Examples

Example 1. Calculate the divergence of $\vec{F} = x\hat{i} + y \hat{j} + z \hat{k}$.

$$\nabla \cdot \vec{F} = \frac{\partial}{\partial x}(x) + \fra... ...partial y}(y) + \frac{\partial}{\partial z}(z) = 1 + 1 + 1 = 3.$$ (4)

This is the vector field shown on the left above. Its divergence is constant throughout all space.

Example 2. What is the divergence of $\vec{F}(\vec{r}) = (k/r^3)\vec{r}$?

Recall that $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$ and $r = \vert\vert\vec{r}\vert\vert$, so

$$\vec{F} = \frac{k}{r^3}(x \hat{i} + y \hat{j} + z \hat{k})$$

$$\qquad = k \left( \frac{x}{(x^2 + y^2 + z^2)^{3/2}}\hat{i} +... .../2}}\hat{j} + \frac{z}{(x^2 + y^2 + z^2)^{3/2}}\hat{k} \right).$$

Thus, the divergence is

$$\nabla \cdot \vec{F} = k \left[ \frac{\partial}{\partial x} ... ...rtial y} \left( \frac{y}{(x^2 + y^2 + z^2)^{3/2}}\right)\right.$$

$$\qquad \qquad \left. +\frac{\partial}{\partial z} \left( \frac{z}{(x^2 + y^2 + z^2)^{3/2}}\right) \right]$$

$$\qquad = k \left[ 3 \frac{1}{(x^2 + y^2 + z^2)^{3/2}} - 3 \frac{x^2 + y^2 + z^2}{(x^2 + y^2 + z^2)^{5/2}} \right] = 0.$$

Thus, the divergence of the vector field here is zero everywhere except at the origin, where the vector field in undefined.