- Exterior Algebra
- Exterior Calculus: Tangent Spaces, Differential Forms, Exterior Derivatives, and Differential of a Map
- Immersions, Submersions, Submanifolds of Euclidean Space
- Differentiable Manifolds
- Vector Bundles and the Tangent Bundle
- Frame Fields and Riemannian Metrics
- Integration on an Oriented Manifold and Stokes' Theorem
- Connections

- Spivak, M.,
**A Comprehensive Introduction to Differential Geometry**, Vols. I-V, Publish or Perish, Berkeley (1979). - Warner, F.,
**Foundations of Differentiable Manifolds and Lie Groups**, Springer, New York (1983).

Read Chapter 1, pgs. 1-23. Try the following exercises:

- Section 1.3, p. 7-8, Exercises: 1*, 2, 4, 5*, 6*, 7*
- Section 1.5, p. 12-13, Exercises: 9*, 10*, 13, 14
- Section 1.8, p. 20-21, Exercises: 16*, 17*, 18, 19*, 21, 23, 24

Continue reading Chapter 1, pgs. 1-23. Also read the handout on Multilinear
Algebra.

Read Chapter 2, pgs. 24-52. Try the following exercises:

- Section 2.4, p. 33-35, Exercises: 1*, 2*, 3, 4, 6, 7
- Section 2.6, p. 39-41, Exercises: 12a*, 13*, 14, 15*, 16, 17
- Section 2.9, p. 47-49, Exercises: 18, 19*, 20*, 21*, 22*, 23*, 24*, 25

Section 2.11 on Maxwell's Equations will be particularly interesting to those with a background in physics. (This section is optional and may be omitted.)

Continue reading Chapter 2, pgs. 24-52.

Do the following exercises for Chapters 1 and 2:

- Section 1.3 6
- Section 1.5 10
- Section 1.8 Show that (*u,*v) = (u,v) where u and v are smooth p-forms on an open set U of n-space. The inner product (u,v) is the one induced on p-forms, etc.
- Section 2.4 1, 6
- Section 2.6 13, 15
- Section 2.9 19, 21, 22

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