### Class Diary for M462, Fall 2017, Jochen Denzler

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Wed Aug 23: **
Smooth parametrized curves; definition and examples; regular curves; arclength
defined in terms of integral

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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.

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Mon Aug 28: **
Quick review of cross product; curvature, torsion, and the Frenet formulas for
curves in R^3.

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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*

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Fri Sep 01: **
Planar curves; signed curvature; special cases; Evolute just started.

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Mon Sep 04: **
LABOR DAY

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Wed Sep 06: **
Some hwk comments; Evolute and evolvent

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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*

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Mon Sep 11: **
Proof of isoperimetric inequality for curves in the plane.

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Wed Sep 13: **
4-vertex theorem for convex curves in the plane

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Fri Sep 15: **
Introduction to surfaces (parametrized surfaces)

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Mon Sep 18: **
Motivation for definition of regular surfaces (in terms of local coordinates)
*Hwk 14-16 due next Monday*

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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.*

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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.

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Wed Sep 27: **
Review of total derivative formalism in R^n.

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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.

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Mon Oct 02: **
First fundamental form: length and angle measurement; area barely started.

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Wed Oct 04: **
area on a surface; sqrt(det of 1st FF); oriented surfaces

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Fri Oct 06: **
FALL BREAK

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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).
*

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Wed Oct 11: ** Examples of Gauss map.
Derivative of Gauss map in detail. Symmetry. 2nd fundamental form: II(v):=
- v . DN(p)v

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Fri Oct 13: **
Geometric interpretation of 2nd FF; normal curvature of a curve

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Mon Oct 16: **
EXAM 1

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Wed Oct 18: **
Normal curvature. Principal curvatures; Gauss and Mean curvature;

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Fri Oct 20: **
Eigenvalues of the 2nd FF resp Gauss map (all with respect to an ONB of TpS);
elliptic, hyperbolic etc points.

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Mon Oct 23: **
Gauss map in arbitrary coordinates; Weingarten equations;

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Wed Oct 25: **
Some examples: torus, surfaces of revolution, graphs. Lines of curvature.
*Hwk 31-36 due next Wed*

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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.

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Fri Nov 03: **
Isometries def'd and explained.

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Mon Nov 06: **
local isometries; examples; invariants: length, distance, area; ad-hoc pf that
sphere ans plane aren't locally isometric; Christoffel symbols defined

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Wed Nov 08: **
Calculating Christoffel symbols; Gauss eqn, Theorema Egregium, Codazzi-Mainardi
equations.

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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*

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Mon Nov 13: **
covariant derivatives; geodesics defined

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Wed Nov 15: **
parallel transport of vectors along curves. No global parallelism: counterexamples

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Fri Nov 17: **
shortest connecting curves are geodesics.

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Mon Nov 20: **
shortest connecting curves are geodesics (II)

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Wed Nov 22: **
geodesic curvature

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Fri Nov 24: **
THANKSGIVING BREAK

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Mon Nov 27: **
EXAM 2

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Wed Nov 29: **
geodesic curvature and turning of tangent angles

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Fri Dec 01: **
... Gauss-Bonnet: simplest case

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Mon Dec 04: **

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Wed Dec 06: ** STUDY DAY

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Wed Dec 13: ** FINAL EXAM 2:45-4:45
(scheduled by university policy)

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