1. Chapter 8 The Tangent Space

2. 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 at a Point 8.5 Local Expression for the Differential 8.6 Curves in a Manifold 8.7 Computing the Differential Using Curves 8.8 Rank, Critical and Regular Points Problems

3. 8.1 The Tangent Space at a Point

7. 8.2 The Differential of a Map

9. 8.3 The Chain Rule

12. 8.4 Bases for the Tangent Space at a Point

16. 8.5 Local Expression for the Differential

19. 8.6 Curves in a Manifold

25. 8.7 Computing the Differential Using Curves

28. 8.8 Rank, Critical and Regular Points

32. Problems

38. Solution:

40. Hint:

42. Solution: