Schwere, Elektricität und Magnetismus:397

Bernhard Riemann: Schwere, Elektricität und Magnetismus
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VECTOR ANALYSIS.


ances within the space ,—then throughout the space


and


     For, if anywhere in the interior of the space has a value different from zero, we may find a point where such is the case, and where has a value different from ,—to fix our ideas we will say less. Imagine a surface enclosing all of the space in which . (This must be possible, since that part of the space does not reach to infinity.) The surface-integral of for this surface has the value zero in virtue of the general condition . But, from the manner in which the surface is defined, no part of the integral can be negative. Therefore no part of the integral can be positive, and the supposition made with respect to the point is untenable. That the supposition that is untenable may be shown in a similar manner. Therefore the value of is constant.

     This proposition may be generalized by substituting the condition for , denoting any positive (or any negative) scalar function of position in space. The conclusion would be the same, and the demonstration similar.

     81. If throughout a certain space (which need not be continuous, and which may extend to infinity,)



and in all the bounding surfaces the normal component of vanishes, and at infinite distances within the space (if such there are) , where denotes the distance from a fixed origin, then throughout the space



and in each continuous portion of the same



     For, if anywhere in the space in question has a value different from zero, let it have such a value at a point , and let be there equal to . Imagine a spherical surface about the above-mentioned origin as center, enclosing the point , and with a radius . Consider that portion of the space to which the theorem relates which is within the sphere and in which . The surface-integral of for this space is equal to zero in virtue of the general condition . That part of the integral (if any) which relates to a portion of the spherical surface has a value numerically not greater than , where denotes the greatest numerical value of in the portion of the spherical surface considered.