Differential Geometry

--- title: Differential Geometry tags: teaching --- # Differential Geometry ## Unit 1: Review (2 weeks) :page_with_curl: 2 homeworks :::info **Linear algebra**: Subspaces, inner products, cross products, orientations - Week 1, [time=Jan 30 and Feb 1] **Read:** Sections 1-1 and 1-3. > may be that _general_ inner products are not really review, matrix $(g_{ij})$ likely not review > Orientations might not be review **HW 1 due on Feb 6.** ::: #### Outline of topics * Vector spaces * Linear independence, basis & dimension * Inner products and Cauchy-Schwarz inequality * Orthonormal bases * $g=(g_{ij})$ matrix * Orientations * Linear transformations (_might extend into Week 2_) :::info **Calculus (and some _analytic geometry_)** - Week 2, [time=Feb 6 and 8, and part of Feb 13] **Read:** Sections: 1-4, 1-5. **HW 2 due on Feb 15.** ::: #### Outline of topics * Lines, planes, ~~spheres~~ (in $\mathbb R^3$) * Parameterization * Vector calculus * Chain rule in this setting * Product rules for inner products, and the cross product ## Unit 2: Regular curves (4-5 weeks) > Mostly, curves will be in $\mathbb R^3$ ("space curves") :page_with_curl: 4 homeworks (3 local theory, 1 global theory) ### Begin with _local theory_. :::info **Definitions & examples** - Week 3, [time=Feb 13 and 15] **Read:** Sections 2-1, 2-2. **HW 3 due on Feb 22.** ::: #### Outline of topics * Regular space curves, examples * Reparameterizations * Arclength (_discussion of section, e.g. unit speed curves, extends to Week 4_) :::info **Arc length parameterized curves & Curvature** - Weeks 4 and 5, [time=Feb 20, 22, and Feb 27] **Read:** Sections 2-3. **HW 4 due on Mar 1.** ::: :::info **The Frenet-Serret apparatus** - Second part of Week 5, [time=Mar 1] **Read:** Section 2-3. **HW 5 (on curvature, torsion and Frenet-Serret) due on Mar 8.** ::: #### Outline of topics * Using a.l.p. curves (unit speed curves) * The curvature, $\kappa$, of a regular curve * The Frenet-Serret apparatus $\{\kappa, \tau, \bf T, N, B\}$ :::info **Frenet-Serret equations & Planes related to curves, Non-arclength formulae** - Week 6, [time=Mar 6 and 8] **Read:** Section 2-4, 2-6. **HW 6 due on Mar. 13.** ::: #### Outline of topics * The Frenet-Serret equations (Theorem 4.2) * The osculating plane and normal plane of a space curve * Calculations with curves that are not a.l.p. ::: success **Test 1, Mar. 13 in class.** On Chap. 1 (not 1-2), Sections 2-1 through 2-4, and Section 2-6. ::: ### Some _global theory_. :::info **Line integrals, plane curvature, rotation index.** - during Week 7, [time=Mar 15] **Read:** Sections 3-1 and 3-2. **HW 7 due on Mar. 27.** ::: #### Outline of topics * Green's Theorem * Simple closed curves in the plane * Plane curvature * Hopf's Umlaufsatz and Rotation index :::info **The isoperimetric inequality** > Might skip this topic > - Week 8 (?), [time=Mar 27] **Read:** Section 3-4. ::: #### Outline of topics * Proving the isoperimetric inequality