## Handouts

This page will consist of the class handouts, in reverse chronological orders by week.

### Week Twelve: April 27, 2009 — May 1, 2009

- Practice Problems for Sections 13.6–13.9
(Janet Chen)
[Friday, May 1st]
Extra problems from Janet Chen.

Revised 4:55 PM on Fri May 8. Janet says: I modified the vector fields in #4 and #8 (#4 was doable but slightly nicer now, and #8 was missing an*x*). Thanks to Eason for both. - The Integral Theorems (Janet Chen) [Friday, May 1st] A worksheet, with solutions, from Janet Chen's section.
- Integral Theorems Review (Peter Garfield) [Friday, May 1st] A worksheet, with answers attached, from Peter Garfield's sections.
- Math 21a Notes (Jun Yin) [Posted Thursday, April 30th] By request, Jun Yin's notes (all 92 pages) from the semester.
- The Divergence Theorem (Janet Chen) [Wednesday, April 29th] A worksheet, with solutions, from Janet Chen's section.
- The Divergence Theorem
(Peter Garfield)
[Wednesday, April 29th]
A worksheet, with answers attached, from Peter Garfield's sections.
(It turns out that
*1–1+2*is*2*, not*1*. This changes the answer to 6(a). Thanks Lilly! Corrected 2:35 PM on Sat May 9.) - Stokes' Theorem (Janet Chen) [Monday, April 27th] A worksheet, with solutions, from Janet Chen's section.
- Stokes' Theorem (Peter Garfield) [Monday, April 27th] A worksheet, with answers attached, from Peter Garfield's sections. This is a slightly modified version of the worksheet used in class, as it has some typos corrected.

### Week Eleven: April 20, 2009 — April 24, 2009

- Flux Integrals (Janet Chen) [Friday, April 24th] A worksheet, with solutions, from Janet Chen's section.
- Flux Integrals (Peter Garfield) [Friday, April 24th] A worksheet, with answers attached, from Peter Garfield's sections. This is a modified version of the worksheet used in John Hall's sections as well.
- Curl and Divergence (Janet Chen) [Wednesday, April 22nd] A worksheet, with solutions, from Janet Chen's section.
- Curl and Divergence (Peter Garfield) [Wednesday, April 22nd] A worksheet, with answers attached, from Peter Garfield's sections. (The answer to 1(a)(iii) was wrong -- it was a vector, not a scalar. Thanks Helen! Corrected 2:35 PM on Wed Apr 22.)
- Green's Theorem (Janet Chen) [Monday, April 20th] A worksheet, with solutions, from Janet Chen's section.
- Green's Theorem
(Peter Garfield)
[Monday, April 20th]
A worksheet, with answers attached, from Peter Garfield's sections.
(The answer to 7(c) was wrong -- it should have been
*2π*, not*π/2*. Thanks John! Corrected 12:05 PM on Thu Apr 30.) - Extra Problems on Sections 13.1–13.4
(Janet Chen)
[Monday, April 20th]
A collection of practice problems from this Sections 13.1 through 13.4, courtesy
of Janet Chen. Solutions will
**not**be posted.

### Week Ten: April 13, 2009 — April 17, 2009

- The Fundamental Theorem for Line Integrals; Gradient Vector Fields (Janet Chen) [Friday, April 17th] A worksheet, with solutions, from Janet Chen's section.
- More On Line Integrals (Peter Garfield) [Friday, April 17th] A worksheet, with answers attached, from Peter Garfield's sections.
- Vector Fields and Line Integrals (Janet Chen) [Wednesday, April 15th] A worksheet, with solutions, from Janet Chen's section.
- Line Integrals (Peter Garfield) [Wednesday, April 15th] Another worksheet, with answers attached, from Peter Garfield's sections.
- Vector Fields (Peter Garfield) [Wednesday, April 15th] A worksheet, with answers attached, from Peter Garfield's sections.
- Extra Problems on Double and Triple Integrals (Janet Chen) [Monday, April 13th] Extra problems, with numerical answers, from Janet Chen.
- Triple Integrals in Cylindrical or Spherical Coordinates (Janet Chen) [Monday, April 13th] A worksheet, with solutions, from Janet Chen's section.
- Cylindrical & Spherical Coordinates (Peter Garfield) [Monday, April 13th] A worksheet, with answers attached, courtesy of Jack Huizenga (an apprentice in Fall 2008) from Peter Garfield's sections.

