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The heat equation in 1-dimension (for example, along a metal wire) is a partial differential equation of the following form:
$$\frac{\partial u}{\partial t} = c^2 \cdot \frac{\partial^2 u}{\partial x^2}$$
also written as
$$u_{t} = c^2 \cdot u_{xx}$$
Where $u:\mathbb{R}^2\to\mathbb{R}$ is the function giving the temperature at time $t$ and position $x$ and $c$ is a real valued constant. This can be easily extended to 2 or 3 dimensions as
$$u_{t} = c^2 \cdot ( u_{xx} + u_{yy} )$$ and $$u_{t} = c^2 \cdot ( u_{xx} + u_{yy} + u_{zz} )$$
Note that in the steady state, that is when $u_{t} = 0$ , we are left with the Laplacian of $u$ :
$$\Delta u = 0 $$
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