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Prove. A fluid moves so that its flux at any point P(x, y, z) in some region R is given by A(x, y, z). Show that the gain of fluid per unit volume per unit time in a small parallelepiped having center at P(x, y, z) and edges parallel to the coordinate axes and having magnitudes Δx, Δy, Δz respectively, is given by div A = •A.

Proof. See Fig. 1. Denote the x
component of flux A at point P by
A_{1}(x, y, z). Then

Consequently,

Thus

Similarly,

Then

total gain in volume per unit volume per unit time

This is true exactly only in the limit as the parallelepiped shrinks to P i.e. as Δx, Δy and Δz approach zero.

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