By Knopp K.

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Quantities such as A or T with˜ two ˜under-tildes are second˜ beneath ˜ order tensors. In general, the number of under-tildes a symbol indicates to you the order of that tensor (for this reason, scalars are sometimes called zeroth-order tensors and vectors are called first-order tensors). Occasionally, we will want to make statements that apply equally well to tensors of any order. In that case, we might use single straight underlines. , x or y ) might represent scalars, vectors, tensors, or other abstract objects.

Individual copies may be made for personal use. No part of this document may be reproduced for profit. 49) ˜ ˜ - = 2v dv ˜ ˜ To understand why we used “d” instead of “ ∂ ” in this equation, refer to the “sidebar” on page 266. The outer product of two column matrices. 50) For this case, the value of the “adjacent” dimension R in Eq. 5) is just 1, so the summation ranges from 1 to 1 (which means that it is just a solitary term). The result of the outer product is an M × N matrix, whose ij component is given by a i b j .

Otherwise it is called curvilinear. If a surface is curved but could be “unrolled” into a flat surface or into a line, then the surface is called Euclidean; qualitatively, a space is Euclidean if it is always possible to set up a coordinate grid covering the space in such a manner that the coordinate grid cells are all equal sized squares or cubes. The surface of a cylinder is both curvilinear and Euclidean. By contrast, the surface of a sphere is curvilinear and non-Euclidean. Mapping a nonEuclidean space to Euclidean space will always involve distortions in shape and/or size.

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