18.101 | Fall 2005 | Undergraduate

Analysis II


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


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

Course Info

As Taught In
Fall 2005
Learning Resource Types
Lecture Notes