Schwere, Elektricität und Magnetismus:406

Bernhard Riemann: Schwere, Elektricität und Magnetismus
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VECTOR ANALYSIS.

Moreover, $\operatorname {Pot} \varepsilon \,$ has in general a definite value, and therefore

$\varepsilon ={\frac {1}{4\pi }}\operatorname {Lap} \nabla \times \varepsilon -{\frac {1}{4\pi }}\operatorname {New} \nabla \cdot \varepsilon =0.\,$

Q. E. D.

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

     Let $\omega _{1}\,$ 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.

     With respect to the solenoidal function $\omega _{1}\,$, ${\frac {1}{4\pi }}\operatorname {Lap} \,$ and $\nabla \times \,$ are inverse operators; i. e.,

${\frac {1}{4\pi }}\operatorname {Lap} \nabla \times \omega _{1}=\nabla \times {\frac {1}{4\pi }}\operatorname {Lap} \omega _{1}=\omega _{1}.\,$

Applied to the irrotational function $\omega _{2}\,$, either of these operators gives zero; i. e.,

$\operatorname {Lap} \omega _{2}=0,\quad \nabla \times \omega _{2}=0.\,$

     With respect to the irrotational function $\omega _{2}\,$ or the scalar function $u\,$, ${\frac {1}{4\pi }}\operatorname {New} \,$ and $-\nabla \cdot \,$ are inverse operators; i. e.,

$-{\frac {1}{4\pi }}\operatorname {New} \nabla \cdot \omega _{2}=\omega _{2},\quad -\nabla \cdot {\frac {1}{4\pi }}\operatorname {New} u=u.\,$

Applied to the solenoidal function $\omega _{1}\,$ the operator $\nabla \cdot \,$ gives zero; i. e.,

$\nabla \cdot \omega _{1}=0.\,$

     Since 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.

     With respect to the solenoidal function $\omega _{1}\,$, ${\frac {1}{4\pi }}\operatorname {Pot} \,$ and $\nabla \times \nabla \times \,$ are inverse operators; i. e.,

${\frac {1}{4\pi }}\operatorname {Pot} \nabla \times \nabla \times \omega _{1}=\nabla \times {\frac {1}{4\pi }}\operatorname {Pot} \nabla \times \omega _{1}=\nabla \times \nabla \times {\frac {1}{4\pi }}\operatorname {Pot} \omega _{1}=\omega _{1}.\,$

     With respect to the irrotational function $\omega _{2}\,$, ${\frac {1}{4\pi }}\operatorname {Pot} \,$ and $-\nabla \nabla \cdot \,$ are inverse operators; i. e.,

$-{\frac {1}{4\pi }}\operatorname {Pot} \nabla \nabla \cdot \omega _{2}=-\nabla {\frac {1}{4\pi }}\operatorname {Pot} \nabla \cdot \omega _{2}=-\nabla \nabla \cdot {\frac {1}{4\pi }}\operatorname {Pot} \omega _{2}=\omega _{2}.\,$

     With respect to any scalar or vector function having in general a definite potential ${\frac {1}{4\pi }}\operatorname {Pot} \,$ and $-\nabla \cdot \nabla \,$ are inverse operators; i. e.,