### Week Nine: April 6, 2009 — April 10, 2009

- Triple Integrals (Janet Chen) [Friday, April 10th] A worksheet, with solutions, from Janet Chen's section.
- Triple Integrals (Peter Garfield) [Friday, April 10th] A worksheet, with answers attached, courtesy of Eric Wofsey from Peter Garfield's noon section.
- Applications of Double Integrals: Center of Mass and Surface Area (Janet Chen) [Wednesday, April 8th] A worksheet, with solutions, from Janet Chen's section.
- Surface Area (Peter Garfield) [Wednesday, April 8th] A worksheet from Peter Garfield's sections. There are no answers yet, but some will be posted in the next few days.

### Week Eight: March 30, 2009 — April 3, 2009

- Double Integrals in Polar Coordinates (Janet Chen) [Friday, April 3rd] A worksheet, with solutions, from Janet Chen's section.
- Double Integrals In Polar Coordinates
(Peter Garfield)
[Friday, April 3rd]
A worksheet, with answers attached, courtesy of Eric Wofsey from
Peter Garfield's noon section.

**Correction:**The answer to Problem 2 was incorrect and has been fixed (Apr 3 2:00 PM). - Double Integrals over General Regions (Janet Chen) [Wednesday, April 1st] A worksheet, with solutions, from Janet Chen's section.
- Double Integrals (Peter Garfield) [Wednesday, April 1st] A worksheet, with answers attached, from Peter Garfield's sections.
- Double Integrals (Janet Chen) [Monday, March 30th] A worksheet, with solutions, from Janet Chen's section.

### Week Seven: March 16, 2009 — March 20, 2009

- Practice Problems For Week 7 (and Earlier)
(Janet Chen)
[Friday, March 20th]
A collection of practice problems from this week (and earlier), courtesy
of Janet Chen. Solutions will
**not**be posted. - More Extremal Problems (Janet Chen) [Friday, March 20th] A worksheet, with solutions, from Janet Chen's section.
- Global Extrema; Lagrange Multipliers II
(Peter Garfield)
[Friday, March 20th]
A worksheet, with answers attached, from Peter Garfield's
sections.

**Correction:**The answer to Problem 4 (Case 2) was incorrect and has been fixed (Apr 2 2:00 PM). Thanks Taylor! - Lagrange Multipliers (Janet Chen) [Wednesday, March 18th] A worksheet, with solutions, from Janet Chen's section.
- Lagrange Multipliers
(Peter Garfield)
[Wednesday, March 18th]
A worksheet, with answers attached, courtesy of Eric Wofsey from Peter Garfield's noon section.

**Correction:**The answer to Problem 4(a) was incorrect. The gradient of*g*was wrong in the*z*component. (Fixed Apr 2 2:00 PM). Thanks Taylor! - Maxima and Minima (Janet Chen) [Monday, March 16th] A worksheet, with solutions, from Janet Chen's section.
- Extremal Points (Peter Garfield) [Monday, March 16th] A worksheet, with answers attached, from Peter Garfield's & John Hall's sections.

### Week Six: March 9, 2009 — March 13, 2009

- Practice Problems For Weeks 5 & 6
(Janet Chen)
[Friday, March 13th]
A collection of practice problems for the past two weeks, courtesy
of Janet Chen. Solutions will
**not**be posted. - Directional Derivatives (Janet Chen) [Friday, March 13th] A worksheet, with solutions, from Janet Chen's section.
- Directional Derivatives (Peter Garfield) [Friday, March 13th] A worksheet, with solutions attached, from Peter Garfield's section.
- The Gradient and Level Sets (Janet Chen) [Wednesday, March 11th] A worksheet, with solutions, from Janet Chen's section.
- Gradients & Level Surfaces (Peter Garfield) [Wednesday, March 11th] A worksheet, with solutions attached, from Peter Garfield's section.
- The Chain Rule (Janet Chen) [Monday, March 9th] A worksheet, with solutions, from Janet Chen's section.

