The Divergence Theorem
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The Divergence Theorem
Let G be a three-dimensional solid bounded by a piecewise smooth closed surface S that has orientation pointing out
of G. Let
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be a vector field whose components have continuous partial derivatives.
The Divergence Theorem states:
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where
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is the divergence of the vector field
(it's also denoted
). The symbol
indicates that the surface integral is
taken over a closed surface.
The Divergence Theorem relates surface integrals of vector fields to volume integrals.
The Divergence Theorem can be also written in coordinate form as
In a particular case, setting
, we obtain a formula for the volume of solid G:
Example 1
Evaluate the surface integral
, where S is the surface of the sphere
that
has upward orientation.
Solution.
Using the Divergence Theorem, we can write:
By changing to spherical coordinates, we have
Example 2
Use the Divergence Theorem to evaluate the surface integral
where S is the surface of the solid bounded by the cylinder
of the vector field
and the planes z = 1, z = 1 (Figure 1).
Solution.
We apply the Divergence Theorem:
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The Divergence Theorem
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By switching to cylindrical coordinates, we have
Fig.1
Fig.2
Example 3
Use the Divergence Theorem to evaluate the surface integral
where S is the surface of a solid bounded by the cone
of the vector field
and the plane z = 1.
Solution.
The solid is sketched in Figure 2. Applying the Divergence Theorem, we can write:
By changing to cylindrical coordinates, we have
Example 4
Using the Divergence Theorem calculate the surface integral
of the vector field
, where S is the surface of tetrahedron with vertices O (0,0,0), A (1,0,0), B (0,1,0),
C (0,0,1) (Figure 3).
Solution.
By Divergence Theorem,
Find the given triple integral. The equation of the line AB has the form:
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The equation of the plane ABC is
So the integral becomes:
Fig.3
Fig.4
Example 5
Calculate the surface integral
of the vector field
, where S is the surface of
the rectangular box bounded by the planes x = 0, x = 1, y = 0, y = 2, z = 0, z = 3 (Figure 4).
Solution.
Using the Divergence Theorem, we can write:
Example 6
, where S is the outer surface of the pyramid
Find the surface integral
(Figure 5).
Solution.
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The Divergence Theorem
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Fig.5
Fig.6
Using the Divergence Theorem, we can write the initial surface integral as
Calculate the triple integral. The region of integration in the xy-plane is shown in Figure 6. Setting z = 0, we find:
Hence, the region D can be represented in the form:
in terms of z:
Rewrite the inequality
Then the triple integral becomes
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