# Princeton MAT 201 Multivariable Calculus: Topics and Practice

> A study guide for Princeton MAT 201 Multivariable Calculus: vectors, partial derivatives, optimization, multiple integrals and vector calculus, with practice.

Source: https://practicecalculus.com/courses/princeton-mat-201
Updated: 2026-11-08

## At a glance

- **Course code:** MAT 201
- **Credits:** Not stated on the department course page
- **Offered:** Fall is described on the course page; other terms are not stated there
- **Prerequisites:** MAT 104 or equivalent
- **Covers:** Vectors, lines, planes, curves and surfaces in 3-space; multivariable limits and derivatives; gradient, chain rule and optimization; double and triple integrals; line and flux integrals; Green's, Stokes's and Gauss's theorems
- **Next course:** Not stated on the course page
- **Practice here:** [Calculus 3](/learn/calculus-3)

## Key takeaways

- MAT 201 is the third semester of Princeton's standard three-semester calculus sequence.
- It covers vectors and geometry in 3-space, multivariable differentiation and optimization, multiple integrals and vector calculus.
- The course ends with the integration theorems that generalize the Fundamental Theorem of Calculus: Green's, Stokes's and the divergence theorem.
- The prerequisite is MAT 104 or equivalent, and no calculators or internet resources are allowed on homework or exams.

**Princeton MAT 201** is Multivariable Calculus, the third and final course in Princeton's standard calculus sequence. The math department describes it as a thorough introduction to the methods needed for real-world problems involving quantities that change in two and three dimensions. This guide maps each MAT 201 topic to free practice with instant feedback.

## What MAT 201 covers

The department's course page lists the topics in order. First come vectors, lines, planes, curves and surfaces in 3-space. Then limits, continuity and differentiation of multivariable functions, followed by the gradient, the chain rule, linear approximation and optimization. Next are double and triple integrals in different coordinate systems. The course finishes with vector fields and vector calculus in 2 and 3-space: line integrals, flux integrals, and the integration theorems that generalize the Fundamental Theorem of Calculus, namely Green's theorem, Stokes's theorem and Gauss's theorem (the divergence theorem).

As with the rest of the sequence, no computational technology or internet resources are allowed on homework or exams. The course page also mentions optional review sessions and homework help in the fall, staffed by instructors and course assistants.

## MAT 201 topic map

| MAT 201 topic | Practice | Read |
| --- | --- | --- |
| Vectors | [Vectors and length](/learn/calculus-3/vectors-and-space/vectors-and-length) | [Dot and cross product](/blog/vectors-dot-and-cross-product) |
| Lines and planes in 3-space | [Planes in space](/learn/calculus-3/vectors-and-space/planes-in-space) | [Lines in space](/learn/calculus-3/vectors-and-space/lines-in-space) |
| Curves in 3-space | [Curves and motion](/learn/calculus-3/vectors-and-space/curves-and-motion) | [Parametric curves](/learn/calculus-2/power-series-parametric-polar/parametric-curves) |
| Surfaces and functions of several variables | [Functions of several variables](/learn/calculus-3/partial-derivatives/functions-of-several-variables) | [Partial derivatives guide](/blog/partial-derivatives) |
| Differentiation | [Partial derivatives](/learn/calculus-3/partial-derivatives/partial-derivatives) | [Clairaut's theorem](/theorems/clairauts-theorem) |
| Gradient | [Gradient and directional derivatives](/learn/calculus-3/partial-derivatives/gradient-and-directional-derivatives) | [Gradient guide](/blog/gradient-and-directional-derivatives) |
| Chain rule | [Multivariable chain rule](/learn/calculus-3/partial-derivatives/multivariable-chain-rule) | [Chain rule](/rules/chain-rule) |
| Linear approximation | [Tangent planes](/learn/calculus-3/partial-derivatives/tangent-planes) | [Linear approximation](/blog/linear-approximation) |
| Optimization | [Critical points and the second derivative test](/learn/calculus-3/partial-derivatives/critical-points-and-second-derivative-test) | [Lagrange multipliers](/blog/lagrange-multipliers) |
| Double and triple integrals | [Iterated integrals](/learn/calculus-3/multiple-integrals/iterated-integrals) | [Double integrals](/blog/double-integrals) |
| Other coordinate systems | [Cylindrical and spherical](/learn/calculus-3/multiple-integrals/cylindrical-spherical) | [Polar double integrals](/learn/calculus-3/multiple-integrals/polar-double-integrals) |
| Line integrals | [Line integrals of vector fields](/learn/calculus-3/vector-calculus/line-integrals-of-vector-fields) | [Conservative fields](/learn/calculus-3/vector-calculus/conservative-fields) |
| Flux integrals | [Surface integrals and flux](/learn/calculus-3/vector-calculus/surface-integrals-and-flux) | [Divergence theorem](/theorems/divergence-theorem) |
| Green's, Stokes's and Gauss's theorems | [Green's theorem](/learn/calculus-3/vector-calculus/greens-theorem) | [Green's theorem](/theorems/greens-theorem), [Stokes' theorem](/theorems/stokes-theorem) |

## Who takes MAT 201

Most students are first or second-year students considering a science or engineering major. The department notes that more mathematically inclined economics majors take MAT 201 instead of MAT 175, and that many students have seen some multivariable calculus in high school, though it is not required. What is required is solid single-variable calculus: the course page asks for quick, accurate work with graphs, derivatives and definite integrals.

## Example problems at the MAT 201 level

**1. A gradient.** For $f(x, y) = x^2 y + y^3$, $\nabla f = \langle 2xy,\; x^2 + 3y^2 \rangle$, so at $(1, 1)$ the gradient is $\langle 2, 4 \rangle$.

**2. A double integral.**

$$\int_0^1 \int_0^2 xy\,dy\,dx = \int_0^1 2x\,dx = 1$$

**3. Green's theorem.** For $\mathbf{F} = \langle -y, x \rangle$ around the unit circle, $\oint \mathbf{F} \cdot d\mathbf{r} = \iint_D 2\,dA = 2\pi$.

## How to do well in MAT 201

- **Get comfortable with vectors early.** Dot products, cross products and planes come up in every later unit.
- **Draw the region first.** Most multiple integral mistakes come from wrong limits, so sketch before you set up, and practice [switching the order](/learn/calculus-3/multiple-integrals/switching-the-order).
- **Connect the big theorems.** Seeing Green's, Stokes's and the divergence theorem as versions of one idea makes the last unit far easier to remember.
- **Review single-variable integration.** Techniques from [MAT 104](/courses/princeton-mat-104) reappear inside every iterated integral.

## Frequently asked questions

### What is Princeton MAT 201?

MAT 201 is Multivariable Calculus, the third semester of Princeton's standard calculus sequence. It covers vectors, curves and surfaces in 3-space, partial derivatives, optimization, multiple integrals and vector calculus.

### What are the prerequisites for MAT 201?

MAT 104 or equivalent. The department stresses solid single-variable skills: graphing functions, computing and interpreting derivatives, and setting up and calculating definite integrals.

### Who takes MAT 201?

Mostly first and second-year students considering a science or engineering major. The department notes that more mathematically inclined economics majors take it instead of MAT 175.

### Do I need multivariable calculus from high school?

No. The course page says many students have seen some multivariable calculus before, but it is not a prerequisite.

### Which theorems does MAT 201 end with?

Green's theorem, Stokes's theorem and Gauss's theorem, also known as the divergence theorem. The department describes them as generalizations of the Fundamental Theorem of Calculus.