# Differential Manifolds

## Overview

Differential Manifold is the framework of particle physics and astrophysics nowadays. It is important for all research physicists to be well accustomed to it and even experimental physicists should be able to manipulate equations and expressions in that framework.

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This book gives a comprehensive description of the basics of differential manifold with a full proof of any element. A large part of the book is devoted to the basic mathematical concepts in which all necessary for the development of the differential manifold is expounded and fully proved.

This book is self-consistent: it starts from first principles. The mathematical framework is the set theory with its axioms and its formal logic. No special knowledge is needed.

**Contents:**

*Manifold:*- Differentiable Manifold
- Smooth Maps
- Vector Fields on a Differentiable Manifold
- Conventions
- Tangent Spaces and Tangent Vectors
- Coordinate Changes
- Metric on a Differentiable Manifold
- One-Form Field and Differential
- Tensorial Field
- Wedge Product of 1-Linear Forms (versus Vector Fields)
- Exterior Differential
- Volume and Integral in Differential Manifold
- Lie Bracket
- Bundles and Differentiable Manifold
- Parallel Transport
- Curvature
- Lagrangian of the Electro-Weak Interactions
- General Relativity
- Notations

*Some Basic Mathematics Needed for Manifolds:*- General Concepts
- Real Numbers, Set
**ℛ** - Euclidean Metric
- Metric and Topology on
**ℛ** - Behavior at a Point
- Some Properties of Continuous Maps from
**ℛ**to**ℛ** - Continuous Maps from Topological Sets to
**ℛ** - Derivable Function
- Group
- Module Over a Commutative Ring
- Vector Spaces
**ℛ**^{n}- Complex Numbers
- Convex Subset
- Topology on
**ℛ**^{n} - Continuous Map on
**ℛ**^{n}to**ℛ**^{p} - Sequence
- Sequence in
**ℛ**_{∞} - Sequence of Maps
- Partial Derivative
- Topology on Convex Subsets
- Path Connected Sets
- Riemann Integral of Maps with Compact Support
- Volume in
**ℛ**^{n} - Integral of a Continuous Map
- Differential Equations
- Lebesgue Integral
- Taylor Expansion of Functions with Derivatives
- Exponentials
- Polynomials
- Useful Smooth Maps Built with Exponentials
- Eigenvectors of a Linear Transformation

*Conventions, Basic Relations and Symbols:*- Logical Theory
- Specifics Terms
- Quantificators
- Specifics Relations
- Sets
- Integers
- Operations on
**Ƶ**=**Ƶ**^{0+}∪**Ƶ**^{–} - Rational Numbers
- Conventions

**Readership:**Undergraduates in particle physics, astrophysics and mathematical physics.