Schwere, Elektricität und Magnetismus:408

Bernhard Riemann: Schwere, Elektricität und Magnetismus
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VECTOR ANALYSIS.


     In the case of a vector function which is discontinuous at a surface, the expressions and , relating to the element of the shell which we substitute for the surface of discontinuity, are easily transformed by the priciple that these expressions are the direct and skew surface-integrals of for the element of the shell. (See Nos. 55, 56.) The part of the surface-integrals relating to the edge of the element may evidently be neglected, and we shall have




     Whenever, therefore, is discontinuous at surfaces, the expressions and must be regarded as implicitly including the surface-integrals


and


respectively, relating to such surfaces, and the expressions and as including the surface-integrals


and


respectively, relating to such surfaces.

     101. We have already seen that if is the curl of any vector function of position, . (No. 68.) The converse is evidently true, whenever the equation holds throughout all space, and has in general a definite potential; for then



     Again, if within any aperiphractic space , contained within finite boundaries, we may suppose that space to be enclosed by a shell having its inner surface coincident with the surface of . We may imagine a function of position , such that in outside of the shell , and the integral for has the least value consistent with the conditions that the normal component of at the outer surface is zero, and at the inner surface is equal to that of . Then throughout all space, (No. 90,) and the potential of will have in general a definite value. Hence,


,


and will have the same value within the space .

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New Haven: Printed by Tuttle, Morehouse & Taylor, 1881.