# 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).

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)