Laplacian

The Laplacian is the second-order differential operator ${\nabla }^2=\nabla \cdot \nabla$. Applied to a scalar field $V$ in Cartesian coordinates:

${\nabla }^{2}V=\frac{{\partial }^{2}V}{\partial x^2}+\frac{{\partial }^{2}V}{\partial y^2}+\frac{{\partial }^{2}V}{\partial z^2}.$

It is the Divergence of the Gradient: take $\nabla V$ (a vector field — the slope at every point), then take $\nabla \cdot$ of that (how much that gradient field spreads out). The result is a scalar.

What it measures

At a point, ${\nabla }^{2}V$ is proportional to the difference between $V$ at that point and the average of $V$ on a small sphere around it. Specifically, for a small radius $r$,

$V_{\text{avg on sphere}}-V(\text{point})\approx \frac{r^2}{6}{\nabla }^{2}V.$

Positive Laplacian: the neighbors are higher on average — the point sits in a “valley.” Negative: a “hill.” Zero: the value equals its spherical average — the function is harmonic (see Harmonic function).

In other coordinate systems

Cylindrical $(r,\varphi ,z)$:

${\nabla }^{2}V=\frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial V}{\partial r}\right)+\frac{1}{r^2}\frac{{\partial }^{2}V}{\partial {\varphi }^2}+\frac{{\partial }^{2}V}{\partial z^2}.$

Spherical $(R,\theta ,\varphi )$:

${\nabla }^{2}V=\frac{1}{R^2}\frac{\partial }{\partial R}\left(R^2\frac{\partial V}{\partial R}\right)+\frac{1}{R^2\sin \theta }\frac{\partial }{\partial \theta }\left(\sin \theta \frac{\partial V}{\partial \theta }\right)+\frac{1}{R^2{\sin }^2\theta }\frac{{\partial }^{2}V}{\partial {\varphi }^2}.$

The extra $r$ and $\sin \theta$ factors are Jacobians — they account for the way coordinate volume elements stretch and contract with position.

Of a vector field

For a vector field $\mathbf{E}$, the Laplacian is applied componentwise in Cartesian coordinates:

${\nabla }^2\mathbf{E}=({\nabla }^2{E}_x)\hat{\mathbf{x}}+({\nabla }^2{E}_y)\hat{\mathbf{y}}+({\nabla }^2{E}_z)\hat{\mathbf{z}}.$

In curvilinear coordinates, ${\nabla }^2\mathbf{E}$ is not the componentwise Laplacian — the position-dependent unit vectors get differentiated too. Instead, use the identity

${\nabla }^2\mathbf{E}=\nabla (\nabla \cdot \mathbf{E})-\nabla \times (\nabla \times \mathbf{E}),$

which holds in any coordinate system. This identity is the engine behind deriving the wave equation from Maxwell’s equations.

In physics

Laplace’s equation ${\nabla }^{2}V=0$ governs equilibrium — electrostatic potential in charge-free regions, steady-state heat, irrotational incompressible flow. Poisson’s equation ${\nabla }^{2}V=-{\rho }_v/ϵ$ adds a source term and is the differential form of Gauss’s law combined with $\mathbf{E}=-\nabla V$.

The wave equation in free space,

${\nabla }^2\mathbf{E}-\mu ϵ\frac{{\partial }^2\mathbf{E}}{\partial t^2}=0,$

falls out of Maxwell’s equations using the vector Laplacian identity. The Laplacian is what makes the spatial structure of a wave propagate.

The diffusion equation ${\partial }_{t}u=D{\nabla }^{2}u$ governs heat conduction and many transport processes — the Laplacian here measures how much $u$ is locally “out of equilibrium,” and that drives the time evolution.