SYLLABUS
MTH 420

Here is a rough summary of what we have done in class. It is not intended to be comprehensive.
I will try to keep it a lecture or two ahead of where we actually are, but this may not be very accurate.
(BG refers to Bishop & Goldberg; F refers to Flanders.)

Week 1:

  • Differential geometry = calculus + algebra.
  • Motivating example 1: dy dx = - dx dy and dx dx =0.
  • Motivating example 2: Vectors, Matrices, Tensors, Forms.
  • Motivating example 3: Two kinds of vectors; derivatives vs. differentials.
  • Definition of a vector space: BG §2.1.
  • Introduction to p-vectors: F §2.1.

    Week 2:

  • Definition of exterior product (wedge product): F §2.3.
  • Invariant definition of determinants: F §2.2 & BG §2.19.
  • Introduction to differential forms: F §2.3.
  • The gradient as a differential form.
  • Example: Parabolic coordinates.
  • Geometric interpretation of 1-forms.

    Week 3:

  • Inner products: F §2.5; BG §2.20 & § 2.21.
  • Line elements (metrics).
  • Inner products of p-vectors: F §2.6.

    Week 4:

  • Orientation.
  • Hodge dual (star operator): F §2.7; BG §2.22; my notes.
    WARNING: Our definition differs slightly from both texts!
  • Example: Dot, cross, and triple products in 3-d Euclidean space.

    Week 5:

  • Differential forms: F §3.1; BG §4.2.
  • Differentials revisited.
  • Exterior derivatives: F §3.2; BG §4.3.
  • Example: Div, grad, curl and all that: BG §4.3.
  • Laplacian: F §4.4.

    Week 6:

  • Maxwell's equations: F §4.6.
  • Review
    ** MIDTERM **

    Week 7:

  • Closed and exact forms.
  • (Converse of the) Poincaré Lemma: BG §4.5.
  • Example: polar coordinates.
  • Pullbacks: F §3.3, BG §3.9.
  • Change of variables.
  • Introduction to integration: BG §4.8.
  • Example: line integral on a circle.

    Week 8:

  • Example: surface integral on a sphere.
  • Summary of integration: BG §4.8.
  • Vector Calculus: line integrals.
  • Vector Calculus: flux in both 2 and 3 dimensions.
  • Orientation and Boundary

    Week 9:

  • Fundamental Theorem of Calculus.
  • Stokes' Theorem: BG §4.9; F §5.8.
  • Green's Theorem, the Divergence Theorem, and all that.
  • Potential theory: F §7.1

    Week 10:

  • The wave equation.
  • Varying a Lagrangian.
    Supplemental reading: The Scalar Field Equation in the Presence of Signature Change,
    Tevian Dray, Corinne A. Manogue, and Robin W. Tucker, Phys. Rev. D48, 2587-2590 (1993).
  • Review
    ** FINAL **

    Looking ahead:

  • Vectors!
  • Tensors!
  • Relativity!