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I have the following question: In an isotropic medium with constant thermal conductivity, the temperature T(x,y) is independent of time. Show that the laplacian of T is zero. (4 marks)

I don't really know where to start with this, I thought this was just what steady heat flow was governed by but I don't know how to show it.

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Hint: look at the heat equation. – Alex R. May 8 '13 at 15:53

The law describing the distribution of heat is the heat equation,namely $$u_t(t,x,y) = D\Delta u(t,x,y).$$ Here the function $u(t,x,y)$ gives the temperature at position $(x,y)$ and time $t$, while $D$ is a constant representing the thermal diffusivity. You can find how to derive the equation here:

Now, if the temperature is constant in time ($u(t,x,y) = u(x,y)$), then obviously $\Delta u = 0$.

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