Course number: CA-MATH-809
Constructor University, Spring 2023
All the most recent information about class can be found on this website.
The class material is similar to the following textbook:
Other good books are:
The total grade is only based on the final exam. There is the possibility for bonus points (up to 10% in the percentage grading scheme).
There will be one final exam and one make-up final exam. All topics discussed in class are relevant for the exams.
Each week there will be a homework assignment. These are an integral part of the coursework and working on the exercise sheets consistently is the best preparation for the exams. Note that
| Date | Sheet Number | Due Date |
|---|---|---|
| Feb. 07, 2023 | Sheet 1 | Feb. 14, 2023 |
| Feb. 14, 2023 | Sheet 2 | Feb. 21, 2023 |
| Feb. 23, 2023 | Sheet 3 | Mar. 02, 2023 |
| Mar. 02, 2023 | Sheet 4 | Mar. 09, 2023 |
| Mar. 09, 2023 | Sheet 5 | Mar. 16, 2023 |
| Mar. 16, 2023 | Sheet 6 | Mar. 23, 2023 |
| Mar. 23, 2023 | Sheet 7 | Mar. 30, 2023 |
| Mar. 30, 2023 | Sheet 8 | Apr. 13, 2023 |
| Apr. 13, 2023 | Sheet 9 | Apr. 20, 2023 |
| Apr. 20, 2023 | Sheet 10 | Apr. 27, 2023 |
| Apr. 27, 2023 | Sheet 11 | May 04, 2023 |
Will be updated while class is progressing.
Below, please click on the date to download the lecture notes of this day.
(Note that the book references given below offer only a rough orientation. Sometimes, only parts of a particular chapter are covered in class.)
| Date | Topics |
|---|---|
| Feb. 02, 2023 | Overview and Motivation; Review of Differentiation in Rn Lee Appendix C: parts of Total and Partial Derivatives |
| Feb. 03, 2023 | Review of Differentiation in Rn (partial derivatives, inverse function theorem) Lee Appendix C: parts of Total and Partial Derivatives, parts of The Inverse and Implicit Function Theorem |
| Feb. 07, 2023 | Review of Topology Lee Appendix A: Topological Spaces, Subspaces, Product Spaces |
| Feb. 10, 2023 | Review of Topology; Definition of Manifolds and Examples Lee Appendix A: Connectedness and Compactness; Lee Chapter 1: Topological Manifolds (Coordinate Charts, Examples of Topological Manifolds) |
| Feb. 16, 2023 | Connectivity and Examples (Real Projective Space) Lee Chapter 1: Topological Manifolds (Connectivity) |
| Feb. 17, 2023 | Differentiable Structures on Manifolds; Smooth Maps between Manifolds Lee Chapter 1: Smooth Structures; Lee Chapter 2: beginning of Smooth Functions and Smooth Maps |
| Feb. 23, 2023 | Diffeomorphisms; Partitions of Unity; Motivation for Tangent Spaces Lee Chapter 2 (most parts, except Applications of Partitions of Unity) |
| Feb. 24, 2023 | Tangent Space (Derivations, Differentials) Lee Chapter 3: Tangent Vectors (Geometric Tangent Vectors, Tangent Vectors on Manifolds), parts of The Differential of a Smooth Map, beginning of Computations in Coordinates |
| Mar. 02, 2023 | Submersions, Immersions, Embeddings, Rank Theorem Lee Chapter 4: beginning of Maps of Constant Rank (statement of The Rank Theorem), beginning of Embeddings |
| Mar. 03, 2023 | Embedded Submanifolds Lee Chapter 5: beginning of Embedded Submanifolds, parts of Slice Charts for Embedded Submanifolds, beginning of Level Sets |
| Mar. 09, 2023 | Sard's Theorem Parts of Lee Chapter 6 |
| Mar. 10, 2023 | Sard's Theorem (proof) Lee Chapter 6 proves Sard's Theorem in full generality in: Sets of Measure Zero and Sard's Theorem; we rather followed the proof in Spivak at the end of Chapter 3. Another good reference for the general case is Chapter 3 of Milnor's book. |
| Mar. 16, 2023 | Whitney Embedding Theorem Lee proves Whitney's embedding theorem also in the non-compact case in Chapter 6: The Whitney Embedding Theorem. Another good reference for a proof are Milnor's lecture notes on differential topology (different from the book referenced above). |
| Mar. 17, 2023 | Lie Groups Parts of Lee Chapter 7, mostly: Basic Definitions, Lie Group Homomorphisms |
| Mar. 23 2023 | Vector Fields Lee Chapter 3: The Tangent Bundle; Lee Chapter 8: Vector Fields on Manifolds |
| Mar. 24, 2023 | Vector Fields continued Lee Chapter 8: Vector Fields on Manifolds; few parts of Local and Global Frames; Vector Fields as Derivations; parts of Vector Fields and Smooth Maps; beginning of Lie Brackets; beginning of The Lie Algebra of a Lie Group |
| Mar. 30, 2023 | Integral Curves, Flows, Lie Derivative Lee Chapter 9: Integral Curves; Flows; beginning of The Fundamental Theorem of Flows; beginning of Complete Vector Fields; Lie Derivatives |
| Mar. 31, 2023 | Covectors, Covector Fields; Multilinear Maps Lee Chapter 11: Covectors; Tangent Covectors on Manifolds; Covector Fields; The Differential of a Function; Pullbacks of Covector Fields; Lee Chapter 12: beginning of Multilinear Algebra |
| Apr. 06, 2023 | No class (spring break) |
| Apr. 07, 2023 | No class (spring break) |
| Apr. 13, 2023 | Alternating Tensors (wedge product, interior multiplication); Differential Forms Lee Chapter 14: The Algebra of Alternating Tensors; Differential Forms on Manifolds |
| Apr. 14, 2023 | Differential Forms (Pullback and Exterior Derivative) Lee Chapter 14: Differential Forms on Manifolds; Exterior Derivatives |
| Apr. 20, 2023 | Orientation Lee Chapter 15: parts of Orientation of Vector Spaces; beginning of Orientations of Manifolds |
| Apr. 21, 2023 | Integration of Differential Forms Lee Chapter 11: some parts of Line Integrals; Lee Chapter 16: The Geometry of Volume Measurement; Integration of Differential Forms (intro and Integration on Manifolds part) |
| Apr. 27, 2023 | Manifolds with Boundary; Stokes' Theorem Lee Chapter 1: Manifolds with Boundary; Lee Chapter 15: Boundary Orientations; Lee Chapter 16: Stokes's Theorem |
| Apr. 28, 2023 | Stokes' Theorem; Symplectic Vector Spaces Lee Chapter 16: Stokes's Theorem; Lee Chapter 22: beginning of Symplectic Tensors |
| May 04, 2023 | Symplectic Manifolds (Symplectic Vector Spaces) Lee Chapter 22: Symplectic Tensors |
| May 05, 2023 | Symplectic Manifolds (Example of Bogoliubov Maps, Darboux, Hamiltonian Systems) Lee Chapter 22: Symplectic Structures on Manifolds and parts of Hamiltonian Vector Fields |
| May 11, 2023 | Riemannian Manifolds (Definition, Orthonormal Frames, Pullback, Flatness) Lee Chapter 13 (Riemannian Metrics) |
| May 12, 2023 | Riemannian Manifolds (Distance, Metric Topology, Musical Notation, Pseudo-Riemannian Manifolds, Riemannian Volume Form) Lee Chapter 13 (Riemannian Metrics). Also Lee Chapter 15: The Riemannian Volume Form; and Lee Chapter 16: Integration on Riemannian Manifolds |