Integration and Differentiation 135is the mass of fluid passing across the surface area element da per unit time.
On page 154 we derive the equation of continuity that governs such flows.
THE VOLUME INTEGRAL. We now take the set G in (3.1.13) to be
a subset of the space K^3. The set is measurable and has volume V if
the boundary of G is a piece-wise smooth surface. In volume integrals,
we use the differential form of the volume element dV. In cartesian
coordinates, dV = dxdydz. In generalized cylindrical coordinates
x = arcos6, y = brsin9,z — z, dV = ab rdrdOdz; the usual cylindri-
cal coordinates correspond to a = b = 1. In generalized spherical
coordinates x — ar cos9 siny>, y = br sinO sine/?, z = cr costp, dV =
abc r^2 sin <p dr d6 d<p; the usual spherical coordinates correspond to a = b =
c = 1. The volume integral is evaluated by reduction to three iterated in-
tegrals; this is why the volume integral is also known as the triple integral
and is often written as JJJ f dV. The integral is defined if / is a continuous
G
scalar field in G. If / is the volume density of G, then JJJ f dV is the mass
a
of G.
EXERCISE 3.1.16. B Find the volume of the ellipsoid (x^2 /a^2 ) + (y^2 /b^2 ) +
(z /c^2 ) < 1. Hint: use generalized spherical coordinates.
Let us summarize the main facts related to the orientation of domains
and their boundaries. For a domain G in the (?, j) plane, we denote by
dG the boundary of G and assume that this boundary consists of finitely
many simple closed piece-wise smooth curves. The positive orientation
of dG means that the domain G stays on your left as you walk around the
boundary in the direction of the orientation of the corresponding curve. An
equivalent mathematical description is as follows. At all but finitely many
points of dG, there exists the unit tangent vector u; the direction of u
defines the orientation. At every point of dG where u exists, consider the
unit vector n that is perpendicular to u and points outside of G. Draw a
picture and convince yourself that the orientation of dG is positive if and
only if fi x u has the same direction as i x j.
For a domain G in R^3 , we denote by dG the boundary of G and assume
that this boundary consists of finitely many piece-wise smooth simple closed
surfaces. The positive orientation of dG means that, when it exists,
the normal vector to dG points outside of G.
For a piece-wise smooth surface S that is not closed, we denote by
dS the boundary of the surface and assume that this boundary consists