Vector fields, linear spaces and tensors in physics. Part 3: Tensor analysis and its applications

Mathematics

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Derivation, Maxwell Equation, Planetary Orbits, Reciprocal Theorems, Schwarzschild Metric, Tensor Analysis, Tensors, Cartesian Coordinates, Matrices (Mathematics), Oblique Coordinates, Rotation, Spherical Coordinates, Symbols

Scientific paper

The tensors, their transformation and other properties, from the viewpoint of their interpretation and use in physics are studied. Tensors are first introduced by showing the need of their existence in physics. Then the tensors are formally defined through the transformation properties of their components upon the rotation of orthogonal Cartesian coordinates; other properties of Cartesian tensors including tensor products and derivatives are also studied. A brief treatment of tensors follow which refer to oblique Cartesian coordinates such that the concepts of covariant and contravariant terms emerge in a natural way. Tensors referred to arbitrary curvilinear coordinates are formally treated studing the Jacobian of the transformation, contraction, the fundamental and reciprocal tensors, the quotient theorem, physical components, principal directions, Christoffel symbols, and covariant derivative among several others. An introduction to application of the tensor theory by defining the concept of absolute derivative, deriving the equation of geodesics and the curvature tensor, is offered. The Einstein's gravitational theory, nonetheless part of the applications, is separately treated solving the equations following the Schwarzschild spherically symmetric solution applied to planetary orbits. General relativistic expressions for the Maxwell's equations are also derived. In every chapter, whenever possible, elucidating examples are worked and problems that help better understand the theory are proposed.

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