Encyclopedia > Divergence

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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:

<math> \nabla \cdot \mathbf{F} </math>

where <math>\nabla</math> 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} </math>

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 <math>\nabla</math> and F if <math>\nabla</math> was:

\left[ \frac {\partial}{\partial x}, \frac {\partial}{\partial y}, \frac {\partial}{\partial z} \right] </math>

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

See also:

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