Differential Manifolds

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.

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 ManifoldSmooth MapsVector Fields on a Differentiable ManifoldConventionsTangent Spaces and Tangent VectorsCoordinate ChangesMetric on a Differentiable ManifoldOne-Form Field and DifferentialTensorial FieldWedge Product of 1-Linear Forms (versus Vector Fields)Exterior DifferentialVolume and Integral in Differential ManifoldLie BracketBundles and Differentiable ManifoldParallel TransportCurvatureLagrangian of the Electro-Weak InteractionsGeneral RelativityNotationsSome Basic Mathematics Needed for Manifolds:General ConceptsReal Numbers, Set ℛEuclidean MetricMetric and Topology on ℛBehavior at a PointSome Properties of Continuous Maps from ℛ to ℛContinuous Maps from Topological Sets to ℛDerivable FunctionGroupModule Over a Commutative RingVector SpacesℛnComplex NumbersConvex SubsetTopology on ℛnContinuous Map on ℛn to ℛpSequenceSequence in ℛ∞Sequence of MapsPartial DerivativeTopology on Convex SubsetsPath Connected SetsRiemann Integral of Maps with Compact SupportVolume in ℛnIntegral of a Continuous MapDifferential EquationsLebesgue IntegralTaylor Expansion of Functions with DerivativesExponentialsPolynomialsUseful Smooth Maps Built with ExponentialsEigenvectors of a Linear TransformationConventions, Basic Relations and Symbols:Logical TheorySpecifics TermsQuantificatorsSpecifics RelationsSetsIntegersOperations on Ƶ = Ƶ 0+ ∪ Ƶ–Rational NumbersConventions
Readership: Undergraduates in particle physics, astrophysics and mathematical physics.