VectorPotential 2009-04-18 06:55:21 2009-04-18 06:55:21 Definition % this is the default PlanetPhysics preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here Let\, $\vec{U} = \vec{U}(x,\,y,\,z)$\, be a vector field in $\mathbb{R}^3$ with continuous partial derivatives.\, Then the following three conditions are \PMlinkname{equivalent}{Equivalent3}: \begin{itemize} \item The surface integrals of $\vec{U}$ over all contractible \PMlinkname{closed surfaces}{Sphere} $S$ vanish: $$\oint_S\vec{U}\cdot d\vec{S} = 0$$ \item The divergence of $\vec{U}$ vanishes everywhere in the \PMlinkname{field}{VectorField}: $$\nabla\!\cdot\!\vec{U} = 0$$ \item There exists the {\em vector potential}\, $\vec{A} = \vec{A}(x,\,y,\,z)$\, of $\vec{U}$: $$\nabla\!\times\!\vec{A} = \vec{U}$$ \end{itemize} \begin{thebibliography}{9} \bibitem{VV}{\sc K. V\"ais\"al\"a:} {\em Vektorianalyysi}. \,Werner S\"oderstr\"om Osakeyhti\"o, Helsinki (1961). \end{thebibliography}