Jacobs University, Fall 2019
Syllabus (as of Aug. 28, 2019) available here. (Note that the Syllabus will not be updated, the most recent information can be found on this website.)
The class material is similar to the following textbook:
Other good books are:
See the Syllabus.
There will be two exams, a midterm and a final. The midterm will cover the material from the first half of the course and the final will cover all material with emphasis on the second half. Note that this class uses gradescope for grading exams, see here for more information. (Note: Just talk to the instructor if you prefer that gradescope is not used for your exam.)
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 |
|---|---|---|
| Sep. 05, 2019 | Sheet 1 (typo in Problem 1 and notation in Problem 2 corrected) | Sep. 12, 2019 |
| Sep. 12, 2019 | Sheet 2 | Sep. 19, 2019 |
| Sep. 21, 2019 | Sheet 3 | Oct. 02, 2019 |
| Oct. 03, 2019 | Sheet 4 | Oct. 10, 2019 |
| Oct. 10, 2019 | Sheet 5 (Problem 2 (hopefully) clarified) | Oct. 22, 2019 |
| Oct. 31, 2019 | Sheet 6 | Nov. 07, 2019 |
| Nov. 07, 2019 | Sheet 7 | Nov. 14, 2019 |
| Nov. 14, 2019 | Sheet 8 | Nov. 21, 2019 |
| Nov. 21, 2019 | Sheet 9 (typo in Problem 2 corrected) | Nov. 28, 2019 |
| Nov. 28, 2019 | Sheet 10 | Dec. 05, 2019 |
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 |
|---|---|
| Sep. 04, 2019 | Overview and Motivation; Review of Differentiation in Rn Lee Appendix C: parts of Total and Partial Derivatives |
| Sep. 05, 2019 | 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 |
| Sep. 11, 2019 | Review of Topology Lee Appendix A: Topological Spaces |
| Sep. 12, 2019 | Review of Topology Lee Appendix A: Subspaces, Product Spaces, Connectedness and Compactness |
| Sep. 18, 2019 | Definition of Manifolds and Examples Lee Chapter 1: Topological Manifolds (Coordinate Charts, Examples of Topological Manifolds) |
| Sep. 19, 2019 | Examples (Real Projective Space); Differentiable Structures on Manifolds Lee Chapter 1: Topological Manifolds (Connectivity) and Smooth Structures |
| Oct. 01, 2019 | Differentiable Structures on Manifolds; Smooth Maps between Manifolds Lee Chapter 1: Smooth Structures; Lee Chapter 2: beginning of Smooth Functions and Smooth Maps |
| Oct. 02, 2019 | Smooth Maps between Manifolds; Partitions of Unity Lee Chapter 2 (most parts, except Applications of Partitions of Unity) |
| Oct. 03, 2019 | no class (German Unity Day) |
| Oct. 08, 2019 | Tangent Space (Derivations in Rn) Lee Chapter 3: Tangent Vectors (Geometric Tangent Vectors) |
| Oct. 09, 2019 | Tangent Space (Space of Derivations, Differential) Lee Chapter 3: Tangent Vectors (Tangent Vectors on Manifolds), parts of The Differential of a Smooth Map, beginning of Computations in Coordinates |
| Oct. 10, 2019 | Submersions, Immersions, Embeddings (briefly: Rank Theorem) Lee Chapter 4: beginning of Maps of Constant Rank (statement of The Rank Theorem), beginning of Embeddings |
| Oct. 16, 2019 | Embedded Submanifolds Lee Chapter 5: beginning of Embedded Submanifolds, parts of Slice Charts for Embedded Submanifolds, beginning of Level Sets |
| Oct. 17, 2019 | Sard's Theorem Lee Chapter 6 proves Sard's Theorem in full generality in: Sets of Measure Zero and Sard's Theorem |
| Oct. 24, 2019 | Midterm exam |
| Oct. 29, 2019 | Sard's Theorem 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. |
| Oct. 30, 2019 | 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). |
| Oct. 31, 2019 | no class (Reformation Day) |
| Nov. 06, 2019 | Lie Groups Parts of Lee Chapter 7, mostly: Basic Definitions, Lie Group Homomorphisms |
| Nov. 07, 2019 | Lie Groups continued; Vector Fields Lee Chapter 3: The Tangent Bundle; Lee Chapter 8: Vector Fields on Manifolds |
| Nov. 13, 2019 | 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 |
| Nov. 14, 2019 | Integral Curves Lee Chapter 9: Integral Curves; Flows; beginning of The Fundamental Theorem of Flows; beginning of Complete Vector Fields |
| Nov. 20, 2019 | Lie Derivative; Covectors Lee Chapter 9: Lie Derivatives; Lee Chapter 11: Covectors |
| Nov. 21, 2019 | Covector Fields; Multilinear Maps Lee Chapter 11: Tangent Covectors on Manifolds; Covector Fields; The Differential of a Function; Pullbacks of Covector Fields; Lee Chapter 12: beginning of Multilinear Algebra |
| Nov. 27, 2019 | Alternating Tensors; Differential Forms Lee Chapter 14: The Algebra of Alternating Tensors; Differential Forms on Manifolds |
| Nov. 28, 2019 | Differential Forms (Pullback and Exterior Derivative); Orientation Lee Chapter 14: Differential Forms on Manifolds; beginning of Exterior Derivatives; Lee Chapter 15: parts of Orientation of Vector Spaces; beginning of Orientations of Manifolds |
| Dec. 03, 2019 | 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) |
| Dec. 04, 2019 | Manifolds with Boundary and Stokes' Theorem Lee Chapter 1: Manifolds with Boundary; Lee Chapter 15: Boundary Orientations; Lee Chapter 16: Stokes's Theorem |