Tensors and Manifolds: With Applications to Physics
This book is a new edition of "Tensors and Manifolds: With Applications to Mechanics and Relativity" which was published in 1992. It is based on courses taken by advanced undergraduate and beginning graduate students in mathematics and physics, giving an introduction to the expanse of modern mathematics and its application in modern physics. It aims to fill the gap between the basic courses and the highly technical and specialized courses which both mathematics and physics students require in their advanced training, while simultaneously trying to promote at an early stage, a better appreciation and understanding of each other's discipline. The book sets forth the basic principles of tensors and manifolds, describing how the mathematics underlies elegant geometrical models of classical mechanics, relativity and elementary particle physics. He existing material from the first edition has been reworked and extended in some sections to provide extra clarity, as well as additional problems. Four new chapters on Lie groups and fibre bundles have been included, leading to an exposition of gauge theory and the standard model of elementary particle physics. Mathematical rigor combined with an informal style makes this a very accessible book and will provide the reader with an enjoyable panorama of interesting mathematics and physics.
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TENSOR PRODUCT SPACES
SYMMETRIC AND SKEWSYMMETRIC TENSORS
EXTERIOR GRASSMANN ALGEBRA
THE TANGENT MAP OF REAL CARTESIAN SPACES
ADDITIONAL TOPICS IN MECHANICS
SOME PHYSICS ON MINKOWSKI SPACETIME
SPACETIMES NEAR AN ISOLATED STAR
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1-form fields 1-parameter atlas basis bilinear function called cartesian chart coefficients component functions concept constant coordinate curves coordinate neighborhood coordinate system coordinate vector Corollary corresponding curvature defined definition described diffeomorphism differentiable map differential equations domain dual electromagnetic elements Euclidean example exterior fiber bundle frame of reference gauge transformation geodesic given Hence horizontal hypersurface integral curves isomorphism Lagrangian lemma Levi-Civita connection Lie algebra Lie group linear mapping linearly independent Lorentz transformations Lorentzian matrix metric tensor Minkowski spacetime n-dimensional nondegenerate notation Note numbers open set orthogonal particle particular principal bundle Proof Problem properties Prove Theorem pseudo-Riemannian manifold relation respect result s-form satisfy Section Show skew-symmetric solution structure submanifold subset subspace symmetric symplectic tangent map tangent space tensor field tensor product Theorem timelike topology unique values vector bundle vector space velocity write