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Chapter 8 The Tangent Space

Chapter 8 The Tangent Space. Contents: 8.1 The Tangent Space at a Point 8.2 The Differential of a Map 8.3 The Chain Rule 8.4 Bases for the Tangent Space

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Page 1: Chapter 8 The Tangent Space. Contents: 8.1 The Tangent Space at a Point 8.2 The Differential of a Map 8.3 The Chain Rule 8.4 Bases for the Tangent Space

Chapter 8The Tangent Space

Page 2: Chapter 8 The Tangent Space. Contents: 8.1 The Tangent Space at a Point 8.2 The Differential of a Map 8.3 The Chain Rule 8.4 Bases for the Tangent Space

Contents:

8.1 The Tangent Space at a Point8.2 The Differential of a Map8.3 The Chain Rule8.4 Bases for the Tangent Space at a Point8.5 Local Expression for the Differential8.6 Curves in a Manifold8.7 Computing the Differential Using Curves8.8 Rank, Critical and Regular Points Problems

Page 3: Chapter 8 The Tangent Space. Contents: 8.1 The Tangent Space at a Point 8.2 The Differential of a Map 8.3 The Chain Rule 8.4 Bases for the Tangent Space

8.1 The Tangent Space at a Point

Page 4: Chapter 8 The Tangent Space. Contents: 8.1 The Tangent Space at a Point 8.2 The Differential of a Map 8.3 The Chain Rule 8.4 Bases for the Tangent Space
Page 5: Chapter 8 The Tangent Space. Contents: 8.1 The Tangent Space at a Point 8.2 The Differential of a Map 8.3 The Chain Rule 8.4 Bases for the Tangent Space
Page 6: Chapter 8 The Tangent Space. Contents: 8.1 The Tangent Space at a Point 8.2 The Differential of a Map 8.3 The Chain Rule 8.4 Bases for the Tangent Space
Page 7: Chapter 8 The Tangent Space. Contents: 8.1 The Tangent Space at a Point 8.2 The Differential of a Map 8.3 The Chain Rule 8.4 Bases for the Tangent Space

8.2 The Differential of a Map

Page 8: Chapter 8 The Tangent Space. Contents: 8.1 The Tangent Space at a Point 8.2 The Differential of a Map 8.3 The Chain Rule 8.4 Bases for the Tangent Space
Page 9: Chapter 8 The Tangent Space. Contents: 8.1 The Tangent Space at a Point 8.2 The Differential of a Map 8.3 The Chain Rule 8.4 Bases for the Tangent Space

8.3 The Chain Rule

Page 10: Chapter 8 The Tangent Space. Contents: 8.1 The Tangent Space at a Point 8.2 The Differential of a Map 8.3 The Chain Rule 8.4 Bases for the Tangent Space
Page 11: Chapter 8 The Tangent Space. Contents: 8.1 The Tangent Space at a Point 8.2 The Differential of a Map 8.3 The Chain Rule 8.4 Bases for the Tangent Space
Page 12: Chapter 8 The Tangent Space. Contents: 8.1 The Tangent Space at a Point 8.2 The Differential of a Map 8.3 The Chain Rule 8.4 Bases for the Tangent Space

8.4 Bases for the Tangent Space at a Point

Page 13: Chapter 8 The Tangent Space. Contents: 8.1 The Tangent Space at a Point 8.2 The Differential of a Map 8.3 The Chain Rule 8.4 Bases for the Tangent Space
Page 14: Chapter 8 The Tangent Space. Contents: 8.1 The Tangent Space at a Point 8.2 The Differential of a Map 8.3 The Chain Rule 8.4 Bases for the Tangent Space
Page 15: Chapter 8 The Tangent Space. Contents: 8.1 The Tangent Space at a Point 8.2 The Differential of a Map 8.3 The Chain Rule 8.4 Bases for the Tangent Space
Page 16: Chapter 8 The Tangent Space. Contents: 8.1 The Tangent Space at a Point 8.2 The Differential of a Map 8.3 The Chain Rule 8.4 Bases for the Tangent Space

8.5 Local Expression for the Differential

Page 17: Chapter 8 The Tangent Space. Contents: 8.1 The Tangent Space at a Point 8.2 The Differential of a Map 8.3 The Chain Rule 8.4 Bases for the Tangent Space
Page 18: Chapter 8 The Tangent Space. Contents: 8.1 The Tangent Space at a Point 8.2 The Differential of a Map 8.3 The Chain Rule 8.4 Bases for the Tangent Space
Page 19: Chapter 8 The Tangent Space. Contents: 8.1 The Tangent Space at a Point 8.2 The Differential of a Map 8.3 The Chain Rule 8.4 Bases for the Tangent Space

