September 9, 1998

PHY 711 - Problem Set # 6

Consider an integral of the form:

$$E = \int \: d^3r \: F[\rho({\bf{r}}),g({\bf{r}})],$$

where $\rho({\bf{r}})$ is a function of position in three dimensions, and $g({\bf{r}}) \equiv \vert{\bf{\nabla}} \rho({\bf{r}})\vert$. Show that the functional variation of E with respect to $\rho({\bf{r}})$ is given by:

$$\delta E = \int \: d^3r \: \left \{\frac{\partial F}{\partia... ...ho \cdot {\bf{\nabla}} g}{g}\right\} {\delta \rho({\bf{r}})}.$$