MATH-UA 377 Calendar

MATH-UA 377 Differential Geometry Spring 2022 Calendar

Week	Date	Topics	Links
1	Tuesday 1/25/2022	Deductive logic Course logistics Abstract linear algebra	Slides Recording
1	Thursday 1/27/2022	Abstract Linear Algebra Linear subspaces Basis and dimension Change of basis and coordinates Affine Geometry Geometry of points and vectors Affine space Affine basis Geometry of vector and affine spaces	Slides Recording
2	Tuesday 2/1/2022	Linear and Affine Maps Linear functions and maps Linear map with respect to basis Spaces of linear functions, maps, and transformations Affine functions and maps Affine map with respect to basis Affine transformations	Slides Recording
2	Thursday 2/3/2022	Inner product space Dot product on $\mathbb{R}^m$ Cauchy-Schwarz inequality Triangle inequality Pythagorean theorem Positive definite symmetric bilinear function Inner product space Euclidean space	Slides Recording
3	Tuesday 2/8/2022	Euclidean geometry Euclidean space Distance between two points Orthogonal maps Rigid motions	Slides Recording
3	Thursday 2/10/2022	Parameterized curves in affine and Euclidean space Velocity, acceleration Speed, arclength Curvature	Slides Recording
4	Tuesday 2/15/2022	Orientation of affine space Geometry of curves in Euclidean $2$-space Orientation of a parameterized curve Curvature Frenet-Serret frame Frenet-Serret equations	Slides Recording
4	Thursday 2/17/2022	Winding and Rotation Numbers of a Closed Planar Curve	Slides Recording
5	Tuesday 2/22/2022	Geometry of curves in Euclidean $3$-space Frenet-Serret frame and equations	Slides Recording
	Thursday 2/24/2022	Geometry of curves in Euclidean $3$-space Frenet-Serret frame and equations Curvature and torsion Spherical Curves	Slides No recording available
	Supplement	Spherical Crofton Formula Fenchel's theorem Fary-Milnor theorem	Slides
6	Tuesday 3/1/2022	Open sets in $\mathbb{R}^2$ $C^1$ maps and their partial derivatives Parameterized surfaces in $\mathbb{A}^3$	Slides Recording
6	Thursday 3/3/2022	Parameterized surface Coordinate chart Global surface Tangent Space of a Surface	Slides Recording
7	Tuesday 3/8/2022	Differential calculus on affine space Directional derivative Differential of a function Differential calculus on a parameterized surface Directional derivative Differential of a function	Slides Recording
7	Thursday 3/10/2022	Dual vector space Linear functions of a vector space Dual basis Multilinear algebra Tensors Symmetric and antisymmetric $2$-tensors	Slides Recording
8	Tuesday 3/15/2022	No class, Spring Break
8	Thursday 3/17/2022	No class, Spring Break
9	Tuesday 3/22/2022	Dual vector space Tensors Symmetric and antisymmetric $2$-tensors Tensor, symmetric, and wedge products	Slides Recording
9	Thursday 3/24/2022	Midterm
10	Tuesday 3/29/2022	Notation Exterior derivative of $0$-form Pullback of $1$-form on parameterized curve
10	Thursday 3/31/2022	Coordinate Vector fields and $1$-forms Exterior derivative of function Line integral	Slides Recording
11	Tuesday 4/5/2022	Line integral Fundamental Theorem of Line Integrals	Slides Recording
11	Thursday 4/7/2022	Integration on a rectangular region in $\mathbb{R}^m$ Integral of differential form on oriented rectangular region Fundamental theorem of calculus on a rectangular region Integral of $2$-form over a parameterized domain	Slides Recording
12	Tuesday 4/12/2022	Orientation of a rectangular surface in $\mathbb{R}^3$ Integration on an oriented rectangular surface Stokes's Theorem for an oriented rectangular suface Integration on a surface	Slides Lecture Recording
12	Thursday 4/14/2022	Moving frames on affine space	Slides Recording
13	Tuesday 4/19/2022	Moving frames on Euclidean space	Slides Recording
13	Thursday 4/21/2022	Surface in Euclidean space First fundamental form Adapted orthonormal frame and co-frame Structure equations Gauss map Weingarten map Second fundamental form	Slides Recording
14	Tuesday 4/26/2022	Local 2-Manifold Atlas of Coordinate Charts Tangent, Cotangent, Tensor Bundles Riemannian metric	Slides Recording
14	Thursday 4/28/2022	Riemannian Metric on a Local 2-Manifold Orthonormal Frame Uniqueness of Connection 1-Form Gauss Curvature Adapted Orthonormal Frame for Curve in a Manifold Geodesic Curvature	Slides
15	Tuesday 5/3/2022	Geometric Meaning of Connection 1-Form Line integral of Connection 1-Form Along Oriented Curve Line integral of Connection 1-Form Around Boundary of Rectangular 2-Manifold Gauss-Bonnet Theorem for Rectangular 2-Manifold	Slides Recording
15	Thursday 5/5/2022	Gauss-Bonnet Theorem	Slides Recording
16	Thursday 5/12/2022 12-1:50pm	Final, CIWW 517