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Laplace operator: \(\Delta = \nabla^2 = (\nabla \cdot \nabla)\).
When applied to a scalar field \(u\): \(\Delta u = \nabla \cdot (\nabla u)\), i.e. the divergence of the gradient.
Laplace generalization to vector or arbitrary tensor fields by gradient generalizations.
Well known equations involving the Laplacian, roughly in increasing order of complexity:
\[ \begin{aligned} \text{Laplace's} & \text{ equation:} \phantom{\frac{1}{1}} \Delta u = 0 \\ \text{Poission's} & \text{ equation:} \phantom{\frac{1}{1}} \Delta u = f \\ \text{Helmholtz} & \text{ equation:} \phantom{\frac{1}{1}} \Delta u = -k^2 u \\ \text{Heat} & \text{ equation:} \phantom{\frac{1}{1}} \Delta u = \frac{1}{\alpha} \dot {u} \\ \text{Wave} & \text{ equation:} \phantom{\frac{1}{1}} \Delta u = \frac{1}{c^2} \ddot{u} \\ \end{aligned} \]
where Newton’s notation is used:
\[ \begin{aligned} \dot{u} ={} & \frac{\partial u}{\partial t } \\ \ddot{u} ={} & \frac{\partial^2 u}{\partial t^2} \\ \end{aligned} \]
Note that one may be interested in steady-state solutions to the Heat equation, i.e. one sets \(\dot{u} = 0\), in which case it reduces to Laplace’s equation. Note also that Helmholtz equation is the eigenvalue problem for the Laplace operator.