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Whilst studying the heat equation, I ran into the equality\[e(x,t)=c(x)\rho(x)u(x,t),\]where \(e\) is the thermal energy density, \(c\) is the specific heat, \(\rho\) is the mass density, and \(u\) is the temperature. Why is this true, who came up with it, and how?

Physics
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Look at the units of each bit and do some dimensional analysis. What does your intuition say about multiplying the density of a metal times its volume for instance? Use this bit of logic to work your way through each bit of it and it'll start to clear up I think. =D
that comes from the definition of specific heat capacity ...
This looks like some kind of excess thermal energy density, something like the deviations from average as a function of time and space, as might describe the flow of a heat pulse through a solid material, for example. So let's say a little bit of the material at location x and time t was heated up by u degrees. What would be the extra energy required? Well, we have to multiply u by the mass in the region near x at time t and by the heat capacity of the material. That would tell us E = m c u. If we divide by the mass of the little region, so we get densities, then we have e(x,t) = p(x,t) c(x) u(x,t), where e is the energy density (energy per gram), p is the mass density (mass per gram), c is the heat capacity per gram (which apparently you are supposed to assume may vary with position, ,but not with time), and u is the temperature deviation.

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Other answers:

what's that ... heat transfer equation?
\[ {\partial u \over \partial t} - k \nabla^2u =0 \] ??
not sure you would get the eqn asked by OP using heat transfer eqn ... i hate it. i might fail because of it.
time and position dependent heat equation ... i thought that would only be for steady state. I have no idea on it's generalizations.
forget it ... i had enough of PDE's

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