| Week |
Date |
Reference |
Topics |
| 1 |
Monday
2/1/2021 |
Notes: B.1-B.4
|
- Abstract linear algebra
- Abstract vector space
- Linear independence
- Basis and Dimension
|
Wednesday
2/3/2021 |
Notes: Chapter 3
|
- Basis and dimension
- Affine space
- Affine geometry
|
| 2 |
Monday
2/8/2021 |
|
- Concrete versus Abstract Linear Algebra
- Linear Maps
- Dot Product on \(\mathbb{R}^m\)
|
Wednesday
2/10/2021 |
Notes, Chapter 6
|
- Euclidean geometry
- Dot product
- Cauchy-Schwarz inequality
- Triangle inequality
- Euclidean vector space
- Length of vectors
- Angle between vectors
|
| 3 |
Monday
2/15/2021 |
|
No class. President's Day
|
Wednesday
2/17/2021 |
Notes, Chapter 6
|
- Euclidean geometry
- Abstract inner product
- Orthonormal basis
- Orthonormal transformation
- Orthogonal matrices
|
Thursday
2/18/2021 |
Notes, Chapter 6
|
- Euclidean geometry
- Euclidean space
- Distance between two points
- Rigid motions
- Isometries
|
| 4 |
Monday
2/22/2021 |
Notes, Chapter 7
|
- Parameterized curves in affine space
- Geometry of curves in Euclidean \(2\)-space
- Velocity, speed, acceleration
- Arclength
- Constant speed parameterization
- Curvature
- Frenet-Serret frame and equations
- Curvature implies uniqueness up to rigid motion
|
Wednesday
2/24/2021 |
Notes, Chapter 7
|
- Geometry of curves in Euclidean \(2\)-space
- Orientation of space and curves
- Frenet-Serret frame and equations of unit speed curve
|
| 5 |
Monday
3/1/2021 |
Notes, Chapter 7
|
- Geometry of curves in Euclidean \(3\)-space
- Frenet-Serret frame and equations of parameterized curve
|
Wednesday
3/3/2021 |
Notes, Chapter 8
|
- Surfaces in \(\mathbb{R}^3\)
|
| 6 |
Monday
3/8/2021 |
Notes, Chapters 8 and 9
|
- Submanifolds of \(\mathbb{R}^n\)
- Surfaces in \(\mathbb{A}^3\)
- Tangent Space of a Surface
|
Wednesday
3/10/2021 |
|
- Differential calculus on a parameterized surface
- Directional derivative
- Differential of a function
- Dual vector space
- Linear functions of a vector space
- Dual basis
|
| 7 |
Monday
3/15/2021 |
Notes, B.9, C.1-8
|
- Multilinear algebra
- Tensors
- Symmetric and antisymmetric tensors
- Tensor and wedge products
- Exterior \(2\)-tensors
|
Wednesday
3/17/2021 |
|
- Oriented area and volume
- Oriented area of parallelogram
- Oriented volume of parallelopiped
- \(m\)-form as measure of oriented \(m\)-dimensional volume
|
| 8 |
Monday
3/22/2021 |
|
- Vector fields on \(\mathbb{R}^m\)
- Differential forms on \(\mathbb{R}^m\)
- Wedge product
- Differential of Function
|
Wednesday
3/24/2021 |
|
|
| 9 |
Monday
3/29/2021 |
|
- Differential forms
- Exterior derivative of functions and \(1\)-forms
- Pullback of \(1)-form on parameterized curve
- Integral of \(1\)-form on oriented curve
- Fundamental Theorem of Line Integrals
|
Wednesday
3/31/2021 |
|
- Pushforward of a vector field
- Pullback of a differential form
|
| 10 |
Monday
4/5/2021 |
|
- 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
|
Wednesday
4/7/2021 |
|
- Orientation of a surface in \(\mathbb{R}^3\)
- Orientation of the boundary of a surface
- Integration on an oriented surface
|
| 11 |
Monday
4/12/2021 |
|
- Integration on an oriented surface
- First fundamental form
- Adapted oriented orthonormal frame along oriented surface
- Adapted oriented orthonormal frame along oriented curve
|
Wednesday
4/14/2021 |
|
- Adapted frame and coframe
- Connection forms
- Structure equations
|
| 12 |
Monday
4/19/2021 |
|
No class, Spring Break
|
Wednesday
4/21/2021 |
|
- Moving frame on Euclidean space
- Structure equations for adapted frame along curve
- Gauss map
- Weingarten map
- Second fundamental form
|
| 13 |
Monday
4/26/2021 |
|
- Moving frames on a surface in Euclidean space
- First and second fundamental forms
- Gauss curvature
- Theorem egregium
|
Wednesday
4/28/2021 |
|
- Geometric meaning of connection 1-form
- Line integral of connection 1-form along curve segment
- Line integral of connection 1-form around rectangle
|
| 14 |
Monday
5/3/2021 |
|
- Stokes's Theorem for a rectangular surface
- Gauss-Bonnet Theorem for a rectangular surface
|
Wednesday
5/5/2021 |
|
- Gauss-Bonnet Theorem for a closed surface
|
| 15 |
Monday
5/10/2021 |
|
- Riemannian \(2\)-manifolds
- Curvature on a Riemannian \(2\)-manifold
- Levi-Civita connection
- Gauss-Bonnet theorem for closed \(2\)-manifold
|
