# Cornell MATH 2220 Multivariable Calculus: Study Guide and Practice

> A study guide for Cornell MATH 2220 Multivariable Calculus: several-variable derivatives and integrals, line and surface integrals and Gauss, with practice.

Source: https://practicecalculus.com/courses/cornell-math-2220
Updated: 2026-11-03

## At a glance

- **Course code:** MATH 2220
- **Credits:** 4
- **Recently offered:** Fall and spring
- **Prerequisites:** MATH 2210 Linear Algebra
- **Covers:** Differential and integral calculus of functions of several variables, line and surface integrals, the theorems of Green, Stokes and Gauss
- **Forbidden overlap:** MATH 1920 and MATH 2240
- **Practice here:** [Calculus 3](/learn/calculus-3)

## Key takeaways

- MATH 2220 is Cornell's multivariable calculus course for students who plan to major or minor in mathematics or a related field.
- It requires MATH 2210 Linear Algebra, which itself follows MATH 1110 and MATH 1120.
- It covers differential and integral calculus in several variables, line and surface integrals, and the theorems of Green, Stokes and Gauss.
- It is a forbidden overlap with MATH 1920, the engineering multivariable course.

**Cornell MATH 2220** is the multivariable calculus course on Cornell's math major track. It is designed for students who plan to major or minor in mathematics or a related field, and it comes after MATH 2210 Linear Algebra, so it can use matrices from day one. This guide maps the MATH 2220 topics to Calculus 3 practice with instant feedback.

## What Cornell MATH 2220 covers

The Cornell class roster describes MATH 2220 as an introduction to multivariable calculus covering differential and integral calculus of functions in several variables, line and surface integrals, and the theorems of Green, Stokes and Gauss.

That is the content of a **Calculus 3** course, taught to students who already know linear algebra. On PracticeCalculus it lines up with the partial derivatives, multiple integrals and vector calculus units, with the vectors unit as background.

## MATH 2220 topic map

| MATH 2220 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 | [Second partial derivatives](/learn/calculus-3/partial-derivatives/second-partial-derivatives) | [Clairaut's theorem](/theorems/clairauts-theorem) |
| Linear approximation and tangent planes | [Tangent planes](/learn/calculus-3/partial-derivatives/tangent-planes) | [Linear approximation](/blog/linear-approximation) |
| Chain rule and gradient | [The gradient](/learn/calculus-3/partial-derivatives/gradient-and-directional-derivatives) | [Gradient and directional derivatives](/blog/gradient-and-directional-derivatives) |
| Extrema and constraints | [Critical points and the second-derivative test](/learn/calculus-3/partial-derivatives/critical-points-and-second-derivative-test) | [Lagrange multipliers](/blog/lagrange-multipliers) |
| Multiple integrals | [Double integrals over general regions](/learn/calculus-3/multiple-integrals/general-regions) | [Double integrals](/blog/double-integrals) |
| 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) | [Green's theorem](/theorems/greens-theorem) |
| Surface integrals | [Surface integrals and flux](/learn/calculus-3/vector-calculus/surface-integrals-and-flux) | [Divergence Theorem](/theorems/divergence-theorem) |
| Green, Stokes and Gauss | [Stokes and the Divergence Theorem](/learn/calculus-3/vector-calculus/stokes-and-divergence-theorems) | [Stokes' theorem](/theorems/stokes-theorem) |

## The math major path to MATH 2220

| Course | Role | Prerequisite on the roster |
| --- | --- | --- |
| [MATH 1110 Calculus I](/courses/cornell-math-1110) | Single-variable calculus | Not listed on the roster page |
| [MATH 1120 Calculus II](/courses/cornell-math-1120) | Integration and series | MATH 1110 |
| MATH 2210 Linear Algebra | Vectors, matrices, eigenvalues, intro to proofs | MATH 1110 and MATH 1120 |
| MATH 2220 Multivariable Calculus | Calculus in several variables | MATH 2210 |

Engineering students take [MATH 1920](/courses/cornell-math-1920) instead, and the two are forbidden overlaps.

## Example problems at the MATH 2220 level

**1. A critical point.** For $f(x, y) = x^2 + xy + y^2 - 3x$, set $f_x = 2x + y - 3 = 0$ and $f_y = x + 2y = 0$. Then $x = -2y$, so $-3y = 3$, giving $y = -1$ and $x = 2$. Since $f_{xx}f_{yy} - f_{xy}^2 = 4 - 1 = 3 > 0$ and $f_{xx} = 2 > 0$, the point $(2, -1)$ is a local minimum with $f(2, -1) = -3$.

**2. Polar coordinates and the Jacobian.** The area of the unit disk, using the Jacobian $r$:

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

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

$$\iiint_B 3\,dV = 3 \cdot \frac{4}{3}\pi = 4\pi$$

## How to do well in MATH 2220

- **Use your linear algebra.** The derivative of a several-variable function is a matrix, and the Jacobian is its determinant. Thinking this way makes the [change of variables](/learn/calculus-3/multiple-integrals/change-of-variables) formula feel obvious.
- **Practice the setups.** Bounds and coordinate choices decide most multiple-integral problems. Work through [switching the order](/learn/calculus-3/multiple-integrals/switching-the-order) and [polar double integrals](/learn/calculus-3/multiple-integrals/polar-double-integrals).
- **See the three theorems as one.** [Green's theorem](/theorems/greens-theorem), [Stokes' theorem](/theorems/stokes-theorem) and the [Divergence Theorem](/theorems/divergence-theorem) all relate a boundary to an interior.
- **Keep single-variable skills fresh** with the [formula sheet](/formulas), and use [practice exams](/practice) before each prelim.

> **Tip:** when the second-derivative test gives $D = 0$, it tells you nothing. Look at the function along a few lines through the point instead.

## Frequently asked questions

### What is Cornell MATH 2220?

MATH 2220 is a 4-credit introduction to multivariable calculus for students planning to major or minor in math or a related field. It covers differential and integral calculus of functions of several variables, line and surface integrals, and the theorems of Green, Stokes and Gauss.

### Why does MATH 2220 require linear algebra?

The roster lists MATH 2210 as the prerequisite, so the course can treat derivatives of several-variable functions with matrices and linear maps. Being comfortable with matrices and determinants makes topics like the Jacobian much more natural.

### Should I take MATH 2220 or MATH 1920?

MATH 1920 is the engineering multivariable course after MATH 1910. MATH 2220 is aimed at math majors and minors and follows linear algebra. You can only receive credit for one of them, so choose based on your major.

### Is MATH 2220 harder than MATH 1920?

It covers similar core topics but for a mathematically inclined audience that has already taken linear algebra, so expect more emphasis on structure and reasoning. The computational skills are the same, and they still need regular practice.

### How should I prepare for MATH 2220?

Review integration techniques and polar coordinates from Calculus II, and make sure matrix multiplication and determinants from MATH 2210 are quick. Then get comfortable computing partial derivatives.