Schwere, Elektricität und Magnetismus:398

Bernhard Riemann: Schwere, Elektricität und Magnetismus
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VECTOR ANALYSIS.

Hence, the value of this part of the surface-integral may be made less (numerically) than any assignable quantity by giving to $r\,$ a sufficiently great value. Hence, the other part of the surface-integral (viz., that relating to the surface in which $u=b\,$, and to the boundary of the space to which the theorem relates,) may be given a value differing from zero by less than any assignable quantity. But no part of the integral relating to this surface can be negative. Therefore no part can be positive, and the supposition relative to the point $P\,$ is untenable.

     This proposition also may be generalized by substituting $\nabla \cdot \lbrack t\nabla u\rbrack =0\,$ for $\nabla \cdot \nabla u=0\,$, and $tr^{2}{\frac {du}{dr}}=0\,$ for $r^{2}{\frac {du}{dr}}=0\,$.

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

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

then throughout the same space

$t=u+{\text{const.}}\,$

The truth of this and the three following theorems will be apparent if we consider the difference $t-u\,$.

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

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

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

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

then throughout the whole space

$\nabla t=\nabla u\,$, and $t=u+{\text{const.}}\,$

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

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

and in all the bounding surfaces

$t=u,\,$

and at infinite distances within the space (if such there are)

$t=u,\,$

then throughout the space

$t=u.\,$

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

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