Curl and Divergence — Question 9

Question 9

On $D=\mathbb R^3\setminus\{\mathbf 0\}$, define $$\mathbf F(x,y,z)=\frac{\langle x,y,z\rangle}{(x^2+y^2+z^2)^{3/2}}.$$

Tasks

1. Compute $\nabla\cdot\mathbf F$ on $D$.

2. Compute or otherwise justify $\nabla\times\mathbf F$ on $D$.

3. Explain why the excluded origin must be stated.

Question 9 – Solution

Strategy. Write $r=(x^2+y^2+z^2)^{1/2}$ and differentiate $xr^{-3}$, $yr^{-3}$, and $zr^{-3}$.

Step 1: Divergence Since $$\frac{\partial}{\partial x}(xr^{-3})=r^{-3}-3x^2r^{-5},$$ with analogous formulas for $y$ and $z$, $$\begin{align*} \nabla\cdot\mathbf F &=3r^{-3}-3(x^2+y^2+z^2)r^{-5}\\ &=3r^{-3}-3r^2r^{-5}=\boxed{0},\qquad r>0. \end{align*}$$

Step 2: Curl Because $$\mathbf F=\nabla\!\left(-\frac 1r\right)$$ on $D$, the curl-of-a-gradient identity gives $$\boxed{\nabla\times\mathbf F=\mathbf 0}.$$

Step 3: Domain caveat The formula is undefined at the origin, and the derivatives used above do not exist there. The conclusions are pointwise statements on $D$, not statements at $\mathbf 0$.

Verification Direct cross differentiation gives, for example, $R_y-Q_z=-3yzr^{-5}-(-3yzr^{-5})=0$, and the other two components cancel similarly.