Theoretical multivariable calculus (Math 25B)

Summary

This is a rigorous course on multivariable calculus with an emphasis on proof-writing and abstract mathematical reasoning. The main topics are differentiation, integration, and rigorous proofs of the classical theorems of calculus in a broad setting.

Textbooks

• Spivak, Calculus on manifolds
• Munkres, Analysis on manifolds

Topic schedule

• Week 1 (Spivak Calculus Chapters 5-7)

  • Mon: Introduction, functions and limits

  • Wed: Limits and continuity (algebra of limits)

  • Fri: Continuity theorems (intermediate value, boundedness, maximum value theorems)

• Week 2 (Spivak Calculus Chapter 8, Spivak CoM Chapter 1, Munkres Sections 1.3 and 1.4)

  • Mon: Least upper bound property (ordered fields, IVT, boundedness theorem)

  • Wed: Compactness (subsets of Euclidean space)

  • Fri: Heine-Borel theorem (onion rings, closed rectangles are compact, boundedness theorem)

• Week 3 (Spivak Calculus Chapters 9-11)

  • Mon: Derivative in 1-dimension (definition, examples, chain rule)

  • Wed: Computing derivatives (chain rule, Rolle’s theorem, mean value theorem)

  • Fri: Polynomial approximations (L’Hospital’s rule, Taylor polynomials)

• Week 4 (Spivak CoM Chapter 2, Munkres Sections 5-7)

  • Mon: Derivatives in R^n (directional derivatives, the derivative, examples)

  • Wed: Partial derivatives, (continuous partials theorem, C^1 functions, higher-order derivatives)

  • Fri: Chain rule, (applications: 1-d derivative rules, multivariable MVT, derivative of inverse)

• Week 5 (Spivak CoM Chapter 2, Munkres Sections 8-9)

  • Mon: Presidents’ day (no class)

  • Wed: Inverse function theorem

  • Fri: Inverse function theorem proof

• Week 6 (for manifolds: Hubbard Sections 3.0-3.2)

  • Mon: Least upper bound property revisited

  • Wed: Midterm

  • Fri: Manifolds, definitions, linked rods in the plane, manifold recognition

• Week 7 (Hubbard 3.7)

  • Mon: Tangent spaces (of a graph, of a level set), implicit function theorem

  • Wed: Lagrange multipliers (intuition and proof)

  • Fri: applications of Lagrange multipliers (proof of spectral theorem)

• Spring break

• Week 8 (Spivak CoM pages 46-56, Munkres sections 10-12)

  • Mon: The integral (Archimedes method of exhaustion, defining the integral over rectangles)

  • Wed: Integrability criteria (measure 0 and content 0 sets)

  • Fri: Computing integrals (fundamental theorem of calculus)

• Week 9 (Spivak CoM pages 56-66, Munkres sections 16-17)

  • Mon: Fubini’s theorem (proof and applications)

  • Wed: Change of variables (introduction, 1-dimensional COV)

  • Fri: Partitions of unity (existence theorem, bump functions, application to integration)

• Week 10 (Spivak CoM 66-74, Munkres sections 18-20)

  • Mon: Diffeomorphisms (properties, local decomposition into primitive diffeomorphisms)

  • Wed: Midterm 2

  • Fri: Change of variables (primitive diffeomorphisms, proof of COV, application)

• Week 11 (Hubbard sections 6.0-6.2, 6.4, 6.7-6.8)

  • Mon: Alternating algebra (elementary k-forms, wedge product)

  • Wed: Differential forms (examples in R^3, exterior derivative)

  • Fri: Forms and integration (integrating a k-form on R^n over a k-cube, pullbacks)

• Week 12 (Spivak CoM 97-105)

  • Mon: Chains (examples, the boundary map, partial^2=0)

  • Wed: Stokes’ theorem (proof, examples)

  • Fri: Stokes’ application (winding numbers, fundamental theorem of algebra)

• Week 13 (Spivak CoM 122-137)

  • Mon: Application of Stokes (Green and divergence theorems; areas, volumes, surface areas)

  • Wed (last class): Stokes application (Green’s theorem and planimeters)

Homework assignments

HW1 HW2 HW3 HW4 HW5 HW6 HW7 HW8 HW9

Lecture notes

A pdf of lecture notes taken by Michele Tienne (course assistant).