Stanford Math 51 Linear Algebra and Multivariable Calculus Guide

Stanford Math 51 is where Stanford students meet calculus in several variables, and it does so together with linear algebra. Taken after Math 21, it is the foundation for Math 52 and Math 53. This guide focuses on the multivariable calculus and vector topics, mapping each one to free practice with instant feedback, and notes where the linear algebra fits in.

What Stanford Math 51 covers

Stanford's catalog describes Math 51 as unified coverage of linear algebra and multivariable differential calculus, supported by a free course e-text. The linear algebra portion includes orthogonality, linear independence, matrix algebra and eigenvalues, with applications such as least squares, linear regression and Markov chains, and an introduction to the singular value decomposition. The calculus portion covers unconstrained optimization with gradients and Hessians, constrained optimization with Lagrange multipliers, gradient descent, the multivariable chain rule and Newton's method. The emphasis is on computation together with an intuitive grasp of the ideas.

Measured against a standard sequence, Math 51 covers the differential half of Calculus 3 plus most of an introductory linear algebra course. Multiple integrals and vector calculus come in Math 52.

Math 51 topic map

Matrix algebra, eigenvalues and the singular value decomposition are linear algebra topics we do not have lessons for yet.

Math 51 and its neighbors

Credit is not granted for both Math 51 and Math 61CM. After Math 51, read the Math 52 guide.

Example problems at the Math 51 level

1. A gradient. For f(x, y) = x^2 y + 3y^2, \nabla f = \langle 2xy, x^2 + 6y \rangle, so \nabla f(1, 2) = \langle 4, 13 \rangle.

2. A critical point with the Hessian. For f(x, y) = x^2 + xy + y^2, the gradient \langle 2x + y, x + 2y \rangle is zero only at (0, 0). The Hessian is

so (0, 0) is a local minimum.

3. Lagrange multipliers. Maximize f = xy subject to x + y = 10. Setting \langle y, x \rangle = \lambda\langle 1, 1 \rangle gives x = y = 5, so the maximum is 25.

How to do well in Stanford Math 51

• Tie every concept to a picture. Gradients point uphill, level sets are perpendicular to them, and eigenvectors are directions a matrix only stretches. Drawing these makes the algebra memorable.
• Practice the calculus separately. The partial derivatives practice set builds speed so exam time goes to the harder linear algebra reasoning.
• Review single-variable optimization with the second derivative lesson, since the Hessian test generalizes it.
• Use the formula sheet, timed practice exams, and the common calculus mistakes list as you prepare for midterms.