Abstract
Vector and dyadic (second-order tensor) fields are basic entities in Fluid Mechanics and MHD. The term “field” in this book means that the quantity is a function of position in three-dimensional space, in addition to any time dependence. This spatial dependence is conveniently represented by the associated position vector \(\mathbf{r}\) that may itself be a function of the time t (in a dynamical system), so any field is an invariant function of the form \({\varvec{\mathsf f}}(\mathbf{r}(t), t)\). On the other hand, vector and tensor representations depend upon the reference coordinate system chosen, and we discuss the representation of vectors as a useful prelude to the following sections on dyadics and their representation. The vector differential operator then introduced is used more extensively, from determining basis sets for coordinate systems to its role in so many mathematical expressions throughout the book. Although there is a brief section on the familiar special case of orthogonal curvilinear coordinates, it is notable that the previous and following sections on the integral theorems apply to any three-dimensional coordinate system (non-orthogonal curvilinear coordinate systems are particularly important in MHD). The remaining three sections on Green identities and Heaviside, Dirac and Green functions complete this chapter on relevant mathematical topics. The associated bibliography provides recommended sources on vectors and tensors, on distribution theory and mathematical methods, and on partial differential equations for further reading.
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Notes
- 1.
Encyclopedia Britannica provides accurate biographies on many of the mathematicians and physicists mentioned throughout this book, and there are various other (sometimes less reliable) sources of information available nowadays on the internet—e.g. Wikipedia.
- 2.
The superposed hat symbol is used to denote unit vectors throughout this book.
- 3.
The adjective “metric” refers to the coefficient \(g^{\textit{ij}}\) defining the length ds of a line element at any point in space, corresponding to \(ds^{2}=d\mathbf{r} \varvec{\,\cdot \,}d\mathbf{r}=dx_{i}\mathbf{e}^{i} \varvec{\,\cdot \,}dx_{j}\mathbf{e}^{j} = g^{\textit{ij}} dx_{i} dx_{j}\) where \(\mathbf{r}\) denotes the position vector of the point. From their definitions, the coefficients are obviously symmetric (i.e. \(g^{\textit{ij}}=g^{\textit{ji}}\) and \(g_{\textit{ij}}=g_{\textit{ji}}\)); and given the unit dyadic projection \({\varvec{\mathsf I}}\varvec{\,\cdot \,}\mathbf{e}^{k} = \mathbf{e}^{k}\) noted in the next section, we also have \(g_{\textit{ij}} g^{\textit{jk}} = \mathbf{e}_{i} \varvec{\,\cdot \,}\mathbf{e}_{j} \mathbf{e}^{j} \varvec{\,\cdot \,}\mathbf{e}^{k} = \mathbf{e}_{i} \varvec{\,\cdot \,}{\varvec{\mathsf I}}\varvec{\,\cdot \,}\mathbf{e}^{k} = \mathbf{e}_{i}\varvec{\,\cdot \,}\mathbf{e}^{k} =\delta _{i}^{k}\), a result which is sometimes collected into the matrix form \([g_{\textit{ij}}] [ g^{\textit{ij}}] = I\) where I is the \(3 \times 3\) unit matrix.
- 4.
Some authors define this product as \({\varvec{\mathsf A}}\varvec{:}{\varvec{\mathsf B}}=A^j_k B_j^k\).
- 5.
Tensor order (or rank) refers to the number of juxtaposed vectors in the entity, so that a scalar is a zeroth-order tensor, a vector is a first-order tensor, a dyadic a second-order tensor, etc.—i.e. in each case equivalent to the number of times the coordinate transformation matrix is applied in the transformation law, or the number of free (non-summation) indices in the tensor representation.
- 6.
A functional is a map between one or more function spaces and the real or complex numbers, typically involving integration. The arguments of functionals are often surrounded by square brackets \([\cdot ]\), rather than the parentheses \((\cdot )\) used for those of “ordinary” functions.
- 7.
