V9.
Surface Integrals
Surface integrals are a natural generalization of line integrals: instead of integrating over
a curve, we integrate over a surface in 3space. Such integrals are important in any of the
subjects that deal with continuous media (solids, fluids, gases), as well as subjects that deal
with force fields, like electromagnetic or gravitational fields.
Though most of our work will be spent seeing how surface integrals can be calculated and
what they are used for, we first want to indicate briefly how they are defined. The surface
integral of the (continuous) function
f (x, y, z) over the surface
S is denoted by
You can think of dS as the area of an infinitesimal piece of the surface S. To define the
integral (I), we subdivide the surface
S into small pieces having area ASi, pick a point
(xi, yi, zi) in the ith piece, and form the Riemann sum
As the subdivision of
S gets finer and finer, the corresponding sums (2) approach a limit
which does not depend on the choice of the points or how the surface was subdivided. The
surface integral (1) is defined to be this limit. (The surface has to be smooth and not infinite
in extent, and the subdivisions have to be made reasonably, otherwise the limit may not
exist, or it may not be unique.)
1.
The surface integral for flux.
The most important type of surface integral is the one which calculates the flux of a
vector field across S. Earlier, we calculated the flux of a plane vector field F(x, y) across a
directed curve in the xyplane. What we are doing now is the analog of this in space.
We assume that
S is oriented: this means that S has two sides and one of them has been
designated to be the positive side. At each point of
S there are two unit normal vectors,
pointing in opposite directions; the positively directed unit normal vector, denoted by
n, is
the one standing with its base (i.e., tail) on the positive side. If
S is a closed surface, like
a sphere or cube

that is, a surface with no boundaries, so that it completely encloses a
portion of 3space

then by convention it is oriented so that the outer side is the positive
one, i.e., so that
n always points towards the outside of S.
Let F(x, y,
z)
be a continuous vector field in space, and
S an oriented surface. We define
(3)
flux of
F
through
S
=
the two integrals are the same, but the second is written using the common
and suggestive abbreviation dS
=
ndS.
If
F represents the velocity field for the flow of an incompressible fluid of density 1, then
F
.
n represents the component of the velocity in the positive perpendicular direction to the
surface, and
F
.
ndS represents the flow rate across the little infinitesimal piece of surface