Vector and Tensor Algebra

Abstract

This notebook and the package tensalgv2.m contain besides of elementary vector algebra a complete tensor algebra as a part of affine geometry. In Mathematica there doesn’t exist a built-in Tensor definition, but some tensor operations, like e.g. TensorProduct are built-in. In the notebook I define a symbolic tensor object which admits to introduce covariant, contravariant and mixed type tensors. The concept can be used for symbolic calculations as well as for calculations with tensors whose coordinates are numerically or functionally specified. Tensor products, the wedge product as the base of exterior algebra, and contraction of tensors are introduced. Their properties are deduced and compared with the corresponding Mathematica built-in tensor functions. Since also in the commercial packages of Harald H. Soleng, Tensors in Physics, and Steven M. Christensen, MathTensor, an abstract tensor concept, as far as I know, is missing I hope that the notebook and the package presented here deliver useful tools for applications of Mathematica to problems in algebra, geometry and physics needing tensor calculus. The analytic part of tensor calculus is treated in connection with pseudo-Riemannian Geometry in the notebook RG.nb.