Differential geometry for physicists and mathematicians : moving frames and differential forms : from Euclid past Riemann / José G. Vargas.

Format:

Book

Author/Creator:

Vargas, José G., author.

Language:

English

Subjects (All):

Mathematical physics.

Geometry, Differential.

Physical Description:

1 online resource (312 p.)

Place of Publication:

Singapore : World Scientific, 2014.

Language Note:

English

Summary:

This is a book that the author wishes had been available to him when he was student. It reflects his interest in knowing (like expert mathematicians) the most relevant mathematics for theoretical physics, but in the style of physicists. This means that one is not facing the study of a collection of definitions, remarks, theorems, corollaries, lemmas, etc. but a narrative - almost like a story being told - that does not impede sophistication and deep results. It covers differential geometry far beyond what general relativists perceive they need to know. And it introduces readers to other areas

Contents:

2.6 Change of basis of differential forms2.7 Differential forms and measurement; 2.8 Differentiable manifolds DEFINED; 2.9 Another definition of differentiable MANIFOLD; 3 VECTOR SPACES AND TENSOR PRODUCTS; 3.1 INTRODUCTION; 3.2 Vector spaces (over the reals); 3.3 Dual vector spaces; 3.4 Euclidean vector spaces; 3.4.1 Definition; 3.4.2 Orthonormal bases; 3.4.3 Reciprocal bases; 3.4.4 Orthogonalization; 3.5 Not quite right concept of VECTOR FIELD; 3.6 Tensor products: theoretical minimum; 3.7 Formal approach to TENSORS; 3.7.1 Definition of tensor space

3.7.2 Transformation of components of tensors3.8 Clifford algebra; 3.8.1 Introduction; 3.8.2 Basic Clifford algebra; 3.8.3 The tangent Clifford algebra of 3-D Euclidean vector space; 3.8.4 The tangent Clifford algebra of spacetime; 3.8.5 Concluding remarks; 4 EXTERIOR DIFFERENTIATION; 4.1 Introduction; 4.2 Disguised exterior derivative; 4.3 The exterior derivative; 4.4 Coordinate independent definition of exterior derivative; 4.5 Stokes theorem; 4.6 Differential operators in language of forms; 4.7 The conservation law for scalar-valuedness; 4.8 Lie Groups and their Lie algebras

III TWO KLEIN GEOMETRIES5 AFFINE KLEIN GEOMETRY; 5.1 Affine Space; 5.2 The frame bundle of affine space; 5.3 The structure of affine space; 5.4 Curvilinear coordinates: holonomic bases; 5.5 General vector basis fields; 5.6 Structure of affine space on SECTIONS; 5.7 Differential geometry as calculus; 5.8 Invariance of connection differential FORMS; 5.9 The Lie algebra of the affine group; 5.10 The Maurer-Cartan equations; 5.11 HORIZONTAL DIFFERENTIAL FORMS; 6 EUCLIDEAN KLEIN GEOMETRY; 6.1 Euclidean space and its frame bundle; 6.2 Extension of Euclidean bundle to affine bundle

6.3 Meanings of covariance6.4 Hodge duality and star operator; 6.5 The Laplacian; 6.6 Euclidean structure and integrability; 6.7 The Lie algebra of the Euclidean group; 6.8 Scalar-valued clifforms: Kahler calculus; 6.9 Relation between algebra and geometry; IV CARTAN CONNECTIONS; 7 GENERALIZED GEOMETRY MADE SIMPLE; 7.1 Of connections and topology; 7.2 Planes; 7.2.1 The Euclidean 2-plane; 7.2.2 Post-Klein 2-plane with Euclidean metric; 7.3 The 2-sphere; 7.3.1 The Columbus connection on the punctured 2-sphere; 7.3.2 The Levi-Civita connection on the 2-sphere

7.3.3 Comparison of connections on the 2-sphere

Notes:

Description based upon print version of record.

Includes bibliographical references and index.

Description based on print version record.

ISBN:

981-4566-40-3