The five volume set by Gel’fand and Shilov on Generalized Functions (Academic Press, 1964) is often referenced. The book by Vladimirov [10] may also be consulted.
- 8.
In passing, we note that any distribution product f(x)g(x) where the underlining is omitted means \({\underline{f(x)g(x)}}\), not \(\underline{f(x)}\; \underline{g(x)}\).
- 9.
Potential theory was developed by many famous mathematicians—including not only Green and Gauss but also Riemann, Poincaré and Hilbert.
- 10.
In Euclidean n-space \(\mathbf{\mathcal R}^n\) we have \(\delta (\mathbf{x})=\delta (x_1)\delta (x_2) \cdots \delta (x_n)\), where the distribution product must be suitably interpreted as discussed in Sect. 1.10. Alternatively, we may choose to define the delta function as the measure such that \(\int _{\mathbf{\mathcal R}^n}\phi (\mathbf{x})\delta (d\mathbf{x})=\phi (\mathbf{0})\;\forall \) compactly supported continuous function \(\phi (\mathbf{x})\), when \(\delta (\mathbf{x})=\delta (x_1)\delta (x_2) \cdots \delta (x_n)\) corresponds to the product measure of the \(\{\delta (x_i)\}\).
Bibliography
G.B. Arfken, H.J. Weber, F.E. Harris, Mathematical Methods for Physicists, 7th edn. (Academic Press, Waltham, 2012). (Popular textbook on essential mathematical methods for graduate students and beginning researchers)
L. Brand, Vector and Tensor Analysis (Wiley, New York, 1947). (Recommended extensive and thorough presentation, which inter alia covers the key topics of dyadics and general differential geometry as discussed in this chapter—available online)
R. Courant, D. Hilbert, Methods of Mathematical Physics (Interscience Publishers, New York, 1953, 1962). (Widely referenced survey of mathematical methods in two volumes, developed by Courant and colleagues in New York from the German original of 1924 derived from Hilbert’s lectures—first volume available online)
L.C. Evans, Partial Differential Equations (American Mathematical Society, Providence, 1998). (Graduate textbook, surveying both classical and weak solutions via Sobolev spaces)
L.P. Lebedev, M.J. Cloud, Tensor Analysis (World Scientific, River Edge, 2003). (Another presentation that includes dyadics and general differential geometry)
M.J. Lighthill, Introduction to Fourier Analysis and Generalised Functions (Cambridge University Press, Cambridge, 1959). (Another approach to distribution theory, in this case via limits of sequences of functions)
J. Mathews, R.L. Walker, Mathematical Methods of Physics (Addison-Wesley, Reading, 1971). (Physical intuition encourages students to further develop their mathematical ability—available online)
P.M. Morse, H. Feshbach, Methods of Theoretical Physics, Part I (McGraw-Hill, New York, 1953). (Another useful source on mathematical methods)
J. Rauch, Partial Differential Equations (Springer, New York, 2001). (Graduate textbook, with notable reference to the theory of distributions)
V.S. Vladimirov, Methods of the Theory of Generalized Functions (CRC Press, Boca Raton, 2002). (On the Sobolev-Schwarz concept of distributions with extensions to Fourier, Laplace, Mellin, Hilbert, Cauchy-Bochner and Poisson integral transforms and operational calculus)
C.E. Weatherburn, An Introduction to Riemannian Geometry and the Tensor Calculus (Cambridge University Press, Cambridge, 2008). (Emphasises tensor representations and coordinate transformations)
E.B. Wilson, Vector Analysis (Dover, New York, 1960). (Based on the lectures of J. Willard Gibbs and first published in 1901, the seven editions of this classic book helped standardise modern vector notation—available online)
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Hosking, R.J., Dewar, R.L. (2016). Vectors and Tensors. In: Fundamental Fluid Mechanics and Magnetohydrodynamics. Springer, Singapore. https://doi.org/10.1007/978-981-287-600-3_1
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DOI: https://doi.org/10.1007/978-981-287-600-3_1
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