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DIVERGENCE of a Vector Field

If we form the scalar (``dot'') product of $\Grad{}$ with a vector function $\Vec{A}(x,y,z)$ we get a scalar result called the DIVERGENCE of $\Vec{A}$:

$$\hbox{\rm div} \Vec{A} \; \equiv \; \Div{A} \; \equiv \; . . . . . . y \over \partial y} \, + \, {\partial{A}_z \over \partial z}$$

This name is actually quite mnemonic: the DIVERGENCE of a vector field is a local measure of its ``outgoingness'' -- i.e. the extent to which there is more exiting an infinitesimal region of space than entering it. If the field is represented as ``flux lines'' of some indestructible ``stuff'' being emitted by ``sources'' and absorbed by ``sinks,'' then a nonzero DIVERGENCE at some point means there must be a source or sink at that position. That is to say,

``What leaves a region is no longer in it.''

For example, consider the divergence of the CURRENT DENSITY $\Vec{J}$, which describes the FLUX of a CONSERVED QUANTITY such as electric charge Q. (Mass, as in the current of a river, would do just as well.)

  

Figure: Flux into and out of a volume element $dV = dx \, dy \, dz$.

To make this as easy as possible, let's picture a cubical volume element $dV = dx \, dy \, dz$. In general, $\Vec{J}$ will (like any vector) have three components (J_x, J_y, J_z), each of which may be a function of position (x,y,z). If we take the lower left front corner of the cube to have coordinates (x,y,z) then the upper right back corner has coordinates $(x+dx, \, y+dy, \, z+dz)$. Let's concentrate first on J_z and how it depends on z.

It may not depend on z at all, of course. In this case, the amount of Q coming into the cube through the bottom surface (per unit time) will be the same as the amount of Q going out through the top surface and there will be no net gain or loss of Q in the volume - at least not due to J_z.

If J_z is bigger at the top, however, there will be a net loss of Q within the volume dV due to the ``divergence'' of J_z. Let's see how much: the difference between J_z(z) at the bottom and J_z(z+dz) at the top is, by definition, $dJ_z = \left(\partial J_z \over \partial z \right) dz$. The flux is over the same area at top and bottom, namely $dx \, dy$, so the total rate of loss of Q due to the z-dependence of J_z is given by

$$\dot{Q}_z \; = \; - dx \, dy \left(\partial J_z \over \part . . . . . . - \left(\partial J_z \over \partial z \right) dx \, dy \, dz$$

$$\mbox{\rm or} \quad \dot{Q} \; = \; - \left(\partial J_z \over \partial z \right) dV .$$

A perfectly analogous argument holds for the x-dependence if J_x and the y-dependence of J_y, giving a total rate of change of Q

$$\dot{Q} \; = \; - \left( {\partial J_x \over \partial x} \, . . . . . . y} \, + \, {\partial J_z \over \partial z} \right) \, dV$$

$$\mbox{\rm or} \quad \dot{Q} \; = \; - \Div{J} \; dV$$

The total amount of Q in our volume element dV at a given instant is just $\rho \, dV$, of course, so the rate of change of the enclosed Q is just

$$\dot{Q} \; = \; \dot{\rho} \; dV$$

which means that we can write

$${\partial \rho \over \partial t} \, dV \; = \; - \Div{J} \, dV$$

or, just cancelling out the common factor dV on both sides of the equation,

$$\fbox{\hbox{$\displaystyle {\partial \rho \over \partial t} \; = \; - \Div{J} $}}$$

which is the compact and elegant ``differential form'' of the EQUATION OF CONTINUITY.

This equation tells us that the ``Q sourciness'' of each point in space is given by the degree to which flux ``lines'' of $\Vec{J}$ tend to radiate away from that point more than they converge toward that point - namely, the DIVERGENCE of $\Vec{J}$ at the point in question. This esoteric-looking mathematical expression is, remember, just a formal way of expressing our original dumb tautology!