8.6 Curves in a Manifold

Page 20: Chapter 8 The Tangent Space. Contents: 8.1 The Tangent Space at a Point 8.2 The Differential of a Map 8.3 The Chain Rule 8.4 Bases for the Tangent Space
Page 21: Chapter 8 The Tangent Space. Contents: 8.1 The Tangent Space at a Point 8.2 The Differential of a Map 8.3 The Chain Rule 8.4 Bases for the Tangent Space
Page 22: Chapter 8 The Tangent Space. Contents: 8.1 The Tangent Space at a Point 8.2 The Differential of a Map 8.3 The Chain Rule 8.4 Bases for the Tangent Space
Page 23: Chapter 8 The Tangent Space. Contents: 8.1 The Tangent Space at a Point 8.2 The Differential of a Map 8.3 The Chain Rule 8.4 Bases for the Tangent Space
Page 24: Chapter 8 The Tangent Space. Contents: 8.1 The Tangent Space at a Point 8.2 The Differential of a Map 8.3 The Chain Rule 8.4 Bases for the Tangent Space
Page 25: Chapter 8 The Tangent Space. Contents: 8.1 The Tangent Space at a Point 8.2 The Differential of a Map 8.3 The Chain Rule 8.4 Bases for the Tangent Space

8.7 Computing the Differential Using Curves

Page 26: Chapter 8 The Tangent Space. Contents: 8.1 The Tangent Space at a Point 8.2 The Differential of a Map 8.3 The Chain Rule 8.4 Bases for the Tangent Space
Page 27: Chapter 8 The Tangent Space. Contents: 8.1 The Tangent Space at a Point 8.2 The Differential of a Map 8.3 The Chain Rule 8.4 Bases for the Tangent Space
Page 28: Chapter 8 The Tangent Space. Contents: 8.1 The Tangent Space at a Point 8.2 The Differential of a Map 8.3 The Chain Rule 8.4 Bases for the Tangent Space

8.8 Rank, Critical and Regular Points

Page 29: Chapter 8 The Tangent Space. Contents: 8.1 The Tangent Space at a Point 8.2 The Differential of a Map 8.3 The Chain Rule 8.4 Bases for the Tangent Space
Page 30: Chapter 8 The Tangent Space. Contents: 8.1 The Tangent Space at a Point 8.2 The Differential of a Map 8.3 The Chain Rule 8.4 Bases for the Tangent Space
Page 31: Chapter 8 The Tangent Space. Contents: 8.1 The Tangent Space at a Point 8.2 The Differential of a Map 8.3 The Chain Rule 8.4 Bases for the Tangent Space
Page 32: Chapter 8 The Tangent Space. Contents: 8.1 The Tangent Space at a Point 8.2 The Differential of a Map 8.3 The Chain Rule 8.4 Bases for the Tangent Space

Problems

Page 33: Chapter 8 The Tangent Space. Contents: 8.1 The Tangent Space at a Point 8.2 The Differential of a Map 8.3 The Chain Rule 8.4 Bases for the Tangent Space
Page 34: Chapter 8 The Tangent Space. Contents: 8.1 The Tangent Space at a Point 8.2 The Differential of a Map 8.3 The Chain Rule 8.4 Bases for the Tangent Space
Page 35: Chapter 8 The Tangent Space. Contents: 8.1 The Tangent Space at a Point 8.2 The Differential of a Map 8.3 The Chain Rule 8.4 Bases for the Tangent Space
Page 36: Chapter 8 The Tangent Space. Contents: 8.1 The Tangent Space at a Point 8.2 The Differential of a Map 8.3 The Chain Rule 8.4 Bases for the Tangent Space
Page 37: Chapter 8 The Tangent Space. Contents: 8.1 The Tangent Space at a Point 8.2 The Differential of a Map 8.3 The Chain Rule 8.4 Bases for the Tangent Space
Page 38: Chapter 8 The Tangent Space. Contents: 8.1 The Tangent Space at a Point 8.2 The Differential of a Map 8.3 The Chain Rule 8.4 Bases for the Tangent Space

Solution:

Page 39: Chapter 8 The Tangent Space. Contents: 8.1 The Tangent Space at a Point 8.2 The Differential of a Map 8.3 The Chain Rule 8.4 Bases for the Tangent Space
Page 40: Chapter 8 The Tangent Space. Contents: 8.1 The Tangent Space at a Point 8.2 The Differential of a Map 8.3 The Chain Rule 8.4 Bases for the Tangent Space

Hint:

Page 41: Chapter 8 The Tangent Space. Contents: 8.1 The Tangent Space at a Point 8.2 The Differential of a Map 8.3 The Chain Rule 8.4 Bases for the Tangent Space
Page 42: Chapter 8 The Tangent Space. Contents: 8.1 The Tangent Space at a Point 8.2 The Differential of a Map 8.3 The Chain Rule 8.4 Bases for the Tangent Space

Solution:


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