# Michigan Math 215 Multivariable Calculus: Study Guide and Practice

> A study guide for University of Michigan Math 215 (Calculus III): vectors, partial derivatives, multiple integrals and vector calculus, with practice.

Source: https://practicecalculus.com/courses/university-of-michigan-math-215
Updated: 2026-11-03

## At a glance

- **Course code:** Math 215
- **Comes after:** Math 116 (Calculus II), which names Math 215 as its natural sequel
- **Covers:** Vectors and 3D geometry, parametric curves, partial derivatives, gradients, constrained optimization, double and triple integrals, vector fields, line and surface integrals, and the theorems of vector calculus
- **Alternatives:** Math 205 (Calculus of Several Variables) and Math 285 (Honors Multivariable and Vector Calculus)
- **Next course:** Math 214 or Math 217 (linear algebra), or Math 216 (differential equations)
- **Practice here:** [Calculus 3](/learn/calculus-3)

## Key takeaways

- Math 215 completes Michigan's standard calculus sequence after Math 115 and Math 116.
- It extends calculus to higher dimensions: vectors, functions of several variables, multiple integrals and vector fields.
- The course ends with Green's theorem, Stokes' theorem and the divergence theorem.
- The emphasis is on concepts and solving problems rather than theory and proof.

**Michigan Math 215** is Calculus III at the University of Michigan, a first course in multivariable and vector calculus. It completes the standard sequence that starts with Math 115 (mostly differential calculus) and Math 116 (mostly integral calculus). This guide maps every Math 215 topic to free practice, so you can build intuition for three dimensions one problem at a time.

## What Michigan Math 215 covers

The official course site describes Math 215 as the extension of calculus to higher dimensions. That requires three-dimensional geometry, vectors and vector functions, functions of two and three variables, vector fields, surfaces and the theorems of vector calculus. The focus is on concepts and problem solving rather than theory and proof.

The published topic list runs from 3D coordinates and vectors through lines, planes and quadric surfaces, parametric curves, partial derivatives and Clairaut's theorem, tangent planes, the chain rule, gradients, optimization with Lagrange multipliers, double and triple integrals in several coordinate systems, line and surface integrals, and finally Green's, Stokes' and the divergence theorems. That is a complete standard **Calculus 3** course, including the full vector calculus unit that some universities move to a separate class.

## Math 215 topic map

| Math 215 topic | Practice | Read |
| --- | --- | --- |
| 3D coordinates and vectors | [Vectors and their length](/learn/calculus-3/vectors-and-space/vectors-and-length) | [Dot and cross product](/blog/vectors-dot-and-cross-product) |
| Lines, planes and surfaces | [Planes in space](/learn/calculus-3/vectors-and-space/planes-in-space) | [Dot and cross product](/blog/vectors-dot-and-cross-product) |
| Parametric curves, velocity, arc length | [Curves and motion](/learn/calculus-3/vectors-and-space/curves-and-motion) | [Arc length of parametric curves](/learn/calculus-2/power-series-parametric-polar/parametric-arc-length) |
| Partial derivatives and Clairaut's theorem | [Second partial derivatives](/learn/calculus-3/partial-derivatives/second-partial-derivatives) | [Partial derivatives](/blog/partial-derivatives), [Clairaut's theorem](/theorems/clairauts-theorem) |
| Tangent planes and linear approximation | [Tangent planes](/learn/calculus-3/partial-derivatives/tangent-planes) | [Linear approximation](/blog/linear-approximation) |
| Chain rule and implicit differentiation | [The 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 guide](/blog/gradient-and-directional-derivatives) |
| Optimization and Lagrange multipliers | [Lagrange multipliers](/learn/calculus-3/partial-derivatives/lagrange-multipliers) | [Lagrange multipliers](/blog/lagrange-multipliers) |
| Double integrals, Cartesian 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) |
| Vector fields, curl and divergence | [Curl and divergence](/learn/calculus-3/vector-calculus/curl-and-divergence) | [Divergence theorem](/theorems/divergence-theorem) |
| Line integrals and conservative fields | [Conservative fields](/learn/calculus-3/vector-calculus/conservative-fields) | [Green's theorem](/theorems/greens-theorem) |
| Surface integrals, Stokes and divergence theorems | [Stokes' theorem and the Divergence Theorem](/learn/calculus-3/vector-calculus/stokes-and-divergence-theorems) | [Stokes' theorem](/theorems/stokes-theorem) |

