Schwere, Elektricität und Magnetismus:408

Bernhard Riemann: Schwere, Elektricität und Magnetismus
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VECTOR ANALYSIS.

     In the case of a vector function which is discontinuous at a surface, the expressions $\nabla \cdot \omega \;dv$ and $\nabla \times \omega \;dv$, relating to the element of the shell which we substitute for the surface of discontinuity, are easily transformed by the priciple that these expressions are the direct and skew surface-integrals of $\omega \;$ for the element of the shell. (See Nos. 55, 56.) The part of the surface-integrals relating to the edge of the element may evidently be neglected, and we shall have

$\nabla \cdot \omega \;dv=\omega _{2}\cdot \;d\sigma -\omega _{1}\cdot \;d\sigma =\Delta \omega \cdot d\sigma ,\!$

$\nabla \times \omega \;dv=\;d\sigma \times \omega _{2}-\;d\sigma \times \omega _{1}=\;d\sigma \times \Delta \omega .$

     Whenever, therefore, $\omega \!$ is discontinuous at surfaces, the expressions $\operatorname {Pot} \nabla \cdot \omega \!$ and $\operatorname {New} \nabla \cdot \omega \!$ must be regarded as implicitly including the surface-integrals

$\iint {1 \over [\rho '-\rho ]_{0}}\Delta \omega '\cdot \;d\sigma '$ and $\iint {\rho '-\rho \over [\rho '-\rho ]_{0}^{3}}\Delta \omega '\cdot d\sigma '$

respectively, relating to such surfaces, and the expressions $\operatorname {Pot} \nabla \times \omega \!$ and $\operatorname {Lap} \nabla \times \omega \!$ as including the surface-integrals

$\iint {1 \over [\rho '-\rho ]_{0}}d\sigma '\times \Delta \omega '$ and $\iint {\rho '-\rho \over [\rho '-\rho ]_{0}^{3}}\times [d\sigma '\times \Delta \omega ']$

respectively, relating to such surfaces.

     101. We have already seen that if $\omega \!$ is the curl of any vector function of position, $\nabla \cdot \omega =0$. (No. 68.) The converse is evidently true, whenever the equation $\nabla \cdot \omega =0$ holds throughout all space, and $\omega \!$ has in general a definite potential; for then

$\omega =\nabla \times {1 \over 4\pi }\operatorname {Lap} \;\omega .$

     Again, if $\nabla \cdot \omega =0\!$ within any aperiphractic space $A\!$, contained within finite boundaries, we may suppose that space to be enclosed by a shell $B\!$ having its inner surface coincident with the surface of $A\!$. We may imagine a function of position $\omega '\!$, such that $\omega '=\omega \!$ in $A,\omega '=0\!$ outside of the shell $B\!$, and the integral $\iiint \omega '\cdot \omega '\;dv$ for $B\!$ has the least value consistent with the conditions that the normal component of $\omega '\!$ at the outer surface is zero, and at the inner surface is equal to that of $\omega \!$. Then $\nabla \cdot \omega =0\!$ throughout all space, (No. 90,) and the potential of $\omega '\!$ will have in general a definite value. Hence,

$\omega '=\nabla \times {1 \over 4\pi }\operatorname {Lap} \;\omega '$,

and $\omega \!$ will have the same value within the space $A\!$.

__________________ New Haven: Printed by Tuttle, Morehouse & Taylor, 1881.