WEBSITE for MATH 426

Gauss	Riemann	Cartan

Instructor

• Office: 328 McAlister
• Office hours: T Th 2:00-3:30
• Email: ghomi@math.gatech.edu

Syllabus

Lecture Notes

• Lecture Notes 0
  Basics of Euclidean Geometry, Cauchy-Schwarz inequality

• Lecture Notes 1
  Definition of curves, examples, reparametrizations, length

• Lecture Notes 2
  Curvature, tangent

• Lecture Notes 3
  Curves of constant curvature, the principal normal, oschulating circle, signed curvature, turning angle, winding number, Hopf's theorem

• Lecture Notes 4
  Radius of Curvature, total curvature, convexity

• Lecture Notes 5
  The four vertex theorem, isoperimetric inequality

• Lecture Notes 6
  Torsion, Frenet-Seret frame, helices, spherical curves

• Lecture Notes 7
  Fenchels theorem on total curvature, and Milnors theorem on total curvature of knots (Not yet typeset)

• Lecture Notes 8
  Definition of surface, differential map

• Lecture Notes 9
  Gaussian curvature, Gauss map, shape operator, coefficients of the first and second fundamental forms, curvature of graphs

• Lecture Notes 10
  Interpretations of Gaussian curvature as a measure of local convexity, ratio of areas, and products of principal curvatures.

• Lecture Notes 11
  Gauss's Theorema Egregium (Not yet typeset)

• Lecture Notes 12
  Gauss-Bonnet theorem (Not yet typeset but contains the exercises)

• Lecture Notes 13
  A few more exercises.

Mathematica Packages

• CrossProducts.nb
  How to prove various identities involving dot product and cross products, including the three dimensional version of the Pythagorean theorem.

• GraysProgs.nb
  This notebook loads up all the miniprograms written by Alfred Gray to accompany his book on Curves and Surfaces.

• PlanarCurves.nb
  Dozens of parametrizations for various curves; programs for computing curvature, length, and winding number; plotting programs for coloring a curve according to its curvature, and programs for plotting curves determined by a given curvature function; several animation programs including cycloid, tractrix and trochoids.

• SpaceCurves.nb
  Dozens of parametrizations for various curves; programs for computing curvature, torsion and length; programs for coloring a curve by its curvature or torsion; programs for plotting the tangential, normal, and binormal, spherical images.

• Surfaces.nb
  Dozens of parametrizations for various surfaces; programs for computing Gauss and mean curvature; programs for coloring a surface by its Gauss or mean curvature.