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Erschienen in:
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2020 | OriginalPaper | Buchkapitel

1. Vector, Tensors, and Related Matters

verfasst von : Ciprian D. Coman

Erschienen in: Continuum Mechanics and Linear Elasticity

Verlag: Springer Netherlands

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Abstract

All mathematical models of Continuum Mechanics are described by using vectorial and tensorial quantities. For this reason, a certain fluency in vector and tensor manipulations is imperative for much of the rest of the book. The purpose of the current chapter is largely twofold. While the first few sections can serve as a brief review of several key topics in Linear Algebra, for the most part the attention will be directed towards the concept of second-order tensor and its higher order generalisations. The last couple of sections indicate how the usual vector calculus can be extended to tensor fields and functions whose arguments are tensors.

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Nächstes Kapitel Kinematics
Fußnoten
1
It must be emphasised that an n-tuple is not a \(1\times {n}\) (row) matrix; the elements of a vector space can be represented as either row or column matrices, depending on the situation at hand.
 
2
The cyclic permutations of the ordered triple (1, 2, 3) are (1, 2, 3), (2, 3, 1) and (3, 1, 2); note that an even number of transpositions is needed to restore the original order in each of these cases.
 
3
In the usual three-dimensional Euclidean point space (to be introduced later), this convention corresponds to the ‘right-hand rule’: if you put the index of your right hand on \(\varvec{u}\) and the middle finger on \(\varvec{v}\), then your thumb should point in the direction of \(\varvec{u}\wedge \varvec{v}\).
 
4
In general, the vector triple product is not associative: \(\varvec{a}\wedge (\varvec{b}\wedge \varvec{c})\ne (\varvec{a}\wedge \varvec{b})\wedge \varvec{c}\). If the first and the last terms in such a product are equal, then it does not matter the order in which the products are carried out, e.g. \(\varvec{n}\wedge (\varvec{a}\wedge \varvec{n}) = \varvec{n}\wedge \varvec{a}\wedge \varvec{n}= (\varvec{n}\wedge \varvec{a})\wedge \varvec{n}\).
 
5
In this notation, \(\varvec{u}\otimes \varvec{v}\) stands for the ordered pair of \(\varvec{u}\) and \(\varvec{v}\), while the multiplication sign is just a reminder of what to expect.
 
6
CAVEAT: in the literature this is usually called the ‘parallel’ projection, while (1.44) is referred to as the ‘orthogonal’ projection.
 
7
We must emphasise that the vector itself does not change, it is just its components that are modified; the use of different labels for our vector is employed here solely for emphasising this distinction.
 
8
CAVEAT: the definition of the angle \(\theta \) for spherical polar coordinates is different from that in the previous section; refer to Figs. 1.8 and 1.9 for the corresponding differences.
 
9
As the notation suggests, \(\varvec{U}\) represents the square root of \(\varvec{S}\).
 
10
If \(P\in \mathbb {E}^3\) and \(\varepsilon >0\), we say that an \(\varepsilon \)-neighbourhood of P is the set of all points that are less than \(\varepsilon \) in distance away from P. An interior point of a given region \(\varOmega \subset \mathbb {E}^3\) is one which has an \(\varepsilon \)-neighbourhood that lies completely in that region; \(\varOmega \) is said to be open if every point in that region is an interior point.
 
11
We use square brackets to indicate that \(\varvec{h}\rightarrow \varvec{f}'(\varvec{x}_0)[\varvec{h}]\) is a linear mapping; by contrast, \(\varvec{x}_0\rightarrow \varvec{f}'(x_0)\) is a non-linear mapping in general.
 
12
This is known as the Riesz Representation theorem and asserts that for every linear functional \(T\in L\,(\mathcal {Z}_1,\mathbb {R})\) there exists a unique \(\varvec{a}\in \mathcal {Z}_1\) such that \(T[\varvec{x}]=\varvec{a}\cdot \varvec{x}\) for all \(\varvec{x}\in \mathcal {Z}_1\).
 
13
Sometimes \(\varvec{u}\wedge \varvec{\nabla }\) is defined for vector fields as well, but in that case \(\varvec{u}\wedge \varvec{\nabla }:=-\varvec{\nabla }\wedge \varvec{u}\).
 
14
The boundary of a subset \(\varOmega \subset \mathbb {E}^3\), denoted \(\partial \varOmega \), consists of those points \(P\in \mathbb {E}^3\) such that every \(\varepsilon \)-neighbourhood of P (\(\varepsilon >0\)) intersects both \(\varOmega \) and the complement of \(\varOmega \). This definition applies to subsets of \(\mathbb {E}^2\) as well.
 
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Metadaten
Titel
Vector, Tensors, and Related Matters
verfasst von
Ciprian D. Coman
Copyright-Jahr
2020
Verlag
Springer Netherlands
Buch
Continuum Mechanics and Linear Elasticity

Print ISBN: 978-94-024-1769-2

Electronic ISBN: 978-94-024-1771-5

Copyright-Jahr: 2020

DOI
https://doi.org/10.1007/978-94-024-1771-5_1