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Examples

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

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

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

\begin{displaymath}
\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).\end{displaymath}

Thus, the divergence is

\begin{displaymath}
\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.\end{displaymath}

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

\begin{displaymath}
\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.\end{displaymath}

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



Vector Calculus
8/19/1998