Laplacian of a Function

Topics: Calculus - Vector-Valued Function

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(definition)

Let $f:{\mathbb{R}}^3\to \mathbb{R}$ be a vector-valued function.

We define the Laplacian of $f$ as:

$${\nabla }^{2}f=\Delta f:=\nabla \cdot (\nabla f)$$

That is, the divergence of the function’s gradient vector.

The Laplacian can be denoted by ${\nabla }^2$ or $\Delta$.

Alternative Notation

Since $\nabla f=⟨\frac{\partial f}{\partial x},\frac{\partial f}{\partial y},\frac{\partial f}{\partial z}⟩$, then:

$$\begin{array}{rl}\nabla \cdot (\nabla f)& =⟨\frac{\partial }{\partial x},\frac{\partial }{\partial y},\frac{\partial }{\partial z}⟩\cdot ⟨\frac{\partial f}{\partial x},\frac{\partial f}{\partial y},\frac{\partial f}{\partial z}⟩\\ & =\frac{{\partial }^{2}f}{\partial x^2}+\frac{{\partial }^{2}f}{\partial y^2}+\frac{{\partial }^{2}f}{\partial z^2}\end{array}$$