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The divergence of a vector field

is defined as

z

F

y

F

x

F

FFFzyx

321

321 )(

div

kjikji

FF

kjiF ),,(),,(),,(),,( 321 zyxFzyxFzyxFzyx

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Divergence

y ux zu

vv vdiv = v

x y z

v v

2 Physical meaning

(x,y,z)v is a differentiable vector field

div v is associated to local conservation laws: for example, we will show how that if the mass of fluid (or of charge) outcoming from a domain is equal to the mass entering, then

is the fluid velocity (or the current) vectorfield

div 0vv

(x,y,z)V (x dx,y,z)V

x x+dx

Geometrical Interpretation.

The name divergence is well chosen, for . F is a

measure of how much the vector F spreads out

(diverges) from the point in question.

The vector function has a large (positive) divergence

at the point P; it is spreading out. (If the arrows

pointed in, it would be a large negative divergence.)

P

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NOTE: P=electric field due to charge (+ ve or ve)

On the other hand, the function has zero divergence at

P; it is not spreading out at all.

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P

So, for example, if the divergence is positive at a point, it

means that, overall, that the tendency is for fluid to move away

from that point (expansion); if the divergence is negative, then

the fluid is tending to move towards that point (compression).