There are two types of assignments given in this course: daily assignments and graded assignments. The daily assignments are not graded, but one problem from each day is usually included in a graded assignment. Not all lectures have assigned daily problems.
Assignments listed in the table below are from the following textbooks and notes:
(M) Munkres, J. Analysis on Manifolds. Cambridge, MA: Perseus Publishing, 1991. ISBN: 0201510359, ISBN: 0201315963 (paperback).
(S) Spivak, M. Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus. Cambridge, MA: Perseus Publishing, 1965. ISBN: 0805390219.
(MLA) Notes on Multi-linear Algebra (PDF)
(SN) Supplementary Notes (PDF)
| Lec # | Topics | DAILY ASSIGNMENTS | GRADED ASSIGNMENTS |
|---|---|---|---|
| 1 | Metric Spaces, Continuity, Limit Points | M, section 3: 2, 3, 4, 8, 9 | |
| 2 | Compactness, Connectedness | M, section 4: 1, 2, 3, 4, concentrate on 3 | |
| 3 | Differentiation in n Dimensions | M, section 5: 2, 3, 4, 5, 7 | |
| 4 | Conditions for Differentiability, Mean Value Theorem | M, section 6: 2, 5, 9, 10 | M, 4.3, 5.3, 6.10, 8.4 S, 2-7 |
| 5 | Chain Rule, Mean-value Theorem in n Dimensions | M, section 7: 1, 2, 3 | |
| 6 | Inverse Function Theorem | M, section 8: 1, 2 | |
| 7 | Inverse Function Theorem (cont.), Reimann Integrals of One Variable | M, section 8: 3, 4, 5 | |
| 8 | Reimann Integrals of Several Variables, Conditions for Integrability | M, section 10: 1, 3, 4, 5 | |
| 9 | Conditions for Integrability (cont.), Measure Zero | M, section 12: 1, 2, 3, 4 | |
| 10 | Fubini Theorem, Properties of Reimann Integrals | M, section 13: 1, 2, 4, 5 | M, 12.2, 13.2, 14.8, 15.4, 16.3 |
| 11 | Integration Over More General Regions, Rectifiable Sets, Volume | M, section 14: 1, 4, 5, 7 (Hint: look at Example 1 of section 14 for help with two of the homework problems.) | |
| 12 | Improper Integrals | M, section 15: 1, 2, 4, 5 | |
| 13 | Exhaustions | ||
| Midterm | |||
| 14 | Compact Support, Partitions of Unity | M, section 16: 2, 3 | |
| 15 | Partitions of Unity (cont.), Exhaustions (cont.) | ||
| 16 | Review of Linear Algebra and Topology, Dual Spaces | MLA, section 2: 1, 2, 3, 4 | |
| 17 | Tensors, Pullback Operators, Alternating Tensors | MLA, section 3: 1, 2, 4, 6, 7 | |
| 18 | Alternating Tensors (cont.), Redundant Tensors | MLA, section 4: 1, 2, 3, 4, 5 | |
| 19 | Wedge Product | MLA, section 5: 1, 2 and section 6: 1 | |
| 20 | Determinant, Orientations of Vector Spaces | MLA, section 6: 2, 3, 4, 5 | (PDF) |
| 21 | Tangent Spaces and k-forms, The d Operator | ||
| 22 | The d Operator (cont.), Pullback Operator on Exterior Forms | M, section 30: 2, 3, 4, 6 | |
| 23 | Integration with Differential Forms, Change of Variables Theorem, Sard's Theorem | SN, section 1: 1, 2, 4, 5 | |
| 24 | Poincare Theorem | SN, section 2: 1, 2, 3 | |
| 25 | Generalization of Poincare Lemma | ||
| 26 | Proper Maps and Degree | SN, section 4: 3, 4, 5, 6, 7 | |
| 27 | Proper Maps and Degree (cont.) | ||
| 28 | Regular Values, Degree Formula | ||
| 29 | Topological Invariance of Degree | M, section 24: 6 SN, section 2: 2, section 4: 8 (need 5-7), section 6: 6 |
|
| 30 | Canonical Submersion and Immersion Theorems, Definition of Manifold | Prove the canonical submersion and immersion theorems for linear maps (as stated in class). | |
| 31 | Examples of Manifolds | M, section 23: 1, 4, 5 and section 24: 5, 6 | |
| 32 | Tangent Spaces of Manifolds | M, section 29: 1, 2, 3, 5 | |
| 33 | Differential Forms on Manifolds | MLA, section 7: 1, 4, 5, 6 | |
| 34 | Orientations of Manifolds | M, section 34: 3, 6 S, problems 5-14 on p. 120 |
|
| 35 | Integration on Manifolds, Degree on Manifolds | ||
| 36 | Degree on Manifolds (cont.), Hopf Theorem | (PDF) | |
| 37 | Integration on Smooth Domains | (PDF) | |
| 38 | Integration on Smooth Domains (cont.), Stokes’ Theorem | ||
| Final Exam |