# UCSD MATH 20C Multivariable Calculus: Study Guide and Practice

> A study guide for UCSD MATH 20C: vector geometry, vector functions, partial derivatives, max and min problems and double integrals, with free practice.

Source: https://practicecalculus.com/courses/uc-san-diego-math-20c
Updated: 2026-11-03

## At a glance

- **Course code:** MATH 20C (formerly numbered MATH 21C)
- **Units:** 4
- **Prerequisites:** AP Calculus BC 4 or 5, or a C- or better in MATH 20B
- **Covers:** Vector geometry, vector functions and their derivatives, partial differentiation, maxima and minima, and double integration
- **Credit note:** Two units if taken after MATH 10C; no credit for both MATH 20C and MATH 31BH
- **Practice here:** [Calculus 3](/learn/calculus-3)

## Key takeaways

- MATH 20C is the third quarter of UC San Diego's science and engineering calculus sequence and its first multivariable course.
- It covers vectors in space, vector functions, partial derivatives, optimization in several variables and double integrals.
- Line integrals and the big theorems of vector calculus are not in the 20C catalog description; MATH 20E is UCSD's separate Vector Calculus course.
- MATH 31BH is the honors alternative, and credit is not given for both.

**MATH 20C** is the third course in UC San Diego's calculus sequence for science and engineering, and the point where calculus moves from one variable to several. Most students reach it after MATH 20B; a strong AP Calculus BC score also qualifies. This guide maps every MATH 20C topic to free practice so you can build intuition for vectors, surfaces and double integrals one problem at a time.

## What UCSD MATH 20C covers

The UC San Diego catalog lists five topics for MATH 20C: vector geometry, vector functions and their derivatives, partial differentiation, maxima and minima, and double integration. The course was formerly numbered MATH 21C.

That matches the first two thirds of a standard **Calculus 3** course. Triple integrals, line integrals and the theorems of Green, Stokes and Gauss are not part of the 20C description. UC San Diego lists those vector calculus ideas in a separate course, MATH 20E (Vector Calculus).

## MATH 20C topic map

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

| MATH 20C topic | Practice | Read |
| --- | --- | --- |
| Points and vectors in space | [Vectors and their length](/learn/calculus-3/vectors-and-space/vectors-and-length) | [Vectors, dot and cross product](/blog/vectors-dot-and-cross-product) |
| Dot and cross product | [The dot product](/learn/calculus-3/vectors-and-space/dot-product) | [Vectors, dot and cross product](/blog/vectors-dot-and-cross-product) |
| Lines and planes | [Planes in space](/learn/calculus-3/vectors-and-space/planes-in-space) | [Lines in space](/learn/calculus-3/vectors-and-space/lines-in-space) |
| Vector functions and their derivatives | [Curves and motion](/learn/calculus-3/vectors-and-space/curves-and-motion) | [Chain rule](/rules/chain-rule) |
| Functions of several variables | [Functions of several variables](/learn/calculus-3/partial-derivatives/functions-of-several-variables) | [Partial derivatives](/blog/partial-derivatives) |
| Partial differentiation | [Partial derivatives](/learn/calculus-3/partial-derivatives/partial-derivatives) | [Partial derivatives](/blog/partial-derivatives), [Clairaut's theorem](/theorems/clairauts-theorem) |
| Chain rule in several variables | [The multivariable chain rule](/learn/calculus-3/partial-derivatives/multivariable-chain-rule) | [Implicit differentiation](/blog/implicit-differentiation) |
| Gradient and directional derivatives | [The gradient](/learn/calculus-3/partial-derivatives/gradient-and-directional-derivatives) | [Gradient and directional derivatives](/blog/gradient-and-directional-derivatives) |
| Maxima and minima | [Critical points and the second-derivative test](/learn/calculus-3/partial-derivatives/critical-points-and-second-derivative-test) | [Lagrange multipliers](/blog/lagrange-multipliers) |
| Double integration | [Double integrals over rectangles](/learn/calculus-3/multiple-integrals/iterated-integrals) | [Double integrals](/blog/double-integrals) |
| Double integrals over regions and in polar form | [Double integrals in polar coordinates](/learn/calculus-3/multiple-integrals/polar-double-integrals) | [Double integrals](/blog/double-integrals) |

