# derivation of heat equation

In deriving the heat equation in the book it says

Fourier's law says that heat flows from hot to cold proportionately to the temperature gradient. Therefore change of heat energy in $D$ is also equals the flux across the boundary, Here u(x,y,z,t) is the temperature.

$\frac{dH}{dt}$=$\iint_{bdyD} k(n.{\nabla u})dS$.

($bdyD$ is the boundary curve)

What I don't understand is why two integral signs are used?Because when writing flux shouldn't there be only one integral sign .
Because flux=$\int F$.$\hat n dS$
$ds$ is the area right?So is $dS$ equivalent to $dxdydz$

In physics, it is common notation to use multiple integral signs to signify the number of dimensions even before the $dS$ (or $dV$ or something like that) is expanded in terms of the parametrization.
In this case, it's telling you that the integral is 2D: it integrates over a surface (closed surface in this case). $dS$ is not $dx\,dy\,dz$. It integrates across two dimensions - the parametrization depends on your surface. For a cube, you could have a sum of integrals over $dS=dxdy$, $dS=dydz$, $dS=dxdz$, for a sphere, you could integrate $dS=r^2\,d\phi\,d(\cos\theta)$ and so on.
$$\iiint \nabla\cdot \vec{F}{\,\rm d}V=\iint \vec{F}\cdot\,{\rm d}\vec{S}$$
Another notation is to show the dimensionality by writing volume integration as ${\rm d}^3 r$ and surface integration as ${\rm d}^2 S$.