The lecture notes were taken by a student in the class. For all of the lecture notes, including a table of contents, download the following file (PDF - 1.6 MB).
| Lec # | Topics |
|---|---|
| 1 | Metric Spaces, Continuity, Limit Points (PDF) |
| 2 | Compactness, Connectedness (PDF) |
| 3 | Differentiation in n Dimensions (PDF) |
| 4 | Conditions for Differentiability, Mean Value Theorem (PDF) |
| 5 | Chain Rule, Mean-value Theorem in n Dimensions (PDF) |
| 6 | Inverse Function Theorem (PDF) |
| 7 | Inverse Function Theorem (cont.), Reimann Integrals of One Variable (PDF) |
| 8 | Reimann Integrals of Several Variables, Conditions for Integrability (PDF) |
| 9 | Conditions for Integrability (cont.), Measure Zero (PDF) |
| 10 | Fubini Theorem, Properties of Reimann Integrals (PDF) |
| 11 | Integration Over More General Regions, Rectifiable Sets, Volume (PDF) |
| 12 | Improper Integrals (PDF) |
| 13 | Exhaustions (PDF) |
| 14 | Compact Support, Partitions of Unity (PDF) |
| 15 | Partitions of Unity (cont.), Exhaustions (cont.) (PDF) |
| 16 | Review of Linear Algebra and Topology, Dual Spaces (PDF) |
| 17 | Tensors, Pullback Operators, Alternating Tensors (PDF) |
| 18 | Alternating Tensors (cont.), Redundant Tensors (PDF) |
| 19 | Wedge Product (PDF) |
| 20 | Determinant, Orientations of Vector Spaces (PDF) |
| 21 | Tangent Spaces and k-forms, The d Operator (PDF) |
| 22 | The d Operator (cont.), Pullback Operator on Exterior Forms (PDF) |
| 23 | Integration with Differential Forms, Change of Variables Theorem, Sard’s Theorem (PDF) |
| 24 | Poincare Theorem (PDF) |
| 25 | Generalization of Poincare Lemma (PDF) |
| 26 | Proper Maps and Degree (PDF) |
| 27 | Proper Maps and Degree (cont.) (PDF) |
| 28 | Regular Values, Degree Formula (PDF) |
| 29 | Topological Invariance of Degree (PDF) |
| 30 | Canonical Submersion and Immersion Theorems, Definition of Manifold (PDF) |
| 31 | Examples of Manifolds (PDF) |
| 32 | Tangent Spaces of Manifolds (PDF) |
| 33 | Differential Forms on Manifolds (PDF) |
| 34 | Orientations of Manifolds (PDF) |
| 35 | Integration on Manifolds, Degree on Manifolds (PDF) |
| 36 | Degree on Manifolds (cont.), Hopf Theorem (PDF) |
| 37 | Integration on Smooth Domains (PDF) |
| 38 | Integration on Smooth Domains (cont.), Stokes’ Theorem (PDF) |
