# Stanford Math 52 Integral Calculus of Several Variables Guide

> A study guide for Stanford Math 52: iterated integrals, line and surface integrals, conservative fields and the theorems of Green, Gauss and Stokes.

Source: https://practicecalculus.com/courses/stanford-math-52
Updated: 2026-11-03

## At a glance

- **Course code:** MATH 52
- **Units:** 5
- **Offered:** Winter and spring quarters (2026-27)
- **Prerequisites:** MATH 21 and MATH 51, or equivalents
- **Covers:** Iterated integrals, line and surface integrals, vector analysis including vector potentials and conservative fields, physical interpretations, and the divergence theorem and the theorems of Green, Gauss and Stokes
- **Equivalent:** MATH 62CM (credit is not granted for both)
- **Practice here:** [Calculus 3](/learn/calculus-3), multiple integrals and vector calculus units

## Key takeaways

- Math 52 is the multivariable integration course at Stanford, taken after Math 51.
- It covers iterated integrals, line and surface integrals, and vector analysis.
- It ends with the divergence theorem and the theorems of Green, Gauss and Stokes.
- It is offered in winter and spring and can be taken before or after Math 53.

**Stanford Math 52** is the integration half of multivariable calculus at Stanford. [Math 51](/courses/stanford-math-51) handles derivatives of functions of several variables, and Math 52 handles their integrals, ending with the great theorems of vector calculus. It is offered in the winter and spring quarters. This guide maps every Math 52 topic to free practice with instant feedback.

## What Stanford Math 52 covers

Stanford's catalog lists iterated integrals, line and surface integrals, and vector analysis with applications to vector potentials and conservative vector fields, along with physical interpretations. The course closes with the divergence theorem and the theorems of Green, Gauss and Stokes. The math department adds that it draws on the linear algebra and matrix derivatives from Math 51.

Compared with a standard **Calculus 3** course, Math 52 is the second half: multiple integrals and vector calculus.

## Math 52 topic map

| Math 52 topic | Practice | Read |
| --- | --- | --- |
| Iterated integrals | [Double integrals over rectangles](/learn/calculus-3/multiple-integrals/iterated-integrals) | [Double integrals guide](/blog/double-integrals) |
| Regions and order of integration | [Switching the order](/learn/calculus-3/multiple-integrals/switching-the-order) | [Double integrals guide](/blog/double-integrals) |
| Polar coordinates | [Polar double integrals](/learn/calculus-3/multiple-integrals/polar-double-integrals) | [Polar area](/learn/calculus-2/power-series-parametric-polar/polar-area) |
| Triple integrals | [Triple integrals](/learn/calculus-3/multiple-integrals/triple-integrals) | [Cylindrical and spherical](/learn/calculus-3/multiple-integrals/cylindrical-spherical) |
| Change of variables | [The Jacobian](/learn/calculus-3/multiple-integrals/change-of-variables) | [U-substitution](/blog/u-substitution) |
| Mass, average value and other applications | [Area, volume, mass and average value](/learn/calculus-3/multiple-integrals/double-integral-applications) | [Center of mass](/learn/calculus-2/improper-integrals-and-applications/center-of-mass) |
| 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) |
| Conservative fields and potentials | [Conservative fields](/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) | [Curl and divergence](/learn/calculus-3/vector-calculus/curl-and-divergence) |
| Divergence theorem and Stokes' theorem | [Stokes and divergence theorems](/learn/calculus-3/vector-calculus/stokes-and-divergence-theorems) | [Divergence theorem](/theorems/divergence-theorem), [Stokes' theorem](/theorems/stokes-theorem) |

## Math 52 vs Math 62CM vs Math 53

| Course | Units | Offered (2026-27) | Focus |
| --- | --- | --- | --- |
| Math 52 | 5 | Winter, spring | Multivariable integration and vector calculus |
| Math 62CM | 5 | Winter | Proof-based: manifolds, differential forms and the general Stokes' theorem |
| Math 53 | 5 | Autumn, winter, spring | Differential equations with linear algebra and Fourier methods |

Math 52 and Math 62CM are equivalent, and credit is not granted for both. Math 53 is independent of Math 52, so the two can go in either order after Math 51. To review the series background these courses assume, see the [Math 21 guide](/courses/stanford-math-21).

## Example problems at the Math 52 level

**1. Polar coordinates.** Integrate $e^{-(x^2 + y^2)}$ over the disk of radius 1:

$$\int_0^{2\pi}\int_0^1 e^{-r^2} r\,dr\,d\theta = 2\pi \cdot \frac{1 - e^{-1}}{2} = \pi(1 - e^{-1})$$

**2. A conservative field.** $\mathbf{F} = \langle 2xy, x^2 \rangle$ equals $\nabla(x^2 y)$, so along any path from $(0, 0)$ to $(2, 3)$:

$$\int_C \mathbf{F}\cdot d\mathbf{r} = x^2 y\Big|_{(0,0)}^{(2,3)} = 12$$

**3. The divergence theorem.** For $\mathbf{F} = \langle x, y, z \rangle$ through the sphere of radius 2, $\nabla\cdot\mathbf{F} = 3$, so the outward flux is

$$\iiint_E 3\,dV = 3 \cdot \frac{4}{3}\pi(2)^3 = 32\pi$$

## How to do well in Stanford Math 52

- **Spend your time on setup.** Sketch the region, write its bounds, and choose coordinates before integrating. Most lost points come from wrong limits, not wrong antiderivatives.
- **Check for shortcuts.** Before computing a line integral, test whether the field is conservative. Before a hard flux integral, ask whether the [divergence theorem](/theorems/divergence-theorem) makes it easier.
- **Keep orientation straight.** Green's and Stokes' theorems assume a specific direction around the boundary. Our [common calculus mistakes](/blog/common-calculus-mistakes) list covers sign errors like these.
- **Drill with [vector calculus practice](/learn/calculus-3/vector-calculus/vector-calculus-practice)**, keep the [formula sheet](/formulas) handy, and finish with timed [practice exams](/practice).

## Frequently asked questions

### What is Stanford Math 52?

Math 52, Integral Calculus of Several Variables, extends integration to several dimensions. It covers iterated integrals, line and surface integrals, conservative vector fields and vector potentials, and the divergence theorem with the theorems of Green, Gauss and Stokes.

### What do I need before Math 52?

The catalog lists Math 21 and Math 51, or equivalents. Math 52 uses the linear algebra and matrix derivatives from Math 51.

### Should I take Math 52 or Math 53 first?

The math department says Math 52 and 53 are independent and can be taken in either order after Math 51. Choose based on which one your major needs first.

### Is Math 52 hard?

Setting up integrals is the hard part: describing a region, choosing coordinates and orienting curves and surfaces. Once the setup is right, the computation is mostly single-variable integration.