# 0.2 Cartesian vectors and tensors: their calculus  (Page 8/9)

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## Irrotational vector fields

The vector field $\mathbf{a}$ is irrotational if its curl vanishes everywhere. By Stokes' theorem the circulation around any closed curve also vanishes. Also, an irrotational vector field can be expressed as the gradient of a scalar.

$\left(\begin{array}{c}\nabla ×\mathbf{a}=0\\ \underset{C}{\oint }\mathbf{a}•\mathbf{t}ds=0\\ \mathbf{a}=\nabla \varphi \end{array}}\mathbf{a}\phantom{\rule{5pt}{0ex}}\text{an irrotational field}$

The velocity field of motions where the viscous effects are insignificant compared to inertial effects and the flow is initially irrotational can be approximated as an irrotational velocity field.

## Solenoidal vector fields

A solenoidal vector field is defined as one in which the divergence vanishes. This implies that the flux across a closed surface must also vanish. A vector identity states that the divergence of the curl of a vector is zero. Thus a continuously differentiable solenoidal vector field has the following three equivalent characteristics.

$\left(\begin{array}{c}\nabla •\mathbf{a}=0\\ \underset{S}{\int \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\int \phantom{\rule{-11.66656pt}{0ex}}◯}\mathbf{a}•\mathbf{n}dS=0\\ \mathbf{a}=\nabla ×\mathbf{A}\end{array}}\mathbf{a}\phantom{\rule{5pt}{0ex}}\text{a solenoidal field}$

The velocity field of motions where the effects of compressibility are insignificant can be approximated as a solenoidal vector field. The surface integral of velocity vanishing over any closed surface means that the net volumetric flow across closed surfaces is zero. Incompressible flow fields can be expressed as the curl of a vector potential. Two-dimensional, incompressible flows have only one nonzero component of the vector potential and this is identified as the stream function.

## Helmholtz' representation

We found that an irrotational vector is the gradient of a scalar potential and a solenoidal vector is the curl of a vector potential. Here we show that any vector field with sufficient continuity is divisible into irrotational and solenoidal parts, and so is expressible in terms of a scalar and a vector potential. The fundamental problem in the analysis of a vector field is the determination of these potentials and their expression in terms of the essential characteristics of the vector, namely divergence, curl, discontinuities, and boundary values. For when the potentials are known the vector itself can be determined by differentiation. The following analysis is taken from H. B. Phillips, Vector Analysis , John Wiley&Sons, 1933. The following nomenclature will differ somewhat in that the vector is expressed as the negative of the gradient of a scalar. Also, the vector field of interest will be denoted as $mathbfF$ . Bold face capital letters will also be used for other vector quantities. Also the equations have the $4\pi$ factor of electromagnetism in mks units rather than the factors ${\epsilon }_{o}$ and ${\epsilon }_{o}\phantom{\rule{0.277778em}{0ex}}{c}^{2}$ of the SI units.

Let $V$ be a region of space where the vector field $F$ has piecewise continuous second derivatives, ${S}_{1}$ be surfaces of discontinuity of $F$ , and $S$ be the bounding surface of $V$ . The Helmholtz's theorem states that $F$ can be expressed in terms of the potentials.

$\begin{array}{c}\mathbf{F}=-\nabla \varphi +\nabla ×\mathbf{A}\hfill \\ \text{where}\hfill \\ \varphi \left({\mathbf{x}}_{P}\right)=\underset{\text{V}}{\phantom{\rule{0.277778em}{0ex}}\int \int \int \phantom{\rule{0.277778em}{0ex}}}\frac{\rho dV}{r}+\underset{{S}_{1}}{\phantom{\rule{0.277778em}{0ex}}\int \int \phantom{\rule{0.277778em}{0ex}}}\frac{\sigma dS}{r}-\frac{1}{4\pi }\underset{S}{\int \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\int \phantom{\rule{-11.66656pt}{0ex}}◯}\frac{\mathbf{n}•\mathbf{F}dS}{r}\hfill \\ \mathbf{A}\left({\mathbf{x}}_{P}\right)=\underset{V}{\phantom{\rule{0.277778em}{0ex}}\int \int \int \phantom{\rule{0.277778em}{0ex}}}\frac{\mathbf{I}dV}{r}+\underset{{S}_{1}}{\phantom{\rule{0.277778em}{0ex}}\int \int \phantom{\rule{0.277778em}{0ex}}}\frac{\mathbf{J}dS}{r}-\frac{1}{4\pi }\underset{S}{\int \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\int \phantom{\rule{-11.66656pt}{0ex}}◯}\frac{\mathbf{n}×\mathbf{F}dS}{r}\hfill \\ \nabla •\mathbf{F}=-{\nabla }^{2}\varphi =4\pi \rho \hfill \\ \nabla ×\mathbf{F}=\nabla ×\left(\nabla ×\mathbf{A}\right)=4\pi \mathbf{I}\hfill \\ \mathbf{n}•\Delta \mathbf{F}=4\pi \sigma \hfill \\ \mathbf{n}×\Delta \mathbf{F}=4\pi \mathbf{J}\hfill \\ r=\left|{\mathbf{x}}_{P}-{\mathbf{x}}_{Q}\right|\text{where}{\mathbf{x}}_{Q}\text{is}\text{coordinate}\text{of}\text{integrand}\hfill \end{array}$

The vectors $\mathbf{I}$ and $\mathbf{J}$ are not arbitrary. They are subject to the equation of continuity $\nabla •\mathbf{A}=0$ . The effect of this condition is to make $\mathbf{I}$ and $\mathbf{J}$ behave like space and surface currents of something which is nowhere created or destroyed. In the electromagnetic field they usually represent currents of electricity. In hydrodynamic fields they represent vorticity. If $\mathbf{A}$ is everywhere solenoidal, the following three equations must then be everywhere satisfied.

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