# UMN MATH 2263 Multivariable Calculus: Topics, Study Guide, Practice

> A study guide for MATH 2263 at the University of Minnesota: partial derivatives, Jacobians, multiple, line and surface integrals, and Stokes, with practice.

Source: https://practicecalculus.com/courses/university-of-minnesota-math-2263
Updated: 2026-11-03

## At a glance

- **Course code:** MATH 2263
- **Credits:** 4
- **Offered:** Typically every fall, spring and summer
- **Prerequisites:** MATH 1272, 1372 or 1572, with a grade of at least C-
- **Covers:** The derivative as a linear map, differential and integral calculus of several variables, Jacobians, line and surface integrals, the theorems of Gauss, Green and Stokes
- **Previous course:** MATH 1272 (Calculus II)
- **Practice here:** [Calculus 3](/learn/calculus-3)

## Key takeaways

- MATH 2263 is the 4-credit Multivariable Calculus course at the University of Minnesota Twin Cities, the third course in the calculus sequence.
- It treats the derivative as a linear map and develops differential and integral calculus for functions of several variables.
- The course finishes with Jacobians, line and surface integrals, and the theorems of Gauss, Green and Stokes.
- It accepts MATH 1272, 1372 or 1572 with at least a C- as the prerequisite.

**MATH 2263** is Multivariable Calculus at the University of Minnesota Twin Cities, the third course in the calculus sequence after MATH 1271 and MATH 1272. It is a 4-credit course typically offered every fall, spring and summer, and it is a core requirement for most engineering, physics and math majors. This guide maps every MATH 2263 topic to free practice, so you can build each skill by solving problems with instant feedback.

## What MATH 2263 covers

The University of Minnesota catalog describes MATH 2263 as starting from the derivative viewed as a linear map, then building differential and integral calculus for functions of several variables. It covers change of coordinates using Jacobians, line integrals and surface integrals, and ends with the theorems of Gauss (the Divergence Theorem), Green and Stokes.

That is a full standard **Calculus 3** course, including the vector calculus that some universities split into a separate fourth course. The "linear map" framing is worth noticing: the derivative of a function of several variables is a matrix, and the tangent plane is the best linear approximation.

## MATH 2263 topic map

| MATH 2263 topic | Practice | Read |
| --- | --- | --- |
| Functions of several variables | [Functions of several variables](/learn/calculus-3/partial-derivatives/functions-of-several-variables) | [Partial derivatives](/blog/partial-derivatives) |
| Partial derivatives | [Partial derivatives](/learn/calculus-3/partial-derivatives/partial-derivatives) | [Clairaut's theorem](/theorems/clairauts-theorem) |
| Derivative as a linear map | [Tangent planes](/learn/calculus-3/partial-derivatives/tangent-planes) | [Linear approximation](/blog/linear-approximation) |
| 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 and directional derivatives](/blog/gradient-and-directional-derivatives) |
| Extrema and constraints | [Lagrange multipliers](/learn/calculus-3/partial-derivatives/lagrange-multipliers) | [Lagrange multipliers](/blog/lagrange-multipliers) |
| Double and triple integrals | [Double integrals over rectangles](/learn/calculus-3/multiple-integrals/iterated-integrals) | [Double integrals](/blog/double-integrals) |
| Polar, cylindrical and spherical coordinates | [Cylindrical and spherical](/learn/calculus-3/multiple-integrals/cylindrical-spherical) | [Polar double integrals](/learn/calculus-3/multiple-integrals/polar-double-integrals) |
| Jacobians and change of variables | [Change of variables and the Jacobian](/learn/calculus-3/multiple-integrals/change-of-variables) | [Double integrals](/blog/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) |
| 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 | [Stokes and the Divergence Theorem](/learn/calculus-3/vector-calculus/stokes-and-divergence-theorems) | [Stokes' theorem](/theorems/stokes-theorem) |

## Where MATH 2263 fits

| Course | Role in the sequence | Credits |
| --- | --- | --- |
| MATH 1271 Calculus I | Single-variable derivatives and first integrals | 4 |
| MATH 1272 Calculus II | Integration techniques, polar, Taylor polynomials, vectors in space | 4 |
| MATH 2263 Multivariable Calculus | Several variables, Jacobians, vector calculus | 4 |

MATH 2263 also accepts MATH 1372 and MATH 1572 in place of MATH 1272, so students arriving from other first-year tracks join the same course. Revisit the [MATH 1272 guide](/courses/university-of-minnesota-math-1272) if vectors and cylindrical coordinates feel rusty, or start from the [MATH 1271 guide](/courses/university-of-minnesota-math-1271).

## Example problems at the MATH 2263 level

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

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

**2. A polar double integral.** Find $\iint_D (x^2 + y^2)\,dA$ where $D$ is the disk of radius 2. In polar coordinates $x^2 + y^2 = r^2$ and $dA = r\,dr\,d\theta$:

$$\int_0^{2\pi}\int_0^2 r^3\,dr\,d\theta = 2\pi \cdot \frac{2^4}{4} = 8\pi$$

**3. Green's theorem.** For $\mathbf{F} = \langle -y, x \rangle$ around the unit circle counterclockwise, $\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} = 1 - (-1) = 2$, so

$$\oint_C \mathbf{F}\cdot d\mathbf{r} = \iint_D 2\,dA = 2\pi$$

## How to do well in MATH 2263

- **Sketch every region.** Setting up limits of integration is where most points are lost. Draw the region, then decide the order. The [switching the order](/learn/calculus-3/multiple-integrals/switching-the-order) lesson trains exactly this.
- **Know when to change coordinates.** Disks suggest polar, cylinders suggest cylindrical, balls suggest spherical. Do not forget the extra $r$ or $\rho^2\sin\phi$.
- **Tie the big theorems together.** Green, Stokes and Gauss all relate a boundary to its interior. Reading the [Green's theorem](/theorems/greens-theorem) and [Stokes' theorem](/theorems/stokes-theorem) pages side by side makes the pattern clear.
- **Keep formulas in view** with the [calculus formula sheet](/formulas), and test yourself with timed [practice](/practice) before each exam.

> **Tip:** before computing a line integral, check whether the field is conservative. If it is, the [potential function](/learn/calculus-3/vector-calculus/conservative-fields) turns the integral into a simple subtraction.

## Frequently asked questions

### What is MATH 2263 at the University of Minnesota?

MATH 2263 is Multivariable Calculus, the Calculus 3 course on the Twin Cities campus. It covers derivatives and integrals of functions of several variables, change of coordinates with Jacobians, line and surface integrals, and the Gauss, Green and Stokes theorems.

### What are the prerequisites for MATH 2263?

You need MATH 1272, MATH 1372 or MATH 1572, completed with a grade of at least C-.

### Is MATH 2263 harder than Calculus II?

It is different rather than simply harder. The algebra is often lighter than in Calculus II integration, but you need to picture surfaces and regions in three dimensions. Sketching every region before setting up an integral helps a lot.

### How should I prepare for MATH 2263?

Review vectors, the dot and cross product, and polar coordinates from Calculus II, then start practicing partial derivatives. The vectors and partial derivatives units of the Calculus 3 path on PracticeCalculus follow that order.