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Schwere, Elektricität und Magnetismus:405

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Bernhard Riemann: Schwere, Elektricität und Magnetismus
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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



where is the element of a spherical surface having center at and radius . We may also set



We thus obtain



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,




     98. If has in general a definite value,




Hence, by No. 71,



That is,



     If we set



we have



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,


and