Tensor Calculus: A Concise Course
"This book will prove to be a good introduction, both for the physicist who wishes to make applications and for the mathematician who prefers to have a short survey before taking up one of the more voluminous textbooks on differential geometry." — MathSciNet (Mathematical Reviews on the Web), American Mathematical Society
A compact exposition of the fundamental results in the theory of tensors, this text also illustrates the power of the tensor technique by its applications to differential geometry, elasticity, and relativity. The first five chapters--comprising tensor algebra, the line element, covariant differentiation, geodesics and parallelism, and curvature tensor--develop their subjects without undue rigor. The final three chapters function independently of each other and cover Euclidean three-dimensional differential geometry, Cartesian tensors and elasticity, and the theory of relativity. Both special and general theories of relativity are reviewed, with introductory material for readers unfamiliar with the concepts.
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angle arbitrary axes calculation called Cartesian tensor Christoffel symbols co-directional constant curvature contravariant tensor contravariant vector coor coordinate curves corresponding covariant derivative covariant differentiation covariant tensor covariant vector deduce defined denote determined Differential Geometry dummy indices Einstein space elasticity Euclidean space field of parallel follows functions fundamental tensor given Hence Inner multiplication intrinsic derivative invariant isotropic tensor Kronecker delta magnitude metric Minkowski space mixed tensor motion necessary and sufficient non-vanishing notation null curve null-geodesic obtain orthogonal transformation parallel propagation parallel vectors parameter partial derivatives polar coordinates pole ponents positive-definite principal directions principal normal Prove quantum quotient law respect rotation satisfy the equation second order sin2 skew-symmetric space VN spherical polar strain tensor substitute sufficient condition summation superscript surface tensor surface vector symbols are zero symmetric tensor tangent vector tensor derivative tensor g tion transformation law unit vector variant yields