Schwere, Elektricität und Magnetismus:398

Vorlage:Bernhard Riemann - Schwere, Elektricität und Magnetismus Vorlage:PageDef2

VECTOR ANALYSIS.

<section begin=t1 /> 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.

Vorlage:IdtThis proposition also may be generalized by substituting $\nabla \cdot [t \nabla u] = 0$ for $\nabla \cdot \nabla u = 0$ , and $t r^{2} \frac{d u}{d r} = 0$ for $r^{2} \frac{d u}{d r} = 0$ .