Divergence

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DIVERGENCE

Def. Divergence of a vector point function. Let A(x, y, z) = A₁ (x, y, z) i + A₂ (x, y, z) j + A₃ (x, y, z) k be a vector point function that is defined and differentiable at each point (x, y, z) in a certain region of space. The divergence of A is a scalar point function defined as

The divergence can also be written, using the del operator, as

Def. Source. In hydrodynamics, potential theory, etc., a point in some region at which fluid (or flux) is created or introduced into the region. The strength m of a point source is the total flux of fluid from it.

Def. Sink. A point in a region at which fluid (or flux) is destroyed or disappears. The strength -m of a sink is the total flux of fluid into it.

Def. Flux strength. A measure of the flow of a fluid. It is equal to the amount of fluid passing through one square unit of cross-sectional area perpendicular to the direction of flow per unit of time.

Example. Water is flowing in a river at a rate of 3 feet per second. The amount passing through a one square foot cross-section perpendicular to the direction of flow is 3 cubic feet per second. The measure of water flow, viewed as flux, would then be given as 3 cu ft/sec per sq. ft.

Divergence. Physical interpretation. The divergence of a vector point function

A(x, y, z) = A₁ (x, y, z) i + A₂ (x, y, z) j + A₃ (x, y, z) k

at some point P in a region R is a measure of the source or sink strength at point P. If the divergence is positive, then point P is a fluid (or flux) source and if it is negative P is a sink. If the divergence is zero, then point P is neither a source nor sink.

Let A be a vector point function defined throughout some region R of space. Then the divergence at any point P is given by

where A represents flux strength at point P, ΔV is the volume enclosed by the surface ΔS centered at P, n is the outward drawn unit normal at dS, and the limit is obtained by shrinking ΔV to point P. See Fig. 1 where ΔV is the volume of the depicted cube centered at P, ΔS is the surface of the cube, and ds is a differential element on ΔS. Physically

represents the net flux outflow per unit volume of the flux A from the surface ΔS. The quantity A∙n ds measures the rate of flow of the flux through the surface element dS.

Proof

If the integral

vanishes, there is no net outward flow. In this case any outward flux of A over part of the surface is balanced by an equal inward flux flow over the rest of the surface. If the integral is positive, the flux flow out of ΔV exceeds that into ΔV, and one says that there are sources within the volume. A negative value of the integral indicates the existence within ΔV of sinks, or points at which flux is being destroyed. If there are both sources and sinks within ΔV, the integral gives a measure of their algebraically combined strength. Thus div A can be taken as a measure of the source or sink strength at point P. If div A = 0 everywhere in a region, then there are no sources or sinks in the region and A is said to be a solenoidal vector field in the region.

Equation 2) can be taken as a definition of the divergence of A , and all the properties may be derived from it.

Continuity equation of an incompressible fluid. The equation

div A = 0

is called the continuity equation of an incompressible fluid. It states the condition that fluid is neither created nor destroyed.

References

Spiegel. Vector Analysis.

Hsu. Vector Analysis.

Angus Taylor. Adv. Calculus.