Cornell MATH 2220 Multivariable Calculus: Study Guide and Practice

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

The math major path to MATH 2220

Engineering students take 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:

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

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 formula feel obvious.
• Practice the setups. Bounds and coordinate choices decide most multiple-integral problems. Work through switching the order and polar double integrals.
• See the three theorems as one. Green's theorem, Stokes' theorem and the Divergence Theorem all relate a boundary to an interior.
• Keep single-variable skills fresh with the formula sheet, and use practice exams 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.