Partial derivatives, \(C^k\) functions, chain rule, inverse function theorem, implicit function theorem.

Charts, atlases, smooth structures, and smooth manifolds.

Smooth functions, diffeomorphisms, bump functions, and partitions of unity.

Inverse images of regular values, products, covering spaces, classical groups, connected sums.

Different definitions of tangent vectors.

The tangent bundle as a smooth manifold.

Differential of a smooth function. Immersions, submersions, and embeddings.

Submanifolds. Examples given by embeddings and level sets.

Vector fields, global derivations, locality.

Flows of vector fields: existence and uniqueness.

Lie brackets of vector fields. Fields commute if and only if their flows commute.

Super quick review of algebras, ideals, and modules.

Tensor products: definition and basic properties.

Vector bundles, trivializations, basic constructions, basic properties.

Projective finitely-generated modules, and Serre-Swan Theorem.

Tensors on manifolds.

Antisymmetrization and forms.

Wedge products, graded commutativity, and associativity.

Exterior derivatives, charts, Leibniz rule, and nilpotency.

Pullbacks: interaction with wedge product and exterior derivative.

Orientations.

Integration of forms on \(\mathbb {R}^n\) and on manifolds.

Everything generalizes to manifolds with boundary.

Stokes Theorem

Due April 10.

Due April 25.

Due May 10.

Due May 24.

Due June 7.