The fundaments of mathematics include:

These mathematical fundaments will employ the orthogonal reference coordinates system introduced in § 2.1 (and visualized in Figure 2.1) and will use extensively the associated unit vectors 11, 12 and 13. These vectors are arranged in a right-hand configuration such that 11 × 12 = 13. Any or all of these will be designated by 1i, with i = 1, 2 and/or 3, as the situation requires. These vectors satisfy the orthogonality relation 1i •1j = δij, where δij is the Kronecker delta.

Vectors and Tensors

This fundament is not a general summary, but instead focuses on vectors and tensors in three-dimensional space, specifically vectors having three elements (or components) and tensors of rank two having nine elements.

Definitions, Notation and Representation

A tensor is an ordered set of nk numbers, where k and n are non-negative integers, that obeys certain tensor transformation rules. The integer k is the rank of the tensor. The A tensor of rank

A tensor of rank 0 is a scalar: a number.

A tensor of rank 1 is a vector: an ordered set of n numbers. Each number is a component of the vector. (Typically n = 3)

Tensors may be thought of as square arrays in k dimensions. A matrix is a related two-dimensional array of size m by n, where m and n are positive integers. A matrix is square if m = n. A tensor of rank two behaves algebraically the same as a square matrix, and the tensor elements may be written out and manipulated in matrix-display form.

In this document, a scalar is denoted by a lower case letter in italics (e.g., x).