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 $dxdydz$


1 Answer 1


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.

So you commonly write the divergence theorem (Gauss' theorem) like this:

$$\iiint \nabla\cdot \vec{F}{\,\rm d}V=\iint \vec{F}\cdot\,{\rm d}\vec{S}$$

Sometimes with a circle over the double integral to signify that the surface is closed.

Another notation is to show the dimensionality by writing volume integration as ${\rm d}^3 r$ and surface integration as ${\rm d}^2 S$.


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