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

> A study guide for UC Davis MAT 21D: multiple integrals, line and surface integrals, and the theorems of Green, Stokes and Gauss, with free practice.

Source: https://practicecalculus.com/courses/uc-davis-mat-21d
Updated: 2026-11-03

## At a glance

- **Course code:** MAT 021D
- **Units:** 4
- **Prerequisites:** MAT 021C or 021CH with a C- or better, or MAT 017C with a B or better
- **Covers:** Multiple integrals over plane and solid regions in several coordinate systems, line and surface integrals, and the theorems of Green, Stokes and Gauss (divergence)
- **Comes after:** MAT 021C (Calculus)
- **Practice here:** [Calculus 3](/learn/calculus-3) multiple integrals and vector calculus units

## Key takeaways

- MAT 21D, Vector Analysis, is the fourth and final course of the UC Davis calculus sequence.
- It covers double and triple integrals in Cartesian and other coordinate systems, then line and surface integrals.
- It ends with the three big theorems of vector calculus: Green's, Stokes' and the divergence theorem.
- Together, MAT 21C and 21D cover what a semester school teaches in Calculus 3.

**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.

| MAT 21D topic | Practice | Read |
| --- | --- | --- |
| Double integrals | [Double integrals over rectangles](/learn/calculus-3/multiple-integrals/iterated-integrals) | [Double integrals](/blog/double-integrals) |
| Changing the order of integration | [Switching the order](/learn/calculus-3/multiple-integrals/switching-the-order) | [Double integrals](/blog/double-integrals) |
| General plane regions | [Double integrals over general regions](/learn/calculus-3/multiple-integrals/general-regions) | [Area between curves](/blog/area-between-curves) |
| Polar coordinates | [Double integrals in polar coordinates](/learn/calculus-3/multiple-integrals/polar-double-integrals) | [Double integrals](/blog/double-integrals) |
| Triple integrals over solid regions | [Triple integrals](/learn/calculus-3/multiple-integrals/triple-integrals) | [Volumes of revolution](/blog/volumes-of-revolution) |
| Cylindrical and spherical coordinates | [Cylindrical and spherical coordinates](/learn/calculus-3/multiple-integrals/cylindrical-spherical) | [Double integrals](/blog/double-integrals) |
| Change of variables | [Change of variables and the Jacobian](/learn/calculus-3/multiple-integrals/change-of-variables) | [U-substitution](/blog/u-substitution) |
| Line integrals | [Work and line integrals of vector fields](/learn/calculus-3/vector-calculus/line-integrals-of-vector-fields) | [Vectors, dot and cross product](/blog/vectors-dot-and-cross-product) |
| Conservative fields | [Conservative fields and potential functions](/learn/calculus-3/vector-calculus/conservative-fields) | [Gradient and directional derivatives](/blog/gradient-and-directional-derivatives) |
| Green's theorem | [Green's theorem](/learn/calculus-3/vector-calculus/greens-theorem) | [Green's theorem](/theorems/greens-theorem) |
| Surface integrals and flux | [Surface integrals and flux](/learn/calculus-3/vector-calculus/surface-integrals-and-flux) | [Divergence theorem](/theorems/divergence-theorem) |
| Stokes' and Gauss's theorems | [Stokes' theorem and the Divergence Theorem](/learn/calculus-3/vector-calculus/stokes-and-divergence-theorems) | [Stokes' theorem](/theorems/stokes-theorem), [divergence theorem](/theorems/divergence-theorem) |

## How MAT 21D fits the UC Davis sequence

| Course | Title | Multivariable content |
| --- | --- | --- |
| [MAT 21A](/courses/uc-davis-mat-21a) | Calculus | None |
| [MAT 21B](/courses/uc-davis-mat-21b) | Calculus | None |
| [MAT 21C](/courses/uc-davis-mat-21c) | Calculus | Vectors, partial derivatives, optimization |
| MAT 21D | Vector Analysis | Multiple, line and surface integrals, vector theorems |

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](/courses/uc-davis-mat-21c) 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.

$$\int_0^{2\pi} \int_0^3 r\,dr\,d\theta = \int_0^{2\pi} \frac{9}{2}\,d\theta = 9\pi$$

**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

$$\int_C \mathbf{F} \cdot d\mathbf{r} = \int_0^1 (t^2 + 2t^2)\,dt = \int_0^1 3t^2\,dt = 1$$

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,

$$\oint_C \mathbf{F} \cdot d\mathbf{r} = \iint_D \left(\frac{\partial}{\partial x}x - \frac{\partial}{\partial y}(-y)\right) dA = \iint_D 2\,dA = 2\pi$$

## 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](/learn/calculus-3/multiple-integrals/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](/formulas) and timed [practice exams](/practice) 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.

## Frequently asked questions

### What is MAT 21D at UC Davis?

MAT 21D, Vector Analysis, is the last course in UC Davis's main calculus sequence. It covers multiple integrals over plane and solid regions in different coordinate systems, line and surface integrals, and the theorems of Green, Stokes and Gauss.

### What do I need before MAT 21D?

The catalog lists MAT 21C or 21CH with a C- or better, or MAT 17C with a B or better.

### Is MAT 21D hard?

The integrals themselves use familiar techniques, but setting up limits of integration for solid regions and parametrizing curves and surfaces take practice. Students who sketch regions carefully and practice setups regularly usually do well.

### What is the difference between MAT 21C and MAT 21D?

MAT 21C covers series plus the differential side of multivariable calculus: vectors, partial derivatives and optimization. MAT 21D covers the integral side: multiple, line and surface integrals and the vector calculus theorems.

### How should I prepare for MAT 21D?

Review integration techniques from MAT 21B and partial derivatives and vectors from MAT 21C. Polar coordinates and the dot and cross product show up constantly.