| 1 | Metric Spaces, Continuity, Limit Points | |
| 2 | Compactness, Connectedness | |
| 3 | Differentiation in n Dimensions | |
| 4 | Conditions for Differentiability, Mean Value Theorem | Graded assignment 1 out |
| 5 | Chain Rule, Mean-value Theorem in n Dimensions | |
| 6 | Inverse Function Theorem | |
| 7 | Inverse Function Theorem | |
| 8 | Reimann Integrals of Several Variables, Conditions for Integrability | |
| 9 | Conditions for Integrability (cont.), Measure Zero | Graded assignment 1 due 2 days after Lec #9 |
| 10 | Fubini Theorem, Properties of Reimann Integrals | Graded assignment 2 out |
| 11 | Integration Over More General Regions, Rectifiable Sets, Volume | |
| 12 | Improper Integrals | |
| 13 | Exhaustions | |
| Midterm | |
| 14 | Compact Support, Partitions of Unity | |
| 15 | Partitions of Unity (cont.), Exhaustions (cont.) | |
| 16 | Review of Linear Algebra and Topology, Dual Spaces | Graded assignment 2 due |
| 17 | Tensors, Pullback Operators, Alternating Tensors | |
| 18 | Alternating Tensors (cont.), Redundant Tensors | |
| 19 | Wedge Product | |
| 20 | Determinant, Orientations of Vector Spaces | Graded assignment 3 out |
| 21 | Tangent Spaces and k-forms, The d Operator | |
| 22 | The d Operator (cont.), Pullback Operator on Exterior Forms | |
| 23 | Integration with Differential Forms, Change of Variables Theorem, Sard's Theorem | |
| 24 | Poincare Theorem | |
| 25 | Generalization of Poincare Lemma | |
| 26 | Proper Maps and Degree | |
| 27 | Proper Maps and Degree (cont.) | |
| 28 | Regular Values, Degree Formula | Graded assignment 3 due |
| 29 | Topological Invariance of Degree | Graded assignment 4 out |
| 30 | Canonical Submersion and Immersion Theorems, Manifolds | |
| 31 | Examples of Manifolds | |
| 32 | Tangent Spaces of Manifolds | |
| 33 | Differential Forms on Manifolds | |
| 34 | Orientations of Manifolds | |
| 35 | Integration on Manifolds, Degree on Manifolds | |
| 36 | Degree on Manifolds (cont.), Hopf Theorem | Graded assignment 4 due |
| 37 | Integration on Smooth Domains | |
| 38 | Integration on Smooth Domains (cont.), Stokes’ Theorem | |
| Final Exam | |