Schwere, Elektricität und Magnetismus:399

Bernhard Riemann: Schwere, Elektricität und Magnetismus
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VECTOR ANALYSIS.


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



and in each continuous part of which the space consists


[1]


     86. If throughout any continuous space (or in all space)


and


and in any finite part of that space, or in any finite surface in or bounding it,



then throughout the whole space



     For, since , we may set , making the space acyclic (if necessary) by diaphragms. Then in the whole space is single-valued and , and in a part of the space, or in a surface in or bounding it, . Hence throughout the space .

     87. If throughout an aperiphractic*[2] space contained within finite boundaries but not necessarily continuous


and


and in all the bounding surfaces the tangential components of and are equal, then throughout the space



     It is evidently sufficient to prove this proposition for a continuous space. Setting , we have for the whole space, and for its boundary, which will be a single surface for a continuous aperiphractic space. Hence throughout the space .

     88. If throughout an acyclic space contained within finite boundaries but not necessarily continuous


and


and in all the bounding surfaces the normal components of and are equal, then throughout the whole space



  1. WS: handschriftliche Ergänzung t-
  2. * If a space encloses within itself another space, it is called periphractic, otherwise aperiphractic.