# Deriving a heat equation

BustedBreaks
Consider heat flow in a long circular cylinder where the temperature depends only on t and on the distance r to the axis of the cylinder. Here $$r=\sqrt{x^{2}+y^{2}}$$ is the cylindrical coordinate. From the three dimensional heat equation derive the equation $$u_{t}=k(u_{rr}+\frac{u_{r}}{r})$$.

My book describes how the general heat equation is derived, but I'm having trouble incorporating the equation for r and in general understanding these concepts.

The book starts out with:

$$H(t)=\int\int\int c\rho u dxdydz$$ which I'm somewhat confident I can change to

$$H(t)=\int\int\int c\rho u dxdydr$$ because of the equation with r

Then the book has
$$\frac{dH}{dt}=\int\int\int c\rho u_{t} dxdydr$$ which makes sense to me

then after this I am a bit confused. I don't really know what to do next in terms of answering the question.

## Answers and Replies

Homework Helper
This can be done very easily, by rewriting the heat equation in spherical coordinates and taking advantage of the symmetry of the situation. A little bit of re-arranging gives you the equation.