## Math 215 in the Michigan sequence

| Course | Main focus |
| --- | --- |
| Math 115 (Calculus I) | Single variable, largely differential calculus |
| Math 116 (Calculus II) | Single variable, largely integral calculus and series |
| Math 215 (Calculus III) | Multivariable and vector calculus |
| Math 205 or Math 285 | Alternatives covering some of the same material in different formats |

After Math 215, students usually choose linear algebra (Math 214 or Math 217) or differential equations (Math 216). Review the earlier courses in the [Math 115 guide](/courses/university-of-michigan-math-115) and the [Math 116 guide](/courses/university-of-michigan-math-116).

## Example problems at the Math 215 level

**1. Partial derivatives.** For $f(x, y) = x^2 y + y^3$:

$$f_x = 2xy, \qquad f_y = x^2 + 3y^2$$

**2. A directional derivative.** For $f(x, y) = x^2 + y^2$ at $(1, 2)$, $\nabla f = \langle 2x, 2y \rangle = \langle 2, 4 \rangle$. In the unit direction $\mathbf{u} = \langle \frac{3}{5}, \frac{4}{5} \rangle$:

$$D_{\mathbf{u}} f = 2 \cdot \frac{3}{5} + 4 \cdot \frac{4}{5} = \frac{22}{5}$$

**3. Green's theorem.** For the counterclockwise unit circle $C$, with $P = -y$ and $Q = x$, $\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} = 2$, so

$$\oint_C -y\,dx + x\,dy = \iint_D 2\,dA = 2\pi$$

## How to do well in Michigan Math 215

- **Draw the region first.** Most errors in double and triple integrals come from wrong limits. Sketch, then practice with [switching the order](/learn/calculus-3/multiple-integrals/switching-the-order).
- **Connect gradients to geometry.** The gradient points in the direction of fastest increase and is perpendicular to level curves. Our [gradient guide](/blog/gradient-and-directional-derivatives) shows why.
- **Know which theorem applies.** Closed curve in the plane: Green. Surface with a boundary curve: Stokes. Closed surface: divergence theorem. Drill this in [vector calculus practice](/learn/calculus-3/vector-calculus/vector-calculus-practice).
- **Keep your single-variable skills sharp.** Use the [formula sheet](/formulas) and [practice exams](/practice) so integration slips do not cost you on multivariable problems.

> **Tip:** before testing whether a field is conservative, compute its curl. If the curl is not zero, no potential function exists and you can stop.

## Frequently asked questions

### What is Math 215 at the University of Michigan?

Math 215 is Calculus III, a first course in multivariable and vector calculus. It covers three-dimensional geometry, vectors and vector functions, functions of two and three variables, multiple integrals, vector fields, surfaces and the theorems of vector calculus.

### What comes after Math 215?

For linear algebra, Math 214 (Applied Linear Algebra) or Math 217 (Linear Algebra) are the suggested next courses, with Math 217 recommended for students considering further mathematics. Students who need differential equations can take Math 216.

### Is Michigan Math 215 hard?

The new challenge is visualizing in three dimensions and keeping track of several variables at once. Students with solid Calculus 1 and 2 skills usually find the computations familiar, so most of the effort goes into setting up integrals and reading geometry correctly.

### How can I prepare for Math 215?

Make sure derivatives, the chain rule and integration techniques from Math 115 and Math 116 are automatic. Then preview vectors, the dot and cross product, and partial derivatives with short practice problems.

### What is the difference between Math 215, Math 205 and Math 285?

All three cover some of the same material from different perspectives and in different formats. Math 285 is the honors version, and Math 205 is Calculus of Several Variables. The department suggests prospective math majors talk to an advisor about which fits best.