5. (mcquarrie, §20.5 #3) determine the partial differential equation that governs the func- tion φ(x,y,z)…

5. (mcquarrie, §20.5 #3) determine the partial differential equation that governs the func- tion φ(x,y,z) for which the average of the square of the gradient over a certain region r is a minimum when φ(x,y,z) is specified on the boundary of r.

5. (mcquarrie, §20.5 #3) determine the partial differential equation that governs the func- tion φ(x,y,z) for which the average of the square of the gradient over a certain region r is a minimum when φ(x,y,z) is specified on the boundary of r.

Answer

Explanation:

Step1: Recall the formula for the square of the gradient

The gradient of $\varphi(x,y,z)$ is $\nabla\varphi=\left(\frac{\partial\varphi}{\partial x},\frac{\partial\varphi}{\partial y},\frac{\partial\varphi}{\partial z}\right)$, and the square of the gradient is $|\nabla\varphi|^{2}=\left(\frac{\partial\varphi}{\partial x}\right)^{2}+\left(\frac{\partial\varphi}{\partial y}\right)^{2}+\left(\frac{\partial\varphi}{\partial z}\right)^{2}$. The average of $|\nabla\varphi|^{2}$ over the region $R$ is given by $\overline{|\nabla\varphi|^{2}}=\frac{1}{V}\int_{R}|\nabla\varphi|^{2}dV$, where $V = \int_{R}dV$ is the volume of the region $R$.

Step2: Use the calculus - of - variations approach

We want to minimize the functional $J[\varphi]=\int_{R}|\nabla\varphi|^{2}dV$ subject to $\varphi$ being specified on $\partial R$. By the Euler - Lagrange equation for a functional of the form $J[\varphi]=\int_{R}F\left(\varphi,\frac{\partial\varphi}{\partial x},\frac{\partial\varphi}{\partial y},\frac{\partial\varphi}{\partial z}\right)dV$ (here $F = \left(\frac{\partial\varphi}{\partial x}\right)^{2}+\left(\frac{\partial\varphi}{\partial y}\right)^{2}+\left(\frac{\partial\varphi}{\partial z}\right)^{2}$), the Euler - Lagrange equation is $\frac{\partial F}{\partial\varphi}-\nabla\cdot\left(\frac{\partial F}{\partial(\nabla\varphi)}\right)=0$.

Step3: Calculate the necessary derivatives

First, $\frac{\partial F}{\partial\varphi}=0$ since $F$ does not depend explicitly on $\varphi$. Second, $\frac{\partial F}{\partial\left(\frac{\partial\varphi}{\partial x}\right)} = 2\frac{\partial\varphi}{\partial x}$, $\frac{\partial F}{\partial\left(\frac{\partial\varphi}{\partial y}\right)} = 2\frac{\partial\varphi}{\partial y}$, $\frac{\partial F}{\partial\left(\frac{\partial\varphi}{\partial z}\right)} = 2\frac{\partial\varphi}{\partial z}$. Then $\nabla\cdot\left(\frac{\partial F}{\partial(\nabla\varphi)}\right)=\nabla\cdot(2\nabla\varphi)=2\nabla^{2}\varphi$.

Step4: Obtain the partial differential equation

Substituting into the Euler - Lagrange equation $\frac{\partial F}{\partial\varphi}-\nabla\cdot\left(\frac{\partial F}{\partial(\nabla\varphi)}\right)=0$, we get $0 - 2\nabla^{2}\varphi=0$, or simply $\nabla^{2}\varphi = 0$.

Answer:

The partial differential equation is $\nabla^{2}\varphi=0$, which in Cartesian coordinates is $\frac{\partial^{2}\varphi}{\partial x^{2}}+\frac{\partial^{2}\varphi}{\partial y^{2}}+\frac{\partial^{2}\varphi}{\partial z^{2}} = 0$.