# Ohio State MATH 2153 Calculus III: Topics, Study Guide, Practice

> A study guide for Ohio State MATH 2153 (Calculus III): vectors, partial derivatives, multiple integrals, Green's, Stokes' and Divergence theorems.

Source: https://practicecalculus.com/courses/ohio-state-math-2153
Updated: 2026-11-03

## At a glance

- **Course code:** MATH 2153
- **Credit hours:** 4
- **Offered:** Autumn, spring and summer
- **Prerequisites:** C- or above in MATH 1152, 1172, 1534, 1544, 1181H or 4181H, or credit for 153.xx, 154, 162.xx or 162.01H
- **Covers:** Multivariable differential and integral calculus: vectors and curves, partial derivatives, gradient, Lagrange multipliers, multiple integrals, line and surface integrals, vector calculus theorems
- **Practice here:** [Calculus 3](/learn/calculus-3)

## Key takeaways

- MATH 2153 is Ohio State's 4-credit Calculus III course, offered in autumn, spring and summer.
- It covers multivariable differential and integral calculus, from vectors and curves to the Divergence Theorem.
- Students can reach it from either MATH 1152 or the engineering course MATH 1172.
- Our Calculus 3 path covers the same four areas: vectors, partial derivatives, multiple integrals and vector calculus.

**Ohio State MATH 2153**, Calculus III, is the multivariable calculus course at Ohio State. It is a 4-credit course offered in autumn, spring and summer, and it is the next step after either MATH 1152 or the engineering course MATH 1172. This guide maps every MATH 2153 topic to practice with instant feedback.

## What Ohio State MATH 2153 covers

The catalog describes MATH 2153 simply as multivariable differential and integral calculus. The department's course description document lists four chapters of topics.

**Vectors and vector-valued functions:** vectors in the plane and in space, dot and cross products, lines and curves in space, calculus of vector-valued functions, motion, arc length and curvature. **Functions of several variables:** planes and surfaces, level curves, limits and continuity, partial derivatives, the chain rule, directional derivatives and the gradient, tangent planes, max/min problems and Lagrange multipliers. **Multiple integration:** double integrals over rectangles and general regions, polar coordinates, triple integrals in cylindrical and spherical coordinates, mass calculations and change of variables. **Vector calculus:** vector fields, line integrals, conservative fields, Green's theorem, divergence and curl, surface integrals, Stokes' theorem and the Divergence Theorem.

That is a complete **Calculus 3** course.

## MATH 2153 topic map

| MATH 2153 topic | Practice | Read |
| --- | --- | --- |
| Vectors, dot and cross products | [The cross product](/learn/calculus-3/vectors-and-space/cross-product) | [Dot and cross product](/blog/vectors-dot-and-cross-product) |
| Lines and curves in space | [Lines in space](/learn/calculus-3/vectors-and-space/lines-in-space) | [Dot and cross product](/blog/vectors-dot-and-cross-product) |
| Motion and arc length | [Curves and motion](/learn/calculus-3/vectors-and-space/curves-and-motion) | [Parametric arc length](/learn/calculus-2/power-series-parametric-polar/parametric-arc-length) |
| Graphs and level curves | [Functions of several variables](/learn/calculus-3/partial-derivatives/functions-of-several-variables) | [Partial derivatives](/blog/partial-derivatives) |
| Partial derivatives and the chain rule | [The multivariable chain rule](/learn/calculus-3/partial-derivatives/multivariable-chain-rule) | [Partial derivatives](/blog/partial-derivatives), [Clairaut's theorem](/theorems/clairauts-theorem) |
| Gradient and tangent planes | [Gradient and directional derivatives](/learn/calculus-3/partial-derivatives/gradient-and-directional-derivatives) | [Gradient and directional derivatives](/blog/gradient-and-directional-derivatives) |
| Max/min problems | [Critical points and the second-derivative test](/learn/calculus-3/partial-derivatives/critical-points-and-second-derivative-test) | [Optimization problems](/blog/optimization-problems) |
| Lagrange multipliers | [Lagrange multipliers](/learn/calculus-3/partial-derivatives/lagrange-multipliers) | [Lagrange multipliers](/blog/lagrange-multipliers) |
| Double integrals and polar | [Double integrals in polar coordinates](/learn/calculus-3/multiple-integrals/polar-double-integrals) | [Double integrals](/blog/double-integrals) |
| Triple integrals, cylindrical and spherical | [Cylindrical and spherical coordinates](/learn/calculus-3/multiple-integrals/cylindrical-spherical) | [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 and conservative fields | [Conservative fields](/learn/calculus-3/vector-calculus/conservative-fields) | [Green's theorem](/theorems/greens-theorem) |
| Divergence and curl | [Curl and divergence](/learn/calculus-3/vector-calculus/curl-and-divergence) | [Divergence theorem](/theorems/divergence-theorem) |
| Surface integrals, Stokes and Divergence | [Surface integrals and flux](/learn/calculus-3/vector-calculus/surface-integrals-and-flux) | [Stokes' theorem](/theorems/stokes-theorem) |

