The Laplacian of a $C^2$ function and of a $C^2$ vector field

Let $n\ge 1$, let $U\subseteq {\mathbb{R}}^n$ be open and let $f:U\to \mathbb{R}$ be $C^2$ in the sense of $C^k$ maps and multi-index derivative notation in Euclidean space. Then $\mathrm{\nabla }f$ is a $C^1$ field on $U$ by $C^k$ Euclidean maps and diffeomorphisms, since each of its components ${\mathrm{\partial }}_{i}f$ has continuous first partial derivatives, so its divergence is defined; the Laplacian of $f$ is

$\mathrm{\Delta }f:=\mathrm{div}\mathrm{\nabla }f={\sum }_{i<n}{\mathrm{\partial }}_i{\mathrm{\partial }}_{i}f,$

with the gradient of The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case and the divergence of Divergence and curl of a $C^1$ vector field. A $C^2$ function with $\mathrm{\Delta }f=0$ on $U$ is called harmonic on $U$.

For a $C^2$ map $F=(F_0,\dots ,F_{q-1}):U\to {\mathbb{R}}^q$, whose components are $C^2$ by $C^k$ Euclidean maps and diffeomorphisms, $\mathrm{\Delta }F$ is the field whose $i$th coordinate is $\mathrm{\Delta }{F}_i$. In the three-coordinate naming of this page, $\mathrm{\Delta }f={\mathrm{\partial }}_x{\mathrm{\partial }}_{x}f+{\mathrm{\partial }}_y{\mathrm{\partial }}_{y}f+{\mathrm{\partial }}_z{\mathrm{\partial }}_{z}f$ and $\mathrm{\Delta }F=(\mathrm{\Delta }{F}_x,\mathrm{\Delta }{F}_y,\mathrm{\Delta }{F}_z)$.

• The vector case is componentwise by convention, and the convention is stated because sources leave it implicit. Nothing forces a single reading of $\mathrm{\Delta }$ on a field; the componentwise one is the one that makes the curl-of-a-curl identity of The curl of a curl is the gradient of the divergence minus the Laplacian true as written, and it is the reading in force everywhere on this page.

• Why $C^2$ and not $C^1$. Forming $\mathrm{\nabla }f$ consumes one degree of differentiability, so $\mathrm{div}\mathrm{\nabla }f$ needs $\mathrm{\nabla }f$ to be $C^1$; that is exactly $f$ being $C^2$. Nothing here interchanges two partial derivatives, so no appeal to a mixed-partials theorem is made in the definition itself, and $\mathrm{\Delta }f$ is defined by the displayed sum in the fixed order ${\mathrm{\partial }}_i{\mathrm{\partial }}_i$.

• The planar equation. For $n=2$ the condition $\mathrm{\Delta }f=0$ reads ${\mathrm{\partial }}_x{\mathrm{\partial }}_{x}f+{\mathrm{\partial }}_y{\mathrm{\partial }}_{y}f=0$. That is the equation written out in The $C^2$ real and imaginary parts of a holomorphic function satisfy Laplace's equation and form a harmonic-conjugate pair ↗ for the real and imaginary parts of a holomorphic function; a reader meeting the word "harmonic" in either place is meeting one notion.