Question

Let \(\mathbf{F}(x, y, z)=x \mathbf{i}+y \mathbf{j}+z \mathbf{k},\) and let \(f(x, y, z)=\|\mathbf{F}(x, y, z)\| .\) The Laplacian is the differential operator $$ \nabla^{2}=\nabla \cdot \nabla=\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}}+\frac{\partial^{2}}{\partial z^{2}} $$ and Laplace's equation is $$ \nabla^{2} w=\frac{\partial^{2} w}{\partial x^{2}}+\frac{\partial^{2} w}{\partial y^{2}}+\frac{\partial^{2} w}{\partial z^{2}}=0 $$ Any function that satisfies this equation is called harmonic. Show that the function \(1 / f\) is harmonic.

Short answer

The function \(1 / f(x, y, z)\) is harmonic because it satisfies Laplace's equation \(\nabla^2 w = 0\).

Step-by-step solution

Compute Function \(f\)

We have \(\mathbf{F}(x, y, z) = x\mathbf{i} + y\mathbf{j} + z\mathbf{k}\). We can compute the magnitude of \(\mathbf{F}\) to find \(f(x, y, z) = \| \mathbf{F}(x, y, z) \| \), which yields \(f(x, y, z) = \sqrt{x^2 + y^2 + z^2}\)

Compute the Second Derivatives

Now, we need to compute the second derivatives of \(1/f(x, y, z)\) with respect to \(x\), \(y\), and \(z\). The derivatives are computed as follows: \(\frac{\partial^2}{\partial x^2} \left(\frac{1}{f}\right) = \frac{\partial}{\partial x}\left(-\frac{x}{(x^2+y^2+z^2)^{\frac{3}{2}}}\right) = \frac{3x^2-y^2-z^2}{(x^2+y^2+z^2)^{\frac{5}{2}}}\),\(\frac{\partial^2}{\partial y^2} \left(\frac{1}{f}\right) = \frac{\partial}{\partial y}\left(-\frac{y}{(x^2+y^2+z^2)^{\frac{3}{2}}}\right) = \frac{3y^2-x^2-z^2}{(x^2+y^2+z^2)^{\frac{5}{2}}}\),\(\frac{\partial^2}{\partial z^2} \left(\frac{1}{f}\right) = \frac{\partial}{\partial z}\left(-\frac{z}{(x^2+y^2+z^2)^{\frac{3}{2}}}\right) = \frac{3z^2-x^2-y^2}{(x^2+y^2+z^2)^{\frac{5}{2}}}\).

Confirm if \(\nabla^2 w = 0\)

The last step is to confirm if the sum of those second derivatives equals to zero. We compute: \(\frac{\partial^2}{\partial x^2} \left(\frac{1}{f}\right) + \frac{\partial^2}{\partial y^2} \left(\frac{1}{f}\right) + \frac{\partial^2}{\partial z^2} \left(\frac{1}{f}\right)= \frac{3x^2-y^2-z^2}{(x^2+y^2+z^2)^{\frac{5}{2}}} + \frac{3y^2-x^2-z^2}{(x^2+y^2+z^2)^{\frac{5}{2}}} + \frac{3z^2-x^2-y^2}{(x^2+y^2+z^2)^{\frac{5}{2}}}= 0. \)This means that the function \(1 / f(x, y, z)\) is indeed harmonic.