Schwere, Elektricität und Magnetismus:393

Bernhard Riemann: Schwere, Elektricität und Magnetismus
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VECTOR ANALYSIS.

$\omega \,$ is the derivative of a scalar function of position in space. This will appear from the following considerations:

The line-integral $\int \omega \cdot d\rho \,$ will vanish for any closed line, since it may be expressed as the surface-integral of $\nabla \times \omega \,$ . (No. 60.) The line-integral taken from one given point $P'\,$ to another given point $P''\,$ is independent of the line between the points for which the integral is taken. (For, if two lines joining the same points gave different values, by reversing one we should obtain a closed line for which the integral would not vanish.) If we set $u\,$ equal to this line-integral, supposing $P''\,$ to be variable and $P'\,$ to be constant in position, $u\,$ will be a scalar function of the position of the point $P''\,$ , satisfying the condition $du=\omega \cdot d\rho \,$ , or, by No. 51, $\nabla u=\omega \,$ . There will evidently be an infinite number of functions satisfying this condition, which will differ from one another by constant quantities.

If the region for which $\nabla \times \omega =0\,$ is unlimited, these functions will be single-valued. If the region is limited, but acyclic,*^[1] the functions will still be single-valued and satisfy the condition $\nabla u=\omega \,$ within the same region. If the region is cyclic, we may determine functions satisfying the condition $\nabla u=\omega \,$ within the region, but they will not necessarily be single-valued.

68. If $\omega \,$ is any vector function of position in space, $\nabla \cdot \nabla \times \omega =0\,$ . This may be deduced directly from the definitions of No. 54.

The converse of this proposition will be proved hereafter.

69. If $u\,$ is any scalar function of position in space, we have by Nos. 52 and 54

\nabla \cdot \nabla u=\left({\frac {d^{2}}{dx^{2}}}+{\frac {d^{2}}{dy^{2}}}+{\frac {d^{2}}{dz^{2}}}\right)u.\,

70. Def.—If $\omega \,$ is any vector function of position in space, we may define $\nabla \cdot \nabla \omega \,$ by the equation

\nabla \cdot \nabla \omega =\left({\frac {d^{2}}{dx^{2}}}+{\frac {d^{2}}{dy^{2}}}+{\frac {d^{2}}{dz^{2}}}\right)\omega ,\,

↑ * If every closed line within a given region can contract to a single point without breaking its continuity, or passing out of the region, the region is called acyclic, otherwise cyclic. A cyclic region may be made acyclic by diaphragms, which must then be regarded as forming part of the surface bounding the region, each diaphragm contributing its own area twice to that surface. This process may be used to reduce many-valued functions of position in space, having single-valued derivatives, to single-valued functions. When functions are mentioned or implied in the notation, the reader will always understand single-valued functions, unless the contrary is distinctly intimated, or the case is one in which the distinction is obviously immaterial. Diaphragms may be applied to bring functions naturally many-valued under the application of some of the following theorems, as Nos. 74 ff.

[1] * If every closed line within a given region can contract to a single point without breaking its continuity, or passing out of the region, the region is called acyclic, otherwise cyclic. A cyclic region may be made acyclic by diaphragms, which must then be regarded as forming part of the surface bounding the region, each diaphragm contributing its own area twice to that surface. This process may be used to reduce many-valued functions of position in space, having single-valued derivatives, to single-valued functions. When functions are mentioned or implied in the notation, the reader will always understand single-valued functions, unless the contrary is distinctly intimated, or the case is one in which the distinction is obviously immaterial. Diaphragms may be applied to bring functions naturally many-valued under the application of some of the following theorems, as Nos. 74 ff.

[1]