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 $\nabla t\,$ and $\nabla u\,$, are equal, and at infinite distances within the space (if such there are) $r^{2}\left({\frac {dt}{dr}}-{\frac {du}{dr}}\right)=0\,$, where $r\,$ denotes the distance from some fixed origin,—then throughout the space

$\nabla t=\nabla u,\,$

and in each continuous part of which the space consists

$t-\,$[1]$u={\text{constant}}.\,$

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

$\nabla \times \tau =\nabla \times \omega \,$ and $\nabla \cdot \tau =\nabla \cdot \omega ,\,$

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

$\tau =\omega ,\,$

then throughout the whole space

$\tau =\omega .\,$

     For, since $\nabla \times (\tau -\omega )=0\,$, we may set $\nabla u=\tau -\omega \,$, making the space acyclic (if necessary) by diaphragms. Then in the whole space $u\,$ is single-valued and $\nabla \cdot \nabla u=0\,$, and in a part of the space, or in a surface in or bounding it, $\nabla u=0\,$. Hence throughout the space $\nabla u=\tau -\omega =0\,$.

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

$\nabla \times \tau =\nabla \times \omega \,$ and $\nabla \cdot \tau =\nabla \cdot \omega ,\,$

and in all the bounding surfaces the tangential components of $\tau \,$ and $\omega \,$ are equal, then throughout the space

$\tau =\omega .\,$

     It is evidently sufficient to prove this proposition for a continuous space. Setting $\nabla u=\tau -\omega \,$, we have $\nabla \cdot \nabla u=0\,$ for the whole space, and $u={\text{constant}}\,$ for its boundary, which will be a single surface for a continuous aperiphractic space. Hence throughout the space $\nabla u=\tau -\omega =0\,$.

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

$\nabla \times \tau =\nabla \times \omega \,$ and $\nabla \cdot \tau =\nabla \cdot \omega ,\,$

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

$\tau =\omega .\,$

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