# Cornell MATH 1920 Multivariable Calculus: Study Guide and Practice

> A study guide for Cornell MATH 1920 Multivariable Calculus for Engineers: partials, multiple integrals, line and surface integrals and Stokes, with practice.

Source: https://practicecalculus.com/courses/cornell-math-1920
Updated: 2026-11-03

## At a glance

- **Course code:** MATH 1920
- **Credits:** 4
- **Recently offered:** Fall, spring and summer
- **Prerequisites:** MATH 1910 or equivalent
- **Covers:** Partial derivatives, double and triple integrals, line and surface integrals, vector fields, Green's theorem, Stokes' theorem, the divergence theorem
- **Forbidden overlap:** MATH 2220 and MATH 2240
- **Practice here:** [Calculus 3](/learn/calculus-3)

## Key takeaways

- MATH 1920 is the multivariable calculus course in Cornell's engineering sequence, taken after MATH 1910.
- It covers partial derivatives, double and triple integrals, vector fields, line and surface integrals, and the theorems of Green, Stokes and Gauss.
- It is a 4-credit course and a forbidden overlap with MATH 2220 and MATH 2240.

**Cornell MATH 1920**, Multivariable Calculus for Engineers, is the second course in Cornell's engineering math sequence. Engineering students take it after MATH 1910, and it supplies the vector calculus used in physics, fluids, electromagnetism and mechanics. This guide maps each MATH 1920 topic to Calculus 3 practice with instant feedback.

## What Cornell MATH 1920 covers

The Cornell class roster describes MATH 1920 as an introduction to multivariable calculus. Topics include partial derivatives, double and triple integrals, line and surface integrals, vector fields, Green's theorem, Stokes' theorem and the divergence theorem.

That is a standard **Calculus 3**, weighted toward integration and vector calculus. It lines up with the partial derivatives, multiple integrals and vector calculus units on PracticeCalculus. Vectors and the geometry of space are the natural warm-up.

## MATH 1920 topic map

| MATH 1920 topic | Practice | Read |
| --- | --- | --- |
| Vectors in space (warm-up) | [The cross product](/learn/calculus-3/vectors-and-space/cross-product) | [Dot and cross product](/blog/vectors-dot-and-cross-product) |
| Partial derivatives | [Partial derivatives](/learn/calculus-3/partial-derivatives/partial-derivatives) | [Partial derivatives guide](/blog/partial-derivatives) |
| Chain rule and gradients | [The multivariable chain rule](/learn/calculus-3/partial-derivatives/multivariable-chain-rule) | [Gradient and directional derivatives](/blog/gradient-and-directional-derivatives) |
| Optimization | [Lagrange multipliers](/learn/calculus-3/partial-derivatives/lagrange-multipliers) | [Lagrange multipliers](/blog/lagrange-multipliers) |
| Double integrals | [Double integrals over 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) | [Double integrals](/blog/double-integrals) |
| Cylindrical and spherical coordinates | [Cylindrical and spherical](/learn/calculus-3/multiple-integrals/cylindrical-spherical) | [Double integrals](/blog/double-integrals) |
| Vector fields | [Vector fields](/learn/calculus-3/vector-calculus/vector-fields) | [Gradient and directional derivatives](/blog/gradient-and-directional-derivatives) |
| Line integrals | [Conservative fields](/learn/calculus-3/vector-calculus/conservative-fields) | [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) |
| Green, Stokes and divergence theorems | [Stokes and the Divergence Theorem](/learn/calculus-3/vector-calculus/stokes-and-divergence-theorems) | [Stokes' theorem](/theorems/stokes-theorem) |

## MATH 1920 vs MATH 2220

| Course | Audience | Prerequisite on the roster |
| --- | --- | --- |
| MATH 1920 Multivariable Calculus for Engineers | Engineering sequence | MATH 1910 or equivalent |
| [MATH 2220 Multivariable Calculus](/courses/cornell-math-2220) | Students planning a math major or minor or a related field | MATH 2210 Linear Algebra |

The two are forbidden overlaps. For the course before MATH 1920, see the [MATH 1910 guide](/courses/cornell-math-1910).

## Example problems at the MATH 1920 level

**1. A triple integral.** Over the unit cube $0 \le x, y, z \le 1$:

$$\iiint (x + y + z)\,dV = \frac{1}{2} + \frac{1}{2} + \frac{1}{2} = \frac{3}{2}$$

since each term integrates to $\frac{1}{2}$ over the cube.

**2. A conservative line integral.** $\mathbf{F} = \langle y, x \rangle$ is the gradient of $f(x, y) = xy$. Along any path from $(0, 0)$ to $(1, 2)$,

$$\int_C \mathbf{F} \cdot d\mathbf{r} = f(1, 2) - f(0, 0) = 2$$

**3. Green's theorem.** For $C$ the circle of radius $2$ traversed counterclockwise,

$$\oint_C (-y\,dx + x\,dy) = \iint_D \left(1 - (-1)\right) dA = 2 \cdot \pi (2)^2 = 8\pi$$

## How to do well in MATH 1920

- **Sketch the region first, every time.** Then practice [switching the order](/learn/calculus-3/multiple-integrals/switching-the-order) so you can choose the easier setup.
- **Learn the theorem map.** [Green's theorem](/theorems/greens-theorem), [Stokes' theorem](/theorems/stokes-theorem) and the [Divergence Theorem](/theorems/divergence-theorem) each turn a boundary integral into an interior one. Know which applies to curves and which to surfaces.
- **Check for conservative fields** before grinding out a line integral. The [conservative fields](/learn/calculus-3/vector-calculus/conservative-fields) lesson shows the test.
- **Review single-variable integration** with the [formula sheet](/formulas), and rehearse with timed [practice exams](/practice).

> **Tip:** the Jacobian factors are the most forgotten piece of multivariable calculus: $r$ in polar and cylindrical, $\rho^2 \sin\phi$ in spherical. Write them before the integrand.

## Frequently asked questions

### What is Cornell MATH 1920?

MATH 1920 Multivariable Calculus for Engineers is an introduction to multivariable calculus. It covers partial derivatives, double and triple integrals, line and surface integrals, vector fields, Green's theorem, Stokes' theorem and the divergence theorem.

### What do I need before MATH 1920?

The roster lists MATH 1910 or equivalent. You should be comfortable with integration techniques and with polar coordinates, which come up in double integrals.

### What is the difference between MATH 1920 and MATH 2220?

Both are multivariable calculus courses and you can only get credit for one. MATH 1920 is part of the engineering sequence after MATH 1910, while MATH 2220 is aimed at students who plan to major or minor in mathematics or a related field and requires MATH 2210 Linear Algebra.

### Is MATH 1920 hard?

The new challenge is visualizing regions and surfaces in three dimensions. The computations themselves reuse single-variable integration, so students who sketch carefully and practice setting up bounds usually do well.

### How do I study for MATH 1920 exams?

Practice setting up integrals more than evaluating them, since bounds and coordinate choices are where points are lost. Then review when each big theorem applies and what it converts into what.