Consider the heat equation $\frac{\partial u}{\partial t}=k\cdot\Delta u$ where $u\colon\Omega\times \mathbb{R}_+\to \mathbb{R}$ maps the space variable $x\in \Omega$ and a time variable $t$ to a temperature.

Is there a name for the quantity $$h_t = \int_\Omega u(x,t)\,\mathrm{d}x \quad? $$

I was under the impression that this quantity is called heat, but the Wikipedia article for heat disagrees. What should this be called?


1 Answer 1


By integrating the heat equation $$\rho c \frac{\partial T}{\partial t} - \nabla \cdot \left( k \nabla T \right) = \dot q$$ over a volume $\Omega$, assuming no heat flux through the boundary $\delta\Omega$, you find that $$\int_\Omega \rho c Tdx=Q,$$ so the heat $Q=\int_\Omega qdx$ equals the spatial integral of the product of temperature $T$, mass density $\rho$, and specific heat capacity $c$. For uniform $\rho$, $c$ you would simply have $\int_\Omega Tdx=Q/\rho c$.

  • 2
    $\begingroup$ Are $\rho$ and $c$ possibly functions of location $x$? Taking them outside of the integral like that seems fine if they are constant throughout space, otherwise I don't get it. $\endgroup$
    – Zach Teitler
    Nov 26, 2017 at 9:21
  • 1
    $\begingroup$ certainly, if these material constants are spatially dependent they should stay inside the integral --- thanks for noting this. $\endgroup$ Nov 26, 2017 at 11:11
  • $\begingroup$ The left hand side of this equation $\int_\Omega \rho c Tdx=Q$ is called the internal energy of the material. $\endgroup$ Nov 26, 2017 at 13:32

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