Schwere, Elektricität und Magnetismus:406

VECTOR ANALYSIS.

<section begin=t1 /> Moreover, $\mathrm{Pot}\epsilon$ has in general a definite value, and therefore

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Vorlage:Idt99. To assist the memory of the student, some of the principal results of Nos. 93-98 may be expressed as follows:

Vorlage:IdtLet ${\omega }_1$ be any solenoidal vector function of position in space, ${\omega }_2$ any irrotational vector function, and $u$ any scalar function, satisfying the conditions that their potentials have in general definite values.

Vorlage:IdtWith respect to the solenoidal function ${\omega }_1$, $\frac{1}{4\pi }\mathrm{Lap}$ and $\mathrm{\nabla }\times$ are inverse operators; i. e.,

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Applied to the irrotational function ${\omega }_2$, either of these operators gives zero; i. e.,

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Vorlage:IdtWith respect to the irrotational function ${\omega }_2$ or the scalar function $u$, $\frac{1}{4\pi }\mathrm{New}$ and $-\mathrm{\nabla }\cdot$ are inverse operators; i. e.,

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Applied to the solenoidal function ${\omega }_1$ the operator $\mathrm{\nabla }\cdot$ gives zero; i. e.,

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Vorlage:IdtSince the most general form of a vector function having in general a definite potential may be written ${\omega }_1+{\omega }_2$, the effect of these operators on such a function needs no especial mention.

Vorlage:IdtWith respect to the solenoidal function ${\omega }_1$, $\frac{1}{4\pi }\mathrm{Pot}$ and $\mathrm{\nabla }\times \mathrm{\nabla }\times$ are inverse operators; i. e.,

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Vorlage:IdtWith respect to the irrotational function ${\omega }_2$, $\frac{1}{4\pi }\mathrm{Pot}$ and $-\mathrm{\nabla }\mathrm{\nabla }\cdot$ are inverse operators; i. e.,

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Vorlage:IdtWith respect to any scalar or vector function having in general a definite potential $\frac{1}{4\pi }\mathrm{Pot}$ and $-\mathrm{\nabla }\cdot \mathrm{\nabla }$ are inverse operators; i. e., <section end=t1 />