VECTOR ANALYSIS.
Moreover,
has in general a definite value, and therefore
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Q. E. D.
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99. To assist the memory of the student, some of the principal results of Nos. 93-98 may be expressed as follows:
Let
be any solenoidal vector function of position in space,
any irrotational vector function, and
any scalar function, satisfying the conditions that their potentials have in general definite values.
With respect to the solenoidal function
,
and
are inverse operators; i. e.,
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Applied to the irrotational function
, either of these operators gives zero; i. e.,
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With respect to the irrotational function
or the scalar function
,
and
are inverse operators; i. e.,
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Applied to the solenoidal function
the operator
gives zero; i. e.,
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Since the most general form of a vector function having in general a definite potential may be written
, the effect of these operators on such a function needs no especial mention.
With respect to the solenoidal function
,
and
are inverse operators; i. e.,
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With respect to the irrotational function
,
and
are inverse operators; i. e.,
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With respect to any scalar or vector function having in general a definite potential
and
are inverse operators; i. e.,