- #1

- 4

- 0

## Homework Statement

The density in 3-D space of a certain kind of conserved substance is given by

[tex]\[\rho (x,y,z, t) = At^{-\frac{3}{2}}e^{-\frac{r^2}{4kt}}\][/tex]

where [tex]\mathbf r = x\mathbf i + y\mathbf j +z\mathbf k[/tex] and [tex] r = |\mathbf r|[/tex]. The corresponding flux vector is given by

[tex]\mathbf J(\mathbf r, t) = Bt^{-\frac{5}{2}}e^{-\frac{r^2}{4kt}}\mathbf r[/tex]

Here, A, B, k and positive constants.

## Homework Equations

Show that [tex]$\rho, \mathbf J$[/tex] satisfy the conservation equation [tex]\frac{\partial \rho}{\partial t}[/tex][tex] + \nabla \cdot \mathbf J = 0[/tex] only if [tex]$ A = 2B$[/tex]

## The Attempt at a Solution

So I've looked at this, found the derivative for the density function, had a fair play with the div function, i'm just wondering if there is a smarter way to solve this then actually deriving the partial derivative and the div function and re-arranging? I have a feeling there is something inherent, for example like the divergence theorum, that i can use?

mind you in the time it took me to get the tex working i could have solved the thing, but i'm still curious

Last edited: