ReynoldsTransportTheorem 2006-07-05 12:34:49 2006-07-06 15:23:55 Theorem Changes for correction #29 ('capitalization, extra article'). % 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 $F(\mathbf{r},t)$ represent the amount of some physical property of a continuous material medium per unit volume. The total amount of this property present in a finite region ${\cal V}$ of the material is obtained through the volume integral. \[ \int_{\cal V} F(\mathbf{r},t) \;dV \] If this property is being transported by the action of the flow of the material with a velocity $\mathbf{u}(\mathbf{r},t)$, then the \emph{Reynolds' transport theorem} states that the rate of change of the total amount of $F$ within the material volume is equal to the volume integral of the instantaneous changes of $F$ occuring within the volume, plus the surface integral of the rate at which $F$ is being transported through the surface ${\cal S}$ (bounding ${\cal V}$) to and from the surrounding region. \[ \frac{d}{d t} \int_{\cal V} F(\mathbf{r},t) \;dV = \int_{\cal V} \frac{\partial F}{\partial t} \;dV + \int_{\cal S} F\mathbf{u} \cdot \mathbf{n} \;dS \] Here, $\mathbf{n}$ is a unit vector indicating the normal direction of the surface (oriented to point out of the volume).