## How students reach MATH 2153

| Path | Courses | Credit hours |
| --- | --- | --- |
| Standard | MATH 1151, then MATH 1152, then MATH 2153 | 5 + 5 + 4 |
| Engineering | MATH 1151, then MATH 1172, then MATH 2153 | 5 + 5 + 4 |

Students coming from MATH 1172 have already seen partial derivatives and the gradient. Students coming from MATH 1152 have seen more series and polar coordinates. See the [MATH 1152 guide](/courses/ohio-state-math-1152) and the [MATH 1172 guide](/courses/ohio-state-math-1172) to fill any gaps.

## Example problems at the MATH 2153 level

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

$$z = 2 + 2(x - 1) + 2(y - 1)$$

**2. Curl of a vector field.** For $\mathbf{F} = \langle -y, x, 0 \rangle$,

$$\nabla \times \mathbf{F} = \left\langle 0 - 0,\ 0 - 0,\ 1 - (-1) \right\rangle = \langle 0, 0, 2 \rangle$$

**3. A triple integral.** The volume of the unit cube is $\int_0^1\int_0^1\int_0^1 1\,dz\,dy\,dx = 1$. With the density $\rho = z$, the mass is

$$\int_0^1\int_0^1\int_0^1 z\,dz\,dy\,dx = \frac{1}{2}$$

## How to do well in Ohio State MATH 2153

- **Sketch every region and surface.** Wrong bounds are the most common lost points. Practice [switching the order](/learn/calculus-3/multiple-integrals/switching-the-order) to build that skill.
- **Learn the coordinate systems together.** Polar, cylindrical and spherical each add a factor to the volume element. The [formula sheet](/formulas) lists them side by side.
- **See the big picture of vector calculus.** Read [Green's theorem](/theorems/greens-theorem) first, then [Stokes' theorem](/theorems/stokes-theorem) and the [Divergence theorem](/theorems/divergence-theorem) as generalizations.
- **Test yourself** with timed [practice exams](/practice), and skim [Calculus 1 vs 2 vs 3](/blog/calculus-1-vs-2-vs-3) to see how the courses connect.

> **Tip:** for a conservative field, a line integral depends only on the endpoints. Checking for a potential function first can save a page of work.

## Frequently asked questions

### What is MATH 2153 at Ohio State?

MATH 2153, Calculus III, is Ohio State's multivariable calculus course. It covers vectors and vector-valued functions, functions of several variables and partial derivatives, multiple integrals, and vector calculus through Green's, Stokes' and the Divergence Theorem.

### What do I need for MATH 2153?

A C- or above in MATH 1152, 1172, 1534, 1544, 1181H or 4181H, or credit for one of the older quarter-system equivalents listed on the course page.

### Is MATH 2153 harder than MATH 1152?

It is different rather than strictly harder. The challenge shifts from choosing integration techniques to visualizing in three dimensions and setting up integrals with the right bounds.

### I took MATH 1172. Am I ready for MATH 2153?

Yes, MATH 1172 is an accepted prerequisite. MATH 1172 already introduces partial derivatives and the gradient, so you will have seen the first part of the differential calculus unit.

### How should I prepare for MATH 2153?

Review integration techniques and polar coordinates, and practice the dot and cross products. The vectors unit of our Calculus 3 path is a good starting point.