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      <text>28.2.4.18 The Laplacian</text>
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      <text><![CDATA[<h3>The Laplacian</h3>
<p>The Laplacian of a function \(\phi\) is written as \({\nabla}^2 \phi\) and is defined as: Laplacian \(\phi\) = div grad \(\phi\), that is \[\begin{aligned}
{\nabla}^2 \phi &= {\underline \nabla} \cdot {\underline \nabla} \phi \quad\quad\quad\quad\quad\quad \\
&= {\underline \nabla} \cdot \left( \frac{\partial \phi}{\partial x} \underline{i} + \frac{\partial \phi}{\partial y} \underline{j} + \frac{\partial \phi}{\partial z} \underline{k} \right) \\
&= \frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} + \frac{\partial^2 \phi}{\partial z^2}
\end{aligned}\]
The equation \({\nabla}^2 \phi = 0\), that is \(\dfrac{\partial^2 \phi}{\partial x^2} + \dfrac{\partial^2 \phi}{\partial y^2} + \dfrac{\partial^2 \phi}{\partial z^2} = 0\) is known as Laplace&rsquo;s equation and has applications in many branches of engineering including Heat Flow, Electrical and Magnetic Fields and Fluid Mechanics.</p>
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<h4 class="HELM_example">Example</h4>
<p>Find the Laplacian of \(u=x^2 y^2 z + 2 x z\).</p>
<h4 class="HELM_solution">Solution</h4>
\[\nabla^2 u = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} = 2y^2z + 2x^2z + 0 = 2(x^2+y^2)z.\]]]></text>
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