Math 143: Course Schedule

Fall 2024

Below is the tentative schedule, which may be adjusted as necessary.

• Week 1:
  
  • Parameterized curves, regular curves, arc-length parameterization
  see Shifrin §1.1 or Donaldson §1.1-§1.2
  • Frenet frame for a curve, curvature, torsion, Fundamental Theorem of Space Curves
  see Donaldson §1.4 (and part of §1.3) or Shifrin §1.2

• Week 2:
  
  • Vector fields, tangent space, 1-forms, differential of a function, line integrals
  see Donaldson §2.1-§2.2
  • Parameterized surfaces, differentials, implicitly defined surfaces
  see Donaldson §3.1 or Shifrin §2.1

• Week 3:
  
  • First and second fundamental forms, curves in surfaces
  see Donaldson §3.2-§3.3 (or Shifrin §2.2)
  • Gauss and mean curvature, principal curvatures and principal directions
  see Donaldson §3.3

• Week 4:
  
  • Euler's formula, elliptic and hyperbolic points
  see Donaldson §3.4 (or Shifrin §2.2)
  • Exterior calculus: wedge product, the exterior derivative
  see Donaldson §2.3

• Week 5:
  
  • Gauss’ Theorem Egregium and Fundamental Theorem of Surfaces via moving frames; Structure equations and adaptive frames
  see Donaldson §3.5 or Shifrin §3.3 (for a "hands on" approach to Structure Eqn see Shifrin §2.3)
  • Riemannian geometry, black holes and relativity
  see Donaldson §3.5

• Week 6:
  
  • MIDTERM
  Material covered: Donaldson §1.1-1.5, §2.1-2.3, §3.1-3.5
  • Geodesics
  see Shifrin §2.4

• Week 7:
  
  • Democracy day: NO CLASS
  • Parallel transport, covariant derivative
  see Shifrin §2.4

• Week 8:
  
  • Geodesic curvature, parallel transport and holonomy; surfaces of constant curvature and Gauss-Bonnet Theorem (motivation)
  see lecture notes (on Canvas) or see Shifrin §3.1 and §3.3 (for the differential forms/adaptive frames approach) but note change in notation: \( \theta_i \) is denoted \(\omega_i \) in Shifrin §3.3, and \( \omega_{ij} \) is denoted \( \omega_{ji}=-\omega_{ij} \) there.
  • Integration, Stokes’ Theorem

• Week 9:
  
  • Gauss-Bonnet theorem
  see Shifrin §3.1
  • Minimal surfaces and area forms; surfaces of constant mean curvature
  see Shifrin §3.4

• Week 10: TBA and Review