 are review from 21C.
The change of variable theorem in the handout was not covered in 21C but is in the textbook for 21C.

Additional information: You can find additional information on the following website additional info. In particular, there are practice exams and their solutions, as well as lecture notes (in pdf format) in the section "Schedule and summary of lectures (tentative)" (click on the section numbers). These notes are from a course taught in a different year, by another instructor, but might still be useful.

Homework assignments

Homeworks need to be turned at the beginning of sections, and selected problems will be graded. It is very important that you do the homework problems as most of the exam problems will be variations of homework problems.

Disclaimer: I will try to get the homework assignment on the net in time. Due to time and other limitations, this may not always be possible. The fact that there is no assignment posted for a particular date does therefore NOT necessarily mean that no homework is due.

for 10/2: Sec. 1.5: 3, 10, Sec. 1.7: 1, Sec. 1.8: 1, 8, Sec. 2.1: 1, Sec. 2.2: 2, 5, 8, Sec. 2.3: 3, 5,

for 10/9: Sec. 3.1: 1, 4, 5, 10b,c, 13, 32, Sec. 3.2: 2,4, Sec. 3.3: 3, 10, 11, Sec. 1.12: 1c, 3, 13, 22

for 10/16: Sec. 3.4: 3, 4, 6, 10, Sec. 3.5: 4, 9, Sec. 4.1: 1, 3, 4, 12,

for 10/23 (relevant for midterm!) Sec. 4.3: 1, 2ac, 3ac, 4, 5, 6, Sec. 4.4: 1ade, 6, 7, 10, Sec. 4.5: 2, 8, 9(no sketching required),

for 11/6: Sec. 4.6: 1, 3(don't worry about geometric interpretation), 5, 6, click on additional homework (updated on Monday 11/3) . Here are answers to some of the homework problems answers.

for 11/13: Sec. 4.7: 3, 4, 10, 11, 12, Sec. 4.8: 1, 4, 5, 6, Sec. 4.9: 3, 5, 21,

for 11/20: Sec. 5.4: 7, 8, 9, 12, Sec. 4.9: 11, 13, 15 16

for 12/4: (but get started, this will be a long assignment!) Sec. 4.9: 20, 21, 25, 26, Sec. 5.1: 6, 9, 10(you need not worry about expressing the vector field in terms of R and e_r), Sec. 5.5: 1,2, Sec. 5.2: 1, 5 (The earlier assigned problem 3 need not be turned in)

Final: Wednesday 12/10, 3-6pm

Office hours for finals week: Monday: 2-3, Tuesday: 1-2

Info about final: The usual rules apply: You are allowed to use one hand-written cheat sheet, but NO calculators, NO books, No other notes. Material goes over the whole term. You can find practice finals with solutions on Lindblad's webpage (click above at "additional info"). For our final, there might be a little bit more emphasis on Stoke's theorem and the divergence theorem. You could also practice doing surface integrals by looking at additional problems in the book, such as Sect. 4.7: 5, 8, 9, Sect. 4.9: 7, 8, 10, 19.

Second midterm: The second midterm takes place on Monday, 11/17 in the class room at the usual time. The rules are the same as for the first one: You are allowed to use one hand-written cheat sheet, but NO calculators, NO books, No other notes. The material goes over the one covered in homework assignments from Sections 4.6-4.9 AND in the additional assignments posted as a pdf-file. This does include e.g. the change of variable formula for double and triple integrals, volume elements for spherical and cylindrical coordinates etc, which are not contained in the Davis-Snider book! It can be found in the Stewart calculus book, in the Shenk handout (see URL listed above under `Texts') or, more concise, at changevar; you need not know more than what we did in class. Material from previous sections is only needed as far as it is necessary for the current material. You can find practice midterms with solutions on Lindblad's webpage (click above at "additional info"). Extra office hour: Friday 11/14 after class.

Midterm: The first midterm takes place on Wednesday, 10/22 in the usual class room. You are allowed to use one hand-written cheat sheet, but NO calculators, NO books, NO other notes besides the cheat sheet. The problems will be similar to homework problems, with the material going up to (including) Section 4.5. You can find practice midterms on Lindblad's webpage which is mentioned above under additional information. Extra office hour: Tuesday 1-2

Homework assignments from previous quarters. These may give you an idea of up-coming material, but can otherwise be ignored.

for 10/7: p95: 1,2 (don't do part (b)), 5(a)-(d), 6, p.112: 4, 7, 9, 12, 14, 17

for 10/14: p.117: 1, 2 p.124: 1, 3, 7(NOT solution in book), 10, p.132: 1, 4, 9 (hint: use formula (3.41) on p.147 to show that a vector field can not be the curl of another vector field) p.135: 2, 3, 4,5

for 10/21: p.140: 5f-j, 7, 8, p.150: 3, 6, 8, 10g, p.169: 3, 4, 6 (only for cylindrical coordinates!) p.190: 1, 2, 3,

Practice problems for midterm: p.191: 12, 14, 19

for 10/28: p.204: 3a,d, 4, p. 212: 1b,d, 4, 7, 9, 10

for 11/4: p.236: 1, 3, 5, 6 (compute area whichever method you prefer; you can ignore (b) and (c)) p.246: 1, 3, 5, 8,

for 11/11: p. 256: 3, 4a-c, 6, p. 262: 1, 3, 5, 6

Note: Solutions for problems 1 and 3 on p. 236 are available at Soft Reserve by 11/15.

for 11/18: p. 294: 6, 8, 9, 10, 12, p. 277: 7, 10

for 12/2: p. 262: 9a, 10, 15, 16, 26, 22, p. 299: 1,2,

additional exercises: p. 263: 17, 18, 25, p: 284: 1, 2 (ignore the `solutions' at the end of the book for 2(a) and (b)), also p. 212: 1a,c, 5,6,8

Final : Monday 12/13, 11:30 in class room. The usual rules apply: 1 cheat sheet allowed with hand-written information, no calculators, books or other notes.

Special office hours: Arif Dowla: Sunday 12-3, 2325 APM, Hans Wenzl, Sunday 3-4, 5256 APM.