Gauss's Theorem 153Hint: Note that divF = V^2 (l/r) = 0 ifr ^ 0. If S encloses the origin, then there
exists a small sphere centered at the origin and lying inside the domain enclosed
by S. Show that the flux through S is equal to the flux through the sphere; then see
Exercise 3.1.23 on page HO. (c) Let dG be an orientable piece-wise smooth
surface enclosing a domain G, and let F be a vector field, continuously
differentiable in an open set containing G. Show that
I FxnGda = - fff curl F dV, (3.2.6)
dG Gwhere fie is the outside normal vector to dG. Hint: the left- and right-hand
sides of (3.2.6) are vectors; denote them by v and w, respectively. The claim is
that v-b = w -b for every constant vector b; in cartesian coordinates, it is enough
to verify this equality for only three vectors, i,J,k instead of every b. By the
properties of the scalar triple product, b • (F x HG) = —(F x b) • HG- By Gauss's
Theorem,
ff{F xb)nada = fff div(F x b) dV,
dG G
and by (3.1.30), div(F x b) = b • curlF.
We will now use Gauss's Theorem to derive the basic equation of fluid
flow, called the EQUATION OF CONTINUITY. Assume that the space M^3 is
filled with a continuum of point masses (particles). This continuum can
represent, for example, liquid, gas, or electric charges; accordingly, "mass"
represents the appropriate characteristic of that continuum, such as the
mass of matter or electric charge. Denote by v = v(P, t) the velocity of the
point mass at the point P and time t, and by p = p(P,t), the density of
the point masses at the point P and time t. By definition,
where AV is the volume of a region containing the point P, AM is the mass
of that region, and limit is assumed to exist at every point P. We also allow
the existence of sources and sinks, that is, in some areas of the space, the
particles can be injected into the space (source) or removed from the space
(sink). The density of the sources and sinks is described by a continuous
scalar field v = v(P, t) so that the net mass per unit time, injected into and
removed from G due to sources and sinks is fff pvdV. We also assume
G
that the functions p, u, v are continuously differentiable as many time as
necessary. Since the total mass M of the particles in G also changes due to