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18.02 Multivariable Calculus
Fall 2007
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The
Divergence
Theorem
1.
Introduction; statement of the theorem.
The divergence theorem is about closed surfaces, so let's start there. By a
closed
surface
S we will mean a surface consisting of one connected piece which doesn't intersect itself, and
which completely encloses a single finite region D of space called its interior. The closed
surface
S is then said to be the boundary of D; we include S in D.
A
sphere, cube, and
torus (an inflated bicycle inner tube) are all examples of closed surfaces. On the other hand,
these are not closed surfaces: a plane, a sphere with one point removed, a tin can whose
crosssection looks like a figure8 (it intersects itself), an infinite cylinder.
h
n
A
closed surface always has two sides, and it has a natural positive
direction

the one for which
n
points away from the interior, i.e., points
toward the outside. We shall always understand that the closed surface has
been oriented this way, unless otherwise specified.
We now generalize to 3space the normal form of Green's theorem (Section V4).
Definition.
Let F(x, y, z)
=
Mi
+
N
j
+
Pk be a vector field differentiable in some
region D. By the
divergence
of
F we mean the scalar function div F of three variables
defined in D by
The divergence theorem.
Let
S be a positivelyoriented closed surface with interior
D, and let
F be a vector field continuously differentiable in a domain contatining D. Then
We write dV on the right side, rather than dxdy dz since the triple integral is
often calculated in other coordinate systems, particularly spherical coordinates.
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 Spring '08
 Auroux
 Multivariable Calculus, Vector Calculus, Flux, Vector field

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