# Duke MATH 212 Multivariable Calculus: Topics and Practice

> A study guide for Duke MATH 212 Multivariable Calculus: partial differentiation, multiple integrals, and the Green's, divergence and Stokes's theorems.

Source: https://practicecalculus.com/courses/duke-math-212
Updated: 2026-11-08

## At a glance

- **Course code:** MATH 212 (also listed with discussion as MATH 212D)
- **Credits:** Not stated on the department course page
- **Offered:** Fall and/or spring
- **Prerequisites:** MATH 22, 112L, 122 or 122L; not open to students who have taken MATH 202, 219 or 222
- **Covers:** Partial differentiation, multiple integrals, and topics in differential and integral vector calculus, including Green's theorem, the divergence theorem and Stokes's theorem
- **Next course:** Not stated on the course page
- **Practice here:** [Calculus 3](/learn/calculus-3)

## Key takeaways

- MATH 212 is Duke's standard multivariable calculus course.
- It covers partial differentiation, multiple integrals and vector calculus through Green's, the divergence and Stokes's theorems.
- The prerequisite is MATH 22, 112L, 122 or 122L.
- MATH 219 covers the same topics for students who have already taken linear algebra, and MATH 202 is a version for economics.

**Duke MATH 212**, Multivariable Calculus, is the standard third calculus course at Duke. The math department's description is short: partial differentiation, multiple integrals, and topics in differential and integral vector calculus, including Green's theorem, the divergence theorem and Stokes's theorem. This guide turns that description into a topic list and links each topic to free practice with instant feedback.

## What MATH 212 covers

The department lists three blocks of material. The first is partial differentiation, which includes partial derivatives, the chain rule, gradients and tangent planes, and optimization. The second is multiple integrals: double and triple integrals and changes of coordinates. The third is vector calculus, ending with Green's theorem, the divergence theorem and Stokes's theorem.

The prerequisite is MATH 22, 112L, 122 or 122L, so students come from either the [laboratory sequence](/courses/duke-math-112l) or the AP route. The course is not open to students who have taken MATH 202, 219 or 222, and it is typically offered in the fall and/or spring.

The description does not list vectors separately, but you will need vectors, lines and planes throughout, so review them early.

## MATH 212 topic map

| MATH 212 topic | Practice | Read |
| --- | --- | --- |
| Vectors, lines and planes | [Vectors and length](/learn/calculus-3/vectors-and-space/vectors-and-length) | [Dot and cross product](/blog/vectors-dot-and-cross-product) |
| Functions of several variables | [Functions of several variables](/learn/calculus-3/partial-derivatives/functions-of-several-variables) | [Partial derivatives guide](/blog/partial-derivatives) |
| Partial differentiation | [Partials with rules](/learn/calculus-3/partial-derivatives/partials-with-rules) | [Partial derivatives practice](/practice-problems/partial-derivatives) |
| Chain rule | [Multivariable chain rule](/learn/calculus-3/partial-derivatives/multivariable-chain-rule) | [Chain rule](/rules/chain-rule) |
| Gradient and directional derivatives | [Gradient and directional derivatives](/learn/calculus-3/partial-derivatives/gradient-and-directional-derivatives) | [Gradient guide](/blog/gradient-and-directional-derivatives) |
| Optimization | [Lagrange multipliers](/learn/calculus-3/partial-derivatives/lagrange-multipliers) | [Lagrange multipliers guide](/blog/lagrange-multipliers) |
| Double integrals | [General regions](/learn/calculus-3/multiple-integrals/general-regions) | [Double integrals](/blog/double-integrals) |
| Triple integrals | [Triple integrals](/learn/calculus-3/multiple-integrals/triple-integrals) | [Cylindrical and spherical](/learn/calculus-3/multiple-integrals/cylindrical-spherical) |
| Vector fields | [Vector fields](/learn/calculus-3/vector-calculus/vector-fields) | [Conservative fields](/learn/calculus-3/vector-calculus/conservative-fields) |
| Line integrals | [Line integrals of vector fields](/learn/calculus-3/vector-calculus/line-integrals-of-vector-fields) | [Line integrals of functions](/learn/calculus-3/vector-calculus/line-integrals-of-functions) |
| Green's theorem | [Green's theorem](/learn/calculus-3/vector-calculus/greens-theorem) | [Green's theorem](/theorems/greens-theorem) |
| Divergence theorem | [Stokes and divergence theorems](/learn/calculus-3/vector-calculus/stokes-and-divergence-theorems) | [Divergence theorem](/theorems/divergence-theorem) |
| Stokes's theorem | [Surface integrals and flux](/learn/calculus-3/vector-calculus/surface-integrals-and-flux) | [Stokes' theorem](/theorems/stokes-theorem) |

## MATH 212, 219 or 202?

| Course | Who it is for |
| --- | --- |
| MATH 212 Multivariable Calculus | The standard route after MATH 22, 112L, 122 or 122L |
| MATH 219 Multivariable Calculus | Students who have already had linear algebra (MATH 216, 218D-1, 218D-2 or 221) |
| MATH 202 Multivariable Calculus for Economics | Economics-focused students |

All three exclude each other for credit, so pick the one that fits your path.

## Example problems at the MATH 212 level

**1. A tangent plane.** For $z = x^2 + y^2$ at $(1, 1, 2)$, $z_x = 2$ and $z_y = 2$, so the tangent plane is $z = 2 + 2(x - 1) + 2(y - 1)$.

**2. A triple integral.**

$$\int_0^1\int_0^1\int_0^1 xyz\,dz\,dy\,dx = \left(\frac{1}{2}\right)^3 = \frac{1}{8}$$

**3. A conservative field.** $\mathbf{F} = \langle 2x, 2y \rangle = \nabla(x^2 + y^2)$, so its line integral around any closed curve is $0$.

## How to do well in MATH 212

- **Picture before you compute.** Sketch surfaces, regions and curves first. It prevents most setup errors.
- **Practice limits of integration.** Work through [switching the order](/learn/calculus-3/multiple-integrals/switching-the-order) until it feels routine.
- **Learn the three big theorems together.** Green's, the divergence and Stokes's theorems all trade a boundary integral for an interior one.
- **Keep single-variable skills sharp.** Integration from [MATH 112L](/courses/duke-math-112l) appears inside every iterated integral.

## Frequently asked questions

### What is Duke MATH 212?

MATH 212 is Multivariable Calculus: partial differentiation, multiple integrals, and topics in differential and integral vector calculus, including Green's theorem, the divergence theorem and Stokes's theorem.

### What are the prerequisites for MATH 212?

MATH 22, 112L, 122 or 122L. The course is not open to students who have taken MATH 202, 219 or 222.

### What is the difference between MATH 212 and MATH 219?

They share the same topic description. MATH 219 is intended for students who have had linear algebra and requires MATH 216, 218D-1, 218D-2 or 221.

### What is MATH 212D?

MATH 212D is a separate listing of Multivariable Calculus with the same description and prerequisites as MATH 212. Check the current class schedule to see how the two listings differ in format.

### Is there a multivariable calculus course for economics at Duke?

Yes. The department lists MATH 202, Multivariable Calculus for Economics. Students who take it cannot also take MATH 212.