UC Davis MAT 21D Vector Analysis: Study Guide and Practice

MAT 21D is the fourth and final course in the UC Davis calculus sequence. Where MAT 21C introduced vectors and partial derivatives, 21D integrates in several variables: over regions, along curves and across surfaces. This guide maps every MAT 21D topic to free practice so you can master each type of integral by working problems with instant feedback.

What UC Davis MAT 21D covers

The UC Davis General Catalog describes MAT 21D, Vector Analysis, as covering multiple integrals over plane and solid regions using various coordinate systems, followed by line and surface integrals. The course culminates in the theorems of Green, Stokes and Gauss, which connect integrals over a region to integrals over its boundary.

That is the integral half of a standard Calculus 3 course. Because UC Davis spreads multivariable calculus across two quarters, 21D has room to treat the vector calculus theorems in depth.

MAT 21D topic map

Each row links to a lesson you can practice and to a reference page or guide that explains the idea.

How MAT 21D fits the UC Davis sequence

MAT 21D finishes the sequence, so it is the course where every earlier skill gets used at once. If your partial derivatives or vectors feel rusty, the MAT 21C guide maps those topics to practice.

Example problems at the MAT 21D level

1. A double integral in polar coordinates. Find the area of the disk x^2 + y^2 \le 9 using polar coordinates.

2. A line integral. Let \mathbf{F} = \langle y, x \rangle and C be \mathbf{r}(t) = \langle t, t^2 \rangle for 0 \le t \le 1. Then \mathbf{F}(\mathbf{r}(t)) = \langle t^2, t \rangle and \mathbf{r}'(t) = \langle 1, 2t \rangle, so

As a check, \mathbf{F} = \nabla(xy), so the answer is xy at (1, 1) minus xy at (0, 0), which is 1.

3. Green's theorem. For \mathbf{F} = \langle -y, x \rangle around the unit circle counterclockwise,

How to do well in UC Davis MAT 21D

• Sketch the region first, every time. Limits of integration come from the picture. Practice switching the order until reading bounds off a sketch is routine.
• Pick coordinates to match the shape. Disks call for polar, cylinders for cylindrical, spheres for spherical. Remember the extra factor r or \rho^2\sin\phi.
• Check for a conservative field before parametrizing. If \mathbf{F} is a gradient, the line integral only depends on the endpoints, which can save a page of work.
• Keep the theorems on one page. The calculus formula sheet and timed practice exams help you recall which theorem applies to which boundary.

Tip: Green's, Stokes' and the divergence theorem all say the same thing in different dimensions: the total change inside a region equals what flows across its boundary.