The Divergence in Cartesian Coordinates

To examine the divergence, let's first compute its form in regular x,y,z coordinates. If we let $\vec{F} = F_1 \hat{i} + F_2 \hat{j} + F_3 \hat{k}$then

$$\mbox{div}\vec{F} = \nabla \cdot \vec{F} = \left( \hat{i} \f... ...rtial z}\right) \cdot (F_1 \hat{i} + F_2 \hat{j} + F_3 \hat{k})$$

$$\qquad = \frac{\partial F_1}{\partial x} + \frac{\partial F_2}{\partial y} + \frac{\partial F_3}{\partial z}.$$

As with any dot product, the divergence is a scalar quantity. Also note that, in general, $\mbox{div}\vec{F}$ is a function and will change in value from point to point.