Michigan Math 215 Multivariable Calculus: Study Guide and Practice

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 in the Michigan sequence

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 and the Math 116 guide.

Example problems at the Math 215 level

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

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:

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

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.
• Connect gradients to geometry. The gradient points in the direction of fastest increase and is perpendicular to level curves. Our gradient guide 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.
• Keep your single-variable skills sharp. Use the formula sheet and practice exams 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.