Math 21a Multivariable Calculus

This page contains some links and information for my section of Math 21a in Fall 2015. It will be updated soon as more information becomes available.

This section meets in Science Center 309 Emerson 305 Sever 202 from 10:00-11:30am1 on Tuesday and Thursday.

Most information about this course can be found on Oliver Knill's (the course head) website. This has crucial information about homework, exams, etc (including the actual homework!).

Candy Hours

Just like Office Hours, except there's always free candy! Come for the candy, stay for the free math help.

Office: Science Center 231

Candy Hours: Monday 4-5pm and Tuesday 3-4pm

More office hours are available from other section heads and CAs and are posted here.

Problem Session

Our CA Michael Hwang holds a weekly problem session.

Room: Science Center 222

Time: Wednesday 7:30-8:30pm


I will be giving you many handouts in class, which you will be able to find here along with solutions.


Date Subject Handout Solutions
9/10 Mathematica Miley by Oliver Knill link N/A
9/10 Points, Vectors, and Geometry (Oh my!) pdf pdf
9/15 Geometry${}^3$ pdf pdf
9/15 Lines and Planes pdf N/A
9/17 Quadrics pdf pdf
9/17 Matching Exercises pdf pdf
9/24 Brief Notes pdf N/A
9/24 Other Coordinate Systems pdf pdf
9/24 Parametric Surfaces pdf pdf
10/1 Partial Derivatives pdf pdf
10/6 Linearization and Estimization pdf pdf
10/15 Directional derivatives pdf p5 of Worksheet
10/20 Extrema to the Extreme pdf pdf
10/22 Lagrane Multipliers pdf pdf
11/03 Midterm 2 Review pdf n/a
11/05 Triple Integrals pdf pdf
11/10 Force Fields pdf pdf
11/12 Line Integrals pdf pdf
11/17 curling up with Green's Theorem pdf pdf
11/19 Curl, Divergence, and Flux (Oh my!) pdf pdf
11/24 Stokes' Theorem pdf pdf
12/1 Divergence Theorem pdf pdf

These handouts are based on ones generously provided by Janet Chen and Peter Garfield.

The following are some Mathematica notebooks used to illustrate some concepts. If you have problems getting any of them to run or with Mathematica in general, let me know. I've included both the .nb files (the ones that open in Mathematica) and .pdf files, but I recommend looking at the .nb's because they're more interactive.

Date Subject Notebook PDF
10/8 Chain Rule and Linearization .nb .pdf
10/8 Tangent Planes .nb .pdf
10/15 Directional Derivatives .nb .pdf
10/20 Extrema .nb .pdf
10/22 Lagrange Multipliers .nb .pdf
11/10 Vector Fields .nb .pdf


The homework is posted here.


The Math Question Center is staffed nightly 8:30pm-10:30pm, Sunday-Thursday in Science Center 309a. More information is available here.

There are a bunch of places that give more information about quadrics, help in visualizing them, etc. Here are some: On the following sites, there are applets that let you visualize the different coordinate systems:
  1. cylindrical
  2. spherical

If you have any questions, please feel free to contact me at
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1. I'm told that classes start 7 minutes after the hour, so it's technically 10:07-11:30am, but we will be starting "on time" at 10:07.Return to text