Class Diary for M462, Fall 2017, Jochen Denzler

Wed Aug 23: Smooth parametrized curves; definition and examples; regular curves; arclength defined in terms of integral
Fri Aug 25: Reparametrization; invariance of arclength under reparametrizations; Every regular C^1-parametric curve can be reparametrized by arclength. Hwk due Wed: numbers 2,4,5,6 in Sec 1.3 (pages 7ff). For item 2, also reparametrize the cycloid by arclength.. Alternative definitions of arclength via partitions (equivalence not proved), just stated.
Mon Aug 28: Quick review of cross product; curvature, torsion, and the Frenet formulas for curves in R^3.
Wed Aug 30: Frenet formulas in matrix formulation. Existence and uniqueness (up to rigid motions) of curves with prescribed curvature (>0) and torsion. Hwk #5-8 as posted on canvas due next Wed
Fri Sep 01: Planar curves; signed curvature; special cases; Evolute just started.
Mon Sep 04: LABOR DAY
Wed Sep 06: Some hwk comments; Evolute and evolvent
Fri Sep 08: Taylor expansion of a curve near a point; projections into (tn), (tb), and (nb) planes respectively; Jordan curve thm stated; isoperimetric inequality stated and explained. Hwk #9-13 as handed out, due next Friday
Mon Sep 11: Proof of isoperimetric inequality for curves in the plane.
Wed Sep 13: 4-vertex theorem for convex curves in the plane
Fri Sep 15: Introduction to surfaces (parametrized surfaces)
Mon Sep 18: Motivation for definition of regular surfaces (in terms of local coordinates) Hwk 14-16 due next Monday
Wed Sep 20: Regular surfaces occur as solutions to f(x,y,z)=a for regular values a. The implicit function theorem mentioned and briefly explained. Hwk 17,18 also due next Monday.
Fri Sep 22: Regular surfaces can be locally desribed as graphs. Quadrics
Mon Sep 25: Differentiability of functions def'd in terms of coordinates. Coordinate change maps. Surfaces of revolution; Tangential developable.
Wed Sep 27: Review of total derivative formalism in R^n.
Fri Sep 29: Tangent spaces; The differential. Hwk 19-24 due Mon after fall break (handed out and posted on canvas) Quick preview 1st fundamental form.
Mon Oct 02: First fundamental form: length and angle measurement; area barely started.
Wed Oct 04: area on a surface; sqrt(det of 1st FF); oriented surfaces
Fri Oct 06: FALL BREAK
Mon Oct 09: oriented surface finished. Curvature properties: rate of change of normal vector should describe curvature properties; Gauss map and its derivative. Hwk 25-30 as posted on canvas; the Hons problems 28,30 are due Wed after the exam; the others (25,26,27,29) on Fri 10/13 (before the exam).
Wed Oct 11: Examples of Gauss map. Derivative of Gauss map in detail. Symmetry. 2nd fundamental form: II(v):= - v . DN(p)v
Fri Oct 13: Geometric interpretation of 2nd FF; normal curvature of a curve
Mon Oct 16: EXAM 1
Wed Oct 18: Normal curvature. Principal curvatures; Gauss and Mean curvature;
Fri Oct 20: Eigenvalues of the 2nd FF resp Gauss map (all with respect to an ONB of TpS); elliptic, hyperbolic etc points.
Mon Oct 23: Gauss map in arbitrary coordinates; Weingarten equations;
Wed Oct 25: Some examples: torus, surfaces of revolution, graphs. Lines of curvature. Hwk 31-36 due next Wed
Fri Oct 27: Hwk hints; Gauss curvature as area mapping ratio of the Gauss map. Hwk 37 (umbilics of ellipsoid); due with following slate of hwk
Mon Oct 30: vector fields and existence of some desirable coordinates; also conformal coords stated without proof (which is deeper). Mercator maps
Wed Nov 01: Minimal surfaces and variation of area.
Fri Nov 03: Isometries def'd and explained.
Mon Nov 06: local isometries; examples; invariants: length, distance, area; ad-hoc pf that sphere ans plane aren't locally isometric; Christoffel symbols defined
Wed Nov 08: Calculating Christoffel symbols; Gauss eqn, Theorema Egregium, Codazzi-Mainardi equations.
Fri Nov 10: existence and uniqueness (up to rigid motion) of surface w/ compatibly prescribed 1st and 2nd FF (no proof). ~~~ Motivation & Intro to geodesics Hwk ..-.. due Fri or Mon
Mon Nov 13: covariant derivatives; geodesics defined
Wed Nov 15: parallel transport of vectors along curves. No global parallelism: counterexamples
Fri Nov 17: shortest connecting curves are geodesics.
Mon Nov 20: shortest connecting curves are geodesics (II)
Wed Nov 22: geodesic curvature
Mon Nov 27: EXAM 2
Wed Nov 29: geodesic curvature and turning of tangent angles
Fri Dec 01: ... Gauss-Bonnet: simplest case
Mon Dec 04:
Wed Dec 06: STUDY DAY
Wed Dec 13: FINAL EXAM 2:45-4:45 (scheduled by university policy)

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