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 |

## Calendar

## Course Info

##### Instructor

##### Departments

##### As Taught In

Fall
2005

##### Level

##### Learning Resource Types

*notes*Lecture Notes