Analysis II

Calendar

Lec #	Topics	KEY DATES
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