Links:    Home     Homework     Calendar     Syllabus     Coordinates    

Math 31BH Winter Quarter 2017

Updated 03/07/17

Note: This calendar is subject to revision during the term.
 The section references are only a guide; your instructor may vary from it.

Discussions:  Thursdays, 7-7:50 and 8-8:50 PM in CENTR 218 (start Jan 12th)

Lecture summaries:

  • Lec. 1: Metrics, open balls, open and closed subsets. (83-85)

  • Lec. 2: Interior, closure, and boundary. Convergent sequences in R^n. (86-89)

  • Lec. 3: Closed sets and limit points, compact sets, convergent subsequences. (90-91, 104-107)

  • Lec. 4: Limits of functions, epsilon-delta, continuous functions. (92-96)

  • Lec. 5: Preimages of open sets, uniform continuity on compact sets, matrix norm. (97-98, 110)

  • Lec. 6: Supremum, infimum, existence of max/min on compact sets. (108-109)

  • Lec. 7: Differentiation in one variable, the mean value theorem. (111, 119-120, 123-124)

  • Lec. 8: Partial derivatives, Jacobi matrix, differentiable functions in several variables. (121-122, 125-126)

  • Lec. 9: Total derivative, differentiable implies continuous, directional derivatives, gradient. (127-129)

  • Lec. 10: Directional derivatives, mean value theorem, a differentiable function which is not C^1. (129-131, 145-148)

  • Lec. 11: Strong mean value theorem (n variables), continuously differentiable implies differentiable. (149-151)

  • Lec. 12: Differentiation rules (sums, products, dot products). Composition, the chain rule. (137-142)

  • Lec. 13: Proof of the chain rule, examples, application to polar coordinates. (715-716)

  • Lec. 14: Geometry of the gradient, critical points, saddle points, fastest increase, level curves. (305-306, 342-343)

  • Lec. 15: Mixed partial derivatives, functions of class C^p, for p=2 second order partials agree. (149, 318)

  • Lec. 16: Proof mixed partials agree, class C^p, multi-index notation D_I etc. (315-318, 732-733)

  • Lec. 17: Summary of Taylor polynomials in one variable, Taylor's remainder theorem (n=1). (740-742)

  • Lec. 18: Proof of Taylor's theorem, intermediate value theorem, mean-value theorem for integrals. (740-742)

  • Lec. 19: Taylor polynomials in n variables, Taylor's theorem generalized to n>1. (319-323, 743-744)

  • Lec. 20: Local extrema, saddle points, quadratic forms, definiteness, Hessian matrix (333-340, 342-348)

  • Lec. 21: Spectral theorem, Hessian matrix, Rayleigh quotient, nature of critical points, the case n=2 (342-348, 363-365)

  • Lec. 22: The inverse function theorem, its statement and background. Local injectivity (258-266 + notes)

  • Lec. 23: Proof of the inverse function theorem (open image, inverse is C^1), examples (258-266 + notes)

  • Lec. 24: The implicit function theorem (with proof) (267-275)

  • Lec. 25: Lagrange multiplies, extrema subject to constraints, examples (349-362 + notes)

  • Lec. 26: Review (or Q&A) for the Final Exam