### Week Five: March 2, 2009 — March 9, 2009

- Tangent Planes and Linear Approximation (Janet Chen) [Friday, March 6th] A worksheet, with solutions, from Janet Chen's section.
- Approximately Tangent Planes (Peter Garfield) [Friday, March 6th] A worksheet, with solutions attached, from Peter Garfield's section. (Updated 5:10 PM on Thursday, March 12th. Fixed the point in Problem 6 and solutions. Thanks Eason!)
- Functions, Partial Derivatives (Janet Chen) [Monday, March 2nd] A worksheet, with solutions, from Janet Chen's section.
- Partial Derivatives (Peter Garfield) [Monday, March 2nd] A worksheet, with solutions attached, from Peter Garfield's section.

### Week Four: February 23, 2009 — February 27, 2009

- Functions, Limits, and Continuity (Janet Chen) [Friday, February 27th] A worksheet, with solutions, from Janet Chen's section.
- Parametric Surfaces (Janet Chen) [Wednesday, February 25th] A worksheet, with solutions, from Janet Chen's section.
- Parametric Surfaces (Peter Garfield) [Monday, February 23rd] A worksheet, with solutions attached, from Peter Garfield's section.
- Polar, Cylindrical, and Spherical Coordinates (Janet Chen) [Monday, February 23rd] A worksheet, with solutions, from Janet Chen's section.
- Other Coordinate Systems (Peter Garfield) [Monday, February 23rd] A worksheet, with solutions attached, from Peter Garfield's section. (Typo for the answer to Problem 3 corrected; Feb 24 11:30 AM. Thanks Eason Hahm!)

### Week Three: February 17, 2009 — February 20, 2009

- Arc Length and Curvature (Janet Chen) [Friday, February 20th] A worksheet, with solutions, from Janet Chen's section.
- Vector-Valued Functions (Janet Chen) [Wednesday, February 18th] A worksheet, with solutions, from Janet Chen's section.
- Vector-Valued Functions (Peter Garfield) [Wednesday, February 18th] A worksheet, with solutions attached, from Peter Garfield's section.

### Week Two: February 9, 2009 — February 13, 2009

- Quadric Surfaces (Janet Chen) [Friday, February 13th] A worksheet, with solutions, from Janet Chen's section.
- Graphs in Space (Peter Garfield) [Wednesday, February 11th] A worksheet, with solutions attached, from Peter Garfield's section.
- Functions and Graphs (Janet Chen) [Wednesday, February 11th] A worksheet, with solutions, from Janet Chen's section.
- Lines and Planes (Peter Garfield) [Monday, February 9th] A worksheet, with solutions attached, from Peter Garfield's section.
- Lines and Planes (Janet Chen) [Monday, February 9th] A worksheet, with solutions, from Janet Chen's section.

### Week One: February 2, 2009 — February 6, 2009

- The Cross Product (Peter Garfield) [Friday, February 6th] A worksheet, with solutions attached, from Peter Garfield's section.
- Cross Product and Triple Product (Janet Chen) [Friday, February 6th] A worksheet, with solutions, from Janet Chen's section.
- Vectors & The Dot Product (Peter Garfield) [Wednesday, February 4th] A worksheet, with solutions attached, from Peter Garfield's section.
- Vectors and the Dot Product (Janet Chen) [Wednesday, February 4th] A worksheet, with solutions, from Janet Chen's section.

### Week Zero: January 28, 2009 — January 30, 2009

- The introductory meeting slides [Wednesday, January 28th] The overheads from the mandatory introductory meeting for students in Math 21a on Wednesday, January 28th are available for download.
- Syllabus [Wednesday, January 28th] The course syllabus. This is really just an abbreviated version of this web site, so browse around here instead.