Matrix Tensor Notation Part II. Skew and curved coordinates

Abstract

In Part I, a notation called Matrix-Tensor Notation was introduced for rectilinear orthogonal coordinates. Part II discusses how the same notation is equally efficient for vectors and 2 nd order tensors in skew bases and skew curved coordinates. Without considering coordinates first, vectors and 2 nd order tensors are described in skew bases, covariant and contravariant descriptions being distinguished similar to tensor notation. A consistent interpretation of deformation and strain in terms of skew bases is given. The use of the transpose symbol complicates the matrix algebra, but integrates it with tensor algebra which makes it possible to interpret customary tensor equations as relations in space with all the advantages that an isomorphism with Euclidean space has. It is demonstrated how different metrics can be assigned arbitrarily to define higher-dimensional orthogonal vectors in abstract higher-dimensional space. A consistent notation is given to distinguish between subspace and subbase. To write higher order tensors as matrices, or 2 nd order tensors as vectors, partial transpose is introduced, which is a formal stretching operation to write any higher order tensor product operation as matrix-vector product. The description of the variable bases in generally curved coordinates is given in this notation, together with the corresponding notation for the partial differentiation rules. The vector operations grad, div and curl are discussed and an almost-physical notation for the Christoffel symbols is given. However, the replacement of tensor notation for higher than second order tensors by Matrix Tensor Notation is not feasible. A version of Matrix Tensor Notation that merges with the customary printed form is presented in an Appendix. Further applications are deferred to Part III, applying the new notation to curved coordinates in function space, and the equations of mechanics projected on finite-dimensional subspaces.