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from Part VIII - Fundaments
Published online by Cambridge University Press: 26 October 2017
The fundaments of mathematics include:
• A.1: a brief introduction to the algebra of vectors and tensors in three dimensions;
• A.2: a summary of the calculus of three-dimensional vectors;
• A.3: an introduction to curvilinear coordinate systems;
• A.4: an introduction to Taylor series;
• A.5: an introduction to Fourier series and integrals;
• A.6: classification of several simple types of linear, second-order partial differential equations;
• A.7: a listing of the Greek symbols that are commonly used in applied mathematics; and
• A.8: an introduction to scalar and vector potentials.
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).
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