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Divergence
Start with a vector field
Divergence represents the net flow out of a point (x,y). This is a scalar field, which is a regular old function.
Think of the vector field as the flow of a fluid.
A source point is a point at which there is more fluid moving away than moving toward:
A sink point is a point at which there is more fluid moving toward than away:
An incompressible point is a point at which there is an equal amount of fluid moving toward and away:
Derivation
Measure fluid produced in a small box around point
. The box is a square from
to
.
Let
represent the difference between fluid out and fluid in (out − in) in the x-direction.
- Right:

- Top:

- Left:

- Bottom:

The sum of all of these equals the area of the box (
) times average production in box (APB). Solve for average production to get
Take the limit as
, and this turns into a derivative:
Example
Explanation of Notation
is a "differential operator":
- gradient

- divergence

Curl
Measures the circulation (imagine a vortex) of
around a point (by the right hand rule). In 3D space,
gives the circulation of
around
axis.
Let
Special Case
When
, the curl is equal to: