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Contents

Co-ordinate independent calculus.

Introduction

Smooth functions

Derivatives as linear operators.

The chain rule.

Diffeomorphisms and the inverse function theorem.

Differentiable manifolds

Co-ordinate charts

Linear manifolds.

Topology of a manifold

Smooth functions on a manifold.

The tangent space.

The derivative of a function.

Co-ordinate tangent vectors and one-forms.

How to calculate.

Submanifolds

Tangent space to a submanifold

Smooth functions between manifolds

The tangent to a smooth map.

Submanifolds again.

Vector fields.

The Lie bracket.

Differential forms.

The exterior algebra of a vector space.

Differential forms and the exterior derivative.

Pulling back differential forms

Integration of differential forms

Orientation.

Integration again

Stokes theorem.

Manifolds with boundary.

Stokes theorem.

Partitions of unity.

Vector fields and the tangent bundle.

Vector fields and derivations.

Tensor products

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# Differential Geometry. Honours 1996

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