Next, we will learn about the divergence theorem named after the German mathematician Carl Friedrich Gauss(1777-1855).
Gauss's divergence theorem
Theorem 4..1
[Gauss's divergence theorem]
For the vector field
,let be the area of the space surrounded by the smooth closed surface , and let
be the normal vector. Then
If you think of and as variables,then
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Figure 4.1:
Gauss's theorem
![\begin{figure}\begin{center}
\includegraphics[width=6cm]{CALCFIG/Fig8-6-3.eps}
\end{center}\end{figure}](img841.png) |
Proof First, suppose
is sandwiched between two curved surfaces
from the bottom and top..Also,
is given by
,
is given by
. Then
For the surface
,normal unit vector for curvilinear coordinates
is equal to
. On the surface
, the unit normal vector is equal to
. Thus,
Therefore,
Similarly, by projecting
onto another plane, we can show
Adding each, we have
If the region
is common, you can prove it by dividing
into subregions
Example 4..1
The surface integral
where, the surface
is the upper sphere
と
.
(1) Evaluate this surface integral by using Gauss's divergence theorem.
(2) Evaluate this surface integral directly.
Answer (1)
より
を求めると
Thus
Now
is an upper sphere with the radius
. Then by the spherical coordinate transformation
, we have
(2)
Since
,projection onto
plane,
implies
. Here,we consier
and
.
Next projection onto
plane. Then
implies
.Thus,for
and
,
Lastly,projection onto
plane. Then
implies
.
Next consider the surface integral for
.First projection onto,
plane.
maps to
.
By Projecting onto
plane,
maps to
.,
By projecting onto
plane,
maps to
.
Adding these,
Here using the polar coorinates,we have
. Thus
Example 4..2
Let
.Show the followings for the region
and the boundary surface
. Here, denote the vaolume of the region
by
.
Answer
(1) Using the arbitrary constant, we express by the surface integral and apply Gauss's divergence theorem. Then
Therefore,
(2) By Gauss's divergence theorem,
(3) Using a constant vector,we express by the surface integral and apply the triple scalar product and Gauss's divergance theorem,
Example 4..3
Prove the following equation for the region
in the scalar field
and its boundary surface
.
(1)
(2) If
is a harmonic function, then
Answer
(1) The problem of surface integral is always rewrite into
.In this case,
isa directional derivative in the direction of the unit normal vector
. Then
.Therefore,,
Now using Gauss's divergence theorem,
(2)
is harmonic function. Then
. Thus,(1) implies
Theorem 4..2
For any region
and its bounary surface
within the common definition of the scalar field
and the vector field
, if
,then
.
Proof By Gauss's divergence theorem,
Then
Thus,
Note that
is a continuous function and for any region
, the followings are true. Thus by the properties of the continuous function, we have
in other words
Theorem 4..3
For about the boundary surface
of any region
in
, prove that if
then
.
Proof If
in the above theorem,then
implies that
.
Exercise4.1
- 1.
- Let
.Prove that for any region
and the boundary surface
, the followings are true.
(1)
(2)
(3)
(4)
(5)
(6)
(7)
- 2.
- Prove the following is true for the boundary surface
of any region
in the scalar field
.
- 3.
- Suppose that
satisfies
.Take the closed curve
that is the boundary of the curved surface
in this vector field.At this time, the surface integral
is always the same value for any curved surface
whose boundary is
. And its value is determined by the closed surface
.Prove the above.
- 4.
- The scalar fiel
and the vector field
are within the comon domain. Prove the following equation for any region
and its bounary surface
.
(1)
(2)
(3)
(4)
ならば,
- 5.
- Prove the following equation for any region
and its boundary surface
within the common definition of the scalar fields
and
.
(1)
(2)
(3)
Green's formula
(4) If
is harmonic function, then
(5) If
are harmonic functions, then
(6) If
on
,then the harmonic function
is 0 in
.
- 6.
- Suppose that the vector field
is defined in all space. Prove that if
for any boundary surface
,then
has a vector potential.
- 7.
- Suppose that
is defined for all space. Prove that if
for any boundary surface
,then
has a scalar potential.