76. From equation (4) we obtain
|
|
where, as elsewhere in these equations, the surface-integral relates to the boundary of the volume-integrals.
From this, by substitution of
for
we derive as a particular case
|
|
which is Green's Theorem. The substitution of
for
gives the more general form of this theorem which is due to Thomson, viz.,
|
|
77. From equation (6) we obtain
|
|
A particular case is
|
|
Integration of Differential Equations.
78. If throughout any continuous space (or in all space)
|
|
then throughout the same space
|
|
79. If throughout any continuous space (or in all space) and in any finite part of that space, or in any finite surface in or bounding it,
|
|
then throughout the whole space
and
|
|
This will appear from the following considerations:
If
in any finite part of the space,
is constant in that part.
If
is not constant throughout, let us imagine a sphere situated principally in the part in which
is constant, but projecting slightly into a part in which
has a greater value, or else into a part in which
has a less. The surface-integral of
for the part of the spherical surface in the region where
is constant will have the value zero: for the other part of the surface, the integral will be either greater than zero, or less than zero. Therefore the whole surface-integral for the spherical surface will not have the value zero, which is required by the general condition,
Again, if
only in a surface in or bounding the space in