## Encyclopedia > Divergence

Article Content

# Divergence

In vector calculus, the divergence is a vector operator that shows a vector field's tendency to originate from or converge upon certain points. For instance, in a vector field that denotes the velocity of water flowing in a draining bathtub, the divergence would have a negative value over the drain because the water flows towards the drain, but does not flow away (if we only consider two dimensions).

Mathematically, the divergence is noted by:

$\nabla \cdot \mathbf{F}$

where $\nabla$ is the vector differential operator del and F is the vector field that the divergence operator is being applied to. Expanded, the notation looks like this:

$\frac {\partial F_x} {\partial x} + \frac {\partial F_y} {\partial y} + \frac {\partial F_z} {\partial z}$

if F = [Fx, Fy, Fz]

A closer examination of the pattern in the expanded divergence reveals that it can be thought of as being like a dot product between $\nabla$ and F if $\nabla$ was:

$\left[ \frac {\partial}{\partial x}, \frac {\partial}{\partial y}, \frac {\partial}{\partial z} \right]$

and its components were thought to apply their respective derivatives to whatever they are multiplied by.

All Wikipedia text is available under the terms of the GNU Free Documentation License

Search Encyclopedia
 Search over one million articles, find something about almost anything!

Featured Article
 East Marion, New York ... 18 and over, there are 90.1 males. The median income for a household in the town is $44,583, and the median income for a family is$52,500. Males have a median income of ...