Implement "nested" derivatives and cylindrical coordinates?

Questions
1

Hello, I am looking to solve the PDE

\frac{1}{r} \frac{\partial}{\partial r} \left( r \frac{\partial T}{\partial r} \right) = \frac{1}{\alpha} \frac{\partial T}{\partial t}

in Mathematica, but I can’t figure out how to input the \frac{\partial}{\partial r}\left( r \frac{\partial T}{\partial r} \right) part. Is it correct to nest the derivatives like below?

(1/r) * D[r * D[T[r,t], r], r] - (1/\[Alpha][r]) * D[T[r,t], t] == 0

Furthermore, how do I indicate to Mathematica that I am working in cylindrical coordinates instead of Cartesian?

Thanks.

2

If you want to go to cylindrical without opening a text book, Mathematica does it for you for the Div and Grad operators, when you specify the operator what coordinate system to use. see the basic examples of the reference pages.

in your above unless you assume cylindrical symmetry, since your already have the derivative form the only thing missing seems to be a independent theta variable.