Schwere, Elektricität und Magnetismus:396

Bernhard Riemann: Schwere, Elektricität und Magnetismus
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VECTOR ANALYSIS.

then throughout the same space

$u={\text{constant}}.\,$

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

$\nabla \cdot \nabla u=0,\,$

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

$\nabla u=0,\,$

then throughout the whole space

$\nabla u=0,\,$ and $u={\text{constant}}.\,$

This will appear from the following considerations.

     If $\nabla u=0\,$ in any finite part of the space, $u\,$ is constant in that part. If $u\,$ is not constant throughout, let us imagine a sphere situated principally in the part in which $u\,$ is constant, but projecting slightly into a part in which $u\,$ has a greater value, or else into a part in which $u\,$ has a less. The surface-integral of $\nabla u\,$ for the part of the spherical surface in the region where $u\,$ is constant will have the value zero : for the other part of the surface, the integral will be either greater than zero, or less than zero. Therefore the whole surface-integral for the spherical surface will not have the value zero, which is required by the general condition, $\nabla \cdot \nabla u=0\,$.

     Again, if $\nabla u=0\,$ only in a surface in or bounding the space in which $\nabla \cdot \nabla u=0\,$, $u\,$ will be constant in this surface, and the surface will be contiguous to a region in which $\nabla \cdot \nabla u=0\,$ and $u\,$ has a greater value than in the surface, or else a less value than in the surface. Let us imagine a sphere lying principally on the other side of the surface, but projecting slightly into this region, and let us particularly consider the surface-integral of $\nabla u\,$ for the small segment cut off by the surface $\nabla u=0\,$. The integral for that part of the surface of the segment which consists of part of the surface $\nabla u=0\,$ will have the value zero, the integral for the spherical part will have a value either greater than zero or else less than zero. Therefore the integral for the whole surface of the segment cannot have the value zero, which is demanded by the general condition, $\nabla \cdot \nabla u=0\,$.

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

$\nabla \cdot \nabla u=0,\,$

and in all the bounding surfaces

$u={\text{constant}}=a,\,$

and (in case the space extends to infinity) if at infinite dist-