# Math Help - Deriving heat diffusion equation with Gauss' Divergence Theorem?

1. ## Deriving heat diffusion equation with Gauss' Divergence Theorem?

Hello!

I've been given a question on an assignment and I'm not quite sure where to start. It's supposed to be one of the harder questions, and I'm really unsure what I'm doing with it!

We're supposed to derive the Heat Diffusion equation ( $u_{t} = ku_{xx}$) using Gauss' Divergence Theorem, which is (as you probably know if you're reading this!):

$\iiint_{V} (\nabla \cdot \textbf{F}) \enspace dV = \iint_{S} \textbf{F} \cdot \textbf{n} \enspace dS$

Any help would be much appreciated!

-Geo

2. ## One way

Can you use continuity of thermal energy:

$
\frac{\partial \rho}{\partial t} + {\nabla} \cdot j = 0
$

and the assumption that

$
j = -k {\nabla} \rho
$

Your result is in 1-D, so you can simplify $\nabla$ to partial derivatives.