| WEEK 1 | W, Sep 5 §1.1 level curves; parametrized curves; tangent vectors; tang vector const ⇒ image is (part of) a straight line |
F, Sep 7 §§1.2, 1.3 arc length, speed, reparametrization, regular curves; reparams of reg curves are reg |
M, Sep 10 §§1.3, 1.4 curve has unit speed reparam &hArr it’s regular; locally, level curves f = 0 with ∇f ≠ 0 are the same as reg param’d curves |
| WEEK 2 | W, Sep 12 §§2.1, 2.2 curvature, signed curvature, tangential & normal cpnts of acceleration, formula for curv |
F, Sep 14 §2.3 Frenet-Serret formulas, calculation of Frenet apparatus |
M, Sep 17 §2.3 examples; if &kappa ≠ 0 then τ = 0 ⇔ curve is planar; κ pos const, τ = 0 ⇒ curve is part of circle |
| WEEK 3 | W, Sep 19 §2.2 local canonical form for space curve; signed curvature |
F, Sep 21 §§2.2, 2.3 signed curv = deriv of turning angle; existence & uniqueness of curves with given curv & torsion; sum formulas for sin & cos via cpx nos; formula for plane rotatns; relatn bet trig, exp, & hyperbolic fcns |
M, Sep 24 Frenet formulas in Rn; loc canon form for plane curve; curve crosses osc circle if κsκ's ≠ 0 (κs = signed curv) Question: Can curve cross osc circle if κsκ's = 0? |
| WEEK 4 | W, Sep 26 §3.1, 3.3, 4.1 simple closed curves; 4 vertex thm; def of surface |
F, Sep 28 §4.1 corrected pf of 4 vertex thm, pics of osculating circles, defs of “open” and “continuous”, patches and charts |
M, Oct 1 Last day to drop without instructor's signature. Assignment 1. §§4.2, 4.3 smooth surfaces, tangent space of a surface |
| WEEK 5 | W, Oct 3 §§4.3, 4.4 normals, orientability, Möbius strips, examples of surfaces |
F, Oct 5 §4.5 quadric surfaces; real sym matrices can be diagonalized by orthog matrices |
M, Oct 8 Assignment 1 due. §§4.5, 4.6 Möbius sandwich, Möbius shorts; quadric surfaces, triply orthogonal systems |
| WEEK 6 | W, Oct 10 §§5.1, 5.2 Lengths of curves on surfaces, isometries |
F, Oct 12 §§5.2, 5.3 isometries, conformal mappings |
M, Oct 15 §§5.3–5.5 stereographic projection, surface area, equiareal maps, Archimedes’ thm |
| WEEK 7 | W, Oct 17 §6.1 2nd fundamental form, shape operator |
F, Oct 19 §6.2 shape operator; curvature of curves on a surface |
M, Oct 22 §§6.1–6.3 rvw of 2nd fund form, κn, and κg; Meusnier's thm; principal curvatures |
| WEEK 8 | W, Oct 24 Returned & rvw’d Assignment 1. §6.3 examples of princ curv & vectors, Euler’s thm |
F, Oct 26 rvw of 2nd deriv test, derivs in general §§6.3, 6.4 Euler’s thm, geometric interpretatn of princ curvatures |
M, Oct 29 §§6.4, 7.1 matrix repn of Sv·w, loc classification of surfaces by principal curvatures; K and H; calc of K for torus |
| WEEK 9 | W, Oct 31 §§6.4, 7.1 classification of umbilic surfaces; calculation of K, H, κ1, κ2 |
F, Nov 2 §§7.1, 7.2 K for ruled surfaces, const curvature surfs of revolution |
M, Nov 5 §§7.2, 7.3 the pseudosphere and tractrix; principal patches, flat surfaces near non-umbilic pts |
| WEEK 10 | W, Nov 7 §7.3 flat surfaces near non-umbilic pts are ruled surfaces; classification of flat ruled surfaces |
F, Nov 9 §7.4 parallel surfaces, surfaces of constant mean curvature |
M, Nov 12 Assignment 2 §7.5 Gaussian curvature of cpct surfaces |
| WEEK 11 | W, Nov 14 §7.6 The Gauss map |
F, Nov 16 §§8.1, 8.2 geodesics: definition and basic properties; geodesic eqs |
M, Nov 19 §8.2 geodesic eqs; geodesics on unit sphere |
| WEEK 12 | W, Nov 21 §§8.2, 8.3 geodesic eqs, geodesics on surfs of revolution |
M, Nov 26 Assignment 2 due. §8.3 geodesics on surfs of revolution, Clairaut’s thm |
W, Nov 28 §8.3 geodesics on the pseudosphere |
| WEEK 13 | F, Nov 30 §§8.3, 8.4 geodesics on a hyperboloid of 1 sheet; geodesics as shortest paths |
M, Dec 3 §§8.5, 10.1 geodesic coords, Gauss’ Theorem Egregium (start of proof) |
W, Dec 5 §10.1 Gauss’ Theorem Egregium (end of proof) |
| WEEK 14 | F, Dec 7 Student evaluations §10.1 classification of const curv surfs up to isometry |
M, Dec 10 §§10.2, 11.1 isometries of surfs; Gauss-Bonnet thm for simple closed curves (start) |
W, Dec 12 §§11.1, 11.2 Gauss-Bonnet thm for simple closed curves & curvilinear polygons |
| FINAL EXAM SLOT |
Mon, Dec 17 Final exam slot 10:40 a.m.–1:10 p.m. 135 State §§11.2, 11.3 Gauss-Bonnet thm for curvilinear polygons and cpct surfaces |