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Ted Shifrin’s Linear algebra and Multivariable calculus

Jul 29, 2022, 3 min read

parent:: inculcation

This fairly formal but very entertaining set of lectures (where the instructor uses blackboard!) explains all the stuff in Multivariable calculus one encounters during the core years of a Fundamental Sciences programme: multiple derivatives, volumous integrals, change of coordinates, and especially does everything in a proper linear algebraic setting (which becomes very important in the long run) and with lots of examples and pictures!

Lecture videos, notes and textbook

Outline

This two-semester class is a theoretical and more challenging amalgamation … Proofs of theorems and additional physical applications will be stressed, and harder computational problems will be included.

The set of lectures is actually a two semester course, with mixed together description of various methods in Linear algebra and Calculus in ℝⁿ.

First semester

  1. Vector algebra and geometry, matrices, linear maps, determinants.
  2. Functions, limits, continuity; the derivative.
  3. Solving linear systems: Gaussian elimination, linear independence, basis and dimension. What is a manifold?
  4. Maximum/minimum problems, quadratic forms, projections.

Second semester

  1. Integration, applications to physics, determinants and the change of variables theorem.
  2. Nonlinear problems and manifolds.
  3. Differential forms and integration on manifolds. Stokes’s Theorem. Applications to physics (div, curl, and all that) and topology.
  4. Eigenvalues and eigenvectors, difference and differential equations, spectral theorem and applications to quadratic forms.

Instructor’s description

This is an integrated year-long course in multivariable calculus and linear algebra. It includes all the material in MATH 2270/2500 and MATH 3000, along with additional applications and theoretical material. There is greater emphasis on proofs, and the pace is quick. Typically the class consists of a blend of sophomores and freshmen who’ve earned a 5 on the AP Calculus BC exam. The text is my book, Multivariable Mathematics: Linear Algebra, Multivariable Calculus, and Manifolds.

Students who would like some guidance in reading and writing proofs might want to look at a wonderful new book called How to Think Like a Mathematician: A Companion to Undergraduate Mathematics, by Kevin Houston, Cambridge University Press, 2009.

Homework

First semester

Second semester

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