## MATH 20C and the courses around it

| Course | Title | How it relates to 20C |
| --- | --- | --- |
| [MATH 20B](/courses/uc-san-diego-math-20b) | Calculus for Science and Engineering | The prerequisite (C- or better) |
| MATH 20C | Calculus and Analytic Geometry for Science and Engineering | Vectors, partials, double integrals |
| MATH 31BH | Honors Multivariable Calculus | Honors alternative; no credit for both |
| MATH 20E | Vector Calculus | Separate course on vector calculus topics |

If you are coming from AP Calculus BC with a 4 or 5, you can start the sequence here. Otherwise, the [MATH 20B guide](/courses/uc-san-diego-math-20b) and the [MATH 20A guide](/courses/uc-san-diego-math-20a) cover the single-variable skills 20C relies on.

## Example problems at the MATH 20C level

**1. The angle between two vectors.** Find the angle between $\mathbf{u} = \langle 1, 1, 0 \rangle$ and $\mathbf{v} = \langle 1, 0, 1 \rangle$.

$\mathbf{u} \cdot \mathbf{v} = 1$ and $|\mathbf{u}| = |\mathbf{v}| = \sqrt{2}$, so

$$\cos\theta = \frac{1}{\sqrt{2}\sqrt{2}} = \frac{1}{2}, \qquad \theta = \frac{\pi}{3}$$

**2. Partial derivatives.** For $f(x, y) = x^2 y + \sin(xy)$,

$$f_x = 2xy + y\cos(xy), \qquad f_y = x^2 + x\cos(xy)$$

**3. A double integral.** Evaluate $\iint_R xy\,dA$ over $R = [0, 2] \times [0, 3]$.

$$\int_0^2 \int_0^3 xy\,dy\,dx = \int_0^2 x \cdot \frac{9}{2}\,dx = \frac{9}{2} \cdot 2 = 9$$

## How to do well in UCSD MATH 20C

- **Sketch everything.** Draw the region before setting up a double integral and the vectors before computing an angle. Most setup errors disappear once you can see the picture.
- **Lean on single-variable rules.** A partial derivative is an ordinary derivative with the other variables held constant, so the [product rule](/rules/product-rule) and [chain rule](/rules/chain-rule) carry over directly.
- **Practice the second-derivative test** until classifying critical points is routine, then mix in constrained problems with [Lagrange multipliers](/blog/lagrange-multipliers).
- **Use the formula sheet and timed practice.** The [calculus formula sheet](/formulas) and [practice exams](/practice) help you get faster before midterms.

> **Tip:** when a double integral looks messy in $x$ and $y$, check whether the region is a disk or a wedge. Switching to polar coordinates often turns a hard integral into a one-line answer.

## Frequently asked questions

### What is MATH 20C at UCSD?

MATH 20C, Calculus and Analytic Geometry for Science and Engineering, is UC San Diego's first course in multivariable calculus. It covers vector geometry, vector functions and their derivatives, partial differentiation, maxima and minima, and double integration.

### What are the prerequisites for MATH 20C?

The catalog lists a 4 or 5 on AP Calculus BC, or a grade of C- or better in MATH 20B.

### Is MATH 20C hard?

The calculus itself is familiar: partial derivatives use the same rules as single-variable derivatives. The new challenge is visualizing surfaces and regions in three dimensions, so sketching often and practicing setup problems pays off.

### What is the difference between MATH 20C and MATH 31BH?

MATH 31BH, Honors Multivariable Calculus, is the second quarter of an honors sequence that combines linear algebra with multivariable calculus and covers topics such as Jacobian matrices. The catalog does not allow credit for both 20C and 31BH.

### How should I prepare for MATH 20C?

Review the chain rule, integration techniques and polar coordinates from MATH 20B, then start with vectors: length, the dot product and the cross product.