Math 4530-1
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Professor Korevaar
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Lectures are listed in reverse chronological order.
Week 15
apr27.pdf 6+ global Gauss-Bonnet Theorem
apr25.pdf 6+ local Gauss-Bonnet Theorem
Week 14
apr22.pdf 5+ geodesics and isometries of hyperbolic space
apr20.pdf 5+ geodesics and isometries of hyperbolic space
apr18.pdf 5 Studying the intrinsic DE's for geodesics
Week 13
apr15.pdf 5 Geodesics!
apr13.pdf 4.6-4.8 Gauss curvature from WE representation. Conjugate surfaces.
apr11.pdf 4.6-4.8 Reconstructing X(u,v) from Phi(u,v) by complex antidifferentiation.
Week 12
apr8.pdf 4.6-4.8 computing the triple composition St(U(X(u,v)))
for conformally parameterized minimal surfaces; we're continuing the notes below.
Also, see the directory minimalsurfaces
minimal.pdf 4.6-4.8 old undergraduate colloquium notes on minimal surfaces
apr6.pdf 4.6-4.8 supplements to the minimal surface notes (above)
apr4.pdf 4.3 Two minimal surface theorems to accompany our soap bubbles
Week 11
apr1.pdf 4.4 Ros' proof of Alexandrov's sphere theorem for compact constant mean curvature surfaces.
mar30.pdf 4.4 the first variation of area computation, and why H=0 surfaces are called minimal.
mar28.pdf 4.1-4.4 balloon science, corrected
Week 10
mar25.pdf 4.1-4.3 pressure, surface tension, and mean curvature
mar23.pdf 3.4-3.5 Finishing Liebmann, and starting the study of constant mean curvature.
mar21.pdf 3.4-3.5 Liebmann's amazing theorem that compact surfaces with constant Gauss curvature must be spheres. Also, more Christoffel fun.
Week 9
mar11.pdf 3.4 computation with indices and the summation convention
mar9.pdf 3.4 isometric surfaces and Gauss' Theorem Egregium
mar7.pdf 3.3 surfaces of revolution with constant K
Week 8
mar2.pdf 3.5 totally umiblic surfaces are parts of spheres or planes
feb28.pdf 1.1-3.3 review for midterm; also, using inverse function theorem to prove local graph property
Week 7
feb25.pdf 2.4, 3.3 normal curvature, lines of curvature, surfaces of revolution
feb23.pdf 2.4 normal curvature
Week 6
feb18.pdf 3.1-3.3 calculating the matrix for the shape operator, K, H.
feb16.pdf 2.3 Mean and Gauss curvature, the shape operator, and the tangential hessian
feb14.pdf 2.2-2.3 Relationship between the shape operator and the Hessian of the function parametrizing the surface about its tangent plane
Week 5
feb11.pdf linalg: matrix of a linear transformation, self-adjoint operators, the spectral theorem; background we apply to the shape operator.
feb9.pdf 2.2 the shape operator
feb7.pdf 2.2 the unit normal and orientability, directional derivatives.
Week 4
feb4.pdf 2.1-2.2 Surfaces!! : local patches, atlas, differentiable surfaces and maps between them. The tangent space.
feb2.pdf extra: the one and only fundamental theorem of calculus
jan31 No notes - we spent the day working on homework in the computer lab.
Week 3
jan28.pdf 1.6: isoperimetric inequality, but not the book's proof.
jan26.pdf 1.4, 1.7: integrating the Frenet system, for space curves and for plane curves. Calculations without p.b.a.l.
Maple documents from today:
pdf and maple worksheet for integrating the Frenet System in space:
frenetsystem.pdf
frenetsystem.mws
planar curves with prescribed planar curvature:
planefrenet.pdf
planefrenet.mws
computing curvature and torsion for arbitrary parametric curves, with computer algebra:
curtor.pdf
curtor.mws
jan24.pdf 1.3: Frenet system and E! theorem for curves of prescribed curvature and torsion
Week 2
jan21.pdf extra: Kepler's and Newton's Laws
jan19.pdf 1.2-1.3 arclength, straight lines minimize length, acceleration decomposition
Week 1
jan14.pdf 1.1-1.3 review of dot and cross product, universal product rule for differentiation.
jan12.pdf 1.1 curves
jan10.pdf intro A global curvature theorem for curves, and one for surfaces