# Lecture Notes

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)