Math 21a: Multivariable Calculus
Handouts
These links are to handouts from Peter Garfield's two sections of Math
21a. The most important resource for the course is
the official
web page. This page should be thought of as a supplement.
The handouts below all contain answers and/or solutions that were
not passed out in class. Please email me if you find any typos or
mistakes in these answers and solutions. Thanks.
Week Thirteen: December 15 to December 19
- Integral Theorems Review
[Monday, December 15th]
A review worksheet, mostly on Stokes' theorem and the Divergence
theorem, with some answers / solutions.
Week Twelve: December 8 to December 12
- Stokes Theorem
[Wednesday, December 10th]
A worksheet on Stokes' theorem, with some answers / solutions.
- Flux Integrals
[Monday, December 8th]
A worksheet on flux integrals, with some answers / solutions.
Partly due to a former preceptor (thanks Tom!) and partly from the
textbook exercises .
Week Eleven: December 1 to December 5
- Curl and Divergence
[Friday, December 5th]
A worksheet on curl and divergence, with some answers and some detailed solutions.
- Green's Theorem
[Wednesday, December 3rd]
A worksheet on Green's theorem, with detailed answers / solutions.
Week Ten: November 24 to November 26
- Vector Fields
[Wednesday, November 26th]
A worksheet on vector fields, with answers / solutions.
- Cylindrical & Spherical Coordinates
[Monday, November 24th]
A worksheet, courtesy of Jack (from the ten o'clock lecture) on triple integrals using cylindrical and spherical coordinates. I've added solutions, so any mistakes you find are my fault.
Week Nine: November 17 to November 21
- Triple Integrals
[Friday, November 21st]
A worksheet on triple integrals, courtesy of my colleague Janet Chen.
This version has answers / solutions.
- Integration Review
[Monday, November 17th]
A review handout on (a) switching the order of integration in double
integrals, and (b) double integrals in polar coordinates. There was
a typo in limits of Problem 4 (the ln(3) should have been
e3) — see this comment [pdf] if you're
interested in changing the order of integration of the problem as
originally posed. (Thanks Tomasz for finding the typo.)
Week Eight: November 10 to November 14
- Double Integrals
[Wednesday, November 12th]
A handout on double integrals. I've made the following changes from
the version handed out in class: the domains for Problems 7 and 8
have been changed from D5 and D3
to D2 and D5, and the lower
limit of integration for y in Problem 10 has been changed to
0 to make the problem more sensible.
Week Seven: November 3 to November 7
- Lagrange Multipliers I
[Wednesday, November 5th]
A handout on Lagrange multipliers. While it's formulated as a
worksheet, it was really used more as simply examples. This version
has extra pages with larger pictures solutions.
Week Six: October 27 to October 31
- Gradients & Level Curves
[Wednesday, October 29th]
A handout on the relationship between the gradient and level curves. This
version has solutions.
Week Five: October 20 to October 24
Week Four: October 13 to October 17
- Limits and Continuity / Partial Derivatives
[Friday, October 17th]
A handout on limits, continuity, and partial derivatives. This
version has solutions.
- Parametric Surfaces
[Wednesday, October 15th]
A handout on parametric surfaces. (Updated almost immediately after
posting to make all the pictures smaller. I've redone them, so they
might look slightly different, but the PDF should be more managable.)
Week Three: October 6 to October 10
Week Two: September 29 to October 3
No handouts.
Week One: September 22 to September 26
- The Cross Product
[Friday, September 26th]
A handout covering the basic properties and calculation of
the cross product.