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:
- <math>
\frac {\partial F_x} {\partial x} +
\frac {\partial F_y} {\partial y} +
\frac {\partial F_z} {\partial z}
</math>
if F = [F_{x}, F_{y}, F_{z}]
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:
- <math>
\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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