Mathematical Methods in Science and Engineering, 2nd Edition

Chapter 9Coordinates and Tensors

Using a coordinate system is probably the fastest way to introduce mathematics into the study of nature. A coordinate system in tune with the symmetries of the physical system at hand, not only simplifies the algebra but also makes the interpretation of the solution easier. Once a coordinate system is defined, physical processes can be studied in terms of operations among mathematical symbols like scalars, vectors, tensors, etc. that represents the physical properties of the system. Regularities and symmetries among the physical phenomena can now be expressed in terms of mathematical expressions as laws of nature. Naturally, the true laws of nature should not depend on what coordinate system is being used. Therefore, it should be possible to express the laws of nature in coordinate independent formalism. In this regard, tensor equations, which preserve their form under general coordinate transformations, have proven to be very useful. In this chapter, we start with the Cartesian coordinates and their transformations. We also introduce Cartesian tensors and their application to the theory of elasticity. We then generalize our discussion to generalized coordinates and general tensors. Curvature, parallel transport, geodesics are other interesting topics discussed in this chapter. The next stop in our discussion is coordinate systems in Minkowski spacetime and their transformation properties. We also introduce four-tensors in spacetime and discuss ...