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!