Smooth manifolds

Calculus review

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

Smooth manifolds

Charts, atlases, smooth structures, and smooth manifolds.

Smooth functions

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

Examples

Inverse images of regular values, products, covering spaces, orthogonal group

Vector fields

Tangent vectors

Different definitions of tangent vectors. A reference for this material is

Tangent bundle

The tangent bundle as a smooth manifold. A reference for this material is

Differentials of smooth maps

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Vector fields

Basic properties of vector fields, including locality. A reference for this material is

Flows

Flows of vector fields: existence and uniqueness. A reference for this material is

Vector bundles

Vector bundles and trivializations

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Vector bundle constructions

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Sections

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Tensors

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Serre–Swan Theorem

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Examples

Differential forms

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Wedge products

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Exterior derivatives

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Oriented manifolds

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Integrals

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Manifolds with boundary

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Stokes Theorem

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Homework

HW 1

Due April 10.

HW 2

Due April 25.

HW 3

Due May 8.

HW 4

Due May 22.

HW 5

Due June 5.