Flux From Flux Density

If you know the mass density of a solid object that fills the region W, how do you calculate the total mass of the object? By integrating the mass density over the volume of the solid region, you get the total mass. Thus,

$$\mbox{total mass} \thinspace = \int_W (\mbox{mass density}) dV.$$

The same idea works for flux. To get the total flux of a vector field $\vec{F}$ through a closed surface S, simply integrate the flux density of $\vec{F}$ over the interior volume, W, of S:

$$\mbox{total flux} \thinspace = \int_W (\mbox{flux density}) dV.$$

Recall that the flux density of a vector field is given by the divergence of the vector field. Since we can also calculate the total flux from the formula

$$\mbox{flux} \thinspace = \int_S \vec{F} \cdot \hat{n} dS,$$

we are led to the Divergence Theorem.

Divergence Theorem

If W is a solid region whose closed boundary S is a piecewise smooth surface, and if $\vec{F}$ is a smooth vector field defined everywhere in W and on S then

$$\int_S \vec{F} \cdot \hat{n} dS = \int_W \mbox{div} \vec{F} dV$$

where S is given the outward orientation.

Note that there are six conditions that must hold for this theorem to be applicable:

1.

W is a finite solid region,

2.

S is the closed boundary of W,

3.

S is piecewise smooth,

4.

$\vec{F}$ is smooth (has all first partial derivatives),

5.

$\vec{F}$ is defined everywhere in W and on S, and

6.

S is oriented outward.