The Laplacian

5 The Laplacian

The Laplacian of a function $ϕ$ is written as $\nabla^{2} ϕ$ and is defined as: Laplacian $ϕ$ = div grad $ϕ$ , that is

\begin{array}{rcl} \nabla^{2} ϕ & = & \underset{̲}{\nabla} \cdot \underset{̲}{\nabla} ϕ \\ = & \underset{̲}{\nabla} \cdot (\frac{\partial ϕ}{\partial x} \underset{̲}{i} + \frac{\partial ϕ}{\partial y} \underset{̲}{j} + \frac{\partial ϕ}{\partial z} \underset{̲}{k}) \\ = & \frac{\partial^{2} ϕ}{\partial x^{2}} + \frac{\partial^{2} ϕ}{\partial y^{2}} + \frac{\partial^{2} ϕ}{\partial z^{2}} \end{array}

The equation $\nabla^{2} ϕ = 0$ , that is $\frac{\partial^{2} ϕ}{\partial x^{2}} + \frac{\partial^{2} ϕ}{\partial y^{2}} + \frac{\partial^{2} ϕ}{\partial z^{2}} = 0$ is known as Laplace’s equation and has applications in many branches of engineering including Heat Flow, Electrical and Magnetic Fields and Fluid Mechanics.

Example 15

Find the Laplacian of $u = x^{2} y^{2} z + 2 x z$ .

Solution

$\nabla^{2} u = \frac{\partial^{2} u}{\partial x^{2}} + \frac{\partial^{2} u}{\partial y^{2}} + \frac{\partial^{2} u}{\partial z^{2}} = 2 y^{2} z + 2 x^{2} z + 0 = 2 (x^{2} + y^{2}) z$