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),
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:
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
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:
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:
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
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:
Fri Nov 24:
Mon Nov 27:
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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