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).
Lecture notes files.
| 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) |