Probably I have not well understood the heat equation: please, can you confirm or correct the followings ? (The question raised in this post is similar to Heat Equation on Manifold but they don't fully overlap , I hope.)
Consider an homogeneous, isolated, flat disk, having radius $R$, whose center lies on the origin of the Cartesian coordinate, in the plane. At $t=0$ the thermal field is known: $\phi_0(x,y)=\Phi(x,y)$.
To compute $\phi=\phi_t(x,y)$ for $t>0$ we must solve the equation $$\frac{\partial \phi}{\partial t} - \alpha \left(\frac{\partial^2 \phi}{\partial x^2}+ \frac{\partial^2\phi}{\partial y^2}\right)=0$$ Now, instead of the disk, consider a surface, having equation $$\sigma : (x,y) \rightarrow (x\,,\,y\,,\,z(x,y))$$ Again the initial thermal field is given, but this time the equation to solve is $$\frac{\partial \phi}{\partial t} -\alpha\frac{1}{h_1h_2h_3}\sum_{i=1}^{3} \frac{\partial}{\partial x^i}\bigg(\frac{h_{a}h_{b}}{h_i}\frac{\partial\phi}{\partial x^i}\bigg)$$ where for $i=1$, we have $a=2$ and $b=3$ etc. cyclically. That's because the Laplacian operator is written in its most general form. In inhomogeneous bodies, $\alpha=\alpha(x,y,z)$.
