VECTOR ANALYSIS.
To find the value of this integral, we may regard the point
, which is constant in the integration, as the center of polar coordinates. Then
becomes the radius vector of the point
, and we may set
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where
is the element of a spherical surface having center at
and radius
. We may also set
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We thus obtain
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where
denotes the average value of
hi a spherical surface of radius
about the point
as center.
Now if
has in general a definite value, we must have
for
. Also,
will have in general a definite value. For
, the value of
is evidently
. We have, therefore,
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98. If
has in general a definite value,
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Hence, by No. 71,
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That is,
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If we set
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we have
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where
, and
are such functions of position that
, and
. This is expressed by saying that
is solenoidal, and
irrotational.
, and
, like
, will have in general definite values.
It is worth while to notice that there is only one way in which a vector function of position in space having a definite potential can be thus divided into solenoidal and irrotational parts having definite potentials. For if
,
are two other such parts,
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and
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