Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/64

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 $\omega _{1},{\frac {1}{4\pi }}{\text{Lap}}$ and $\nabla \times$ are inverse operators; i.e.,

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

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

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

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

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

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

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

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 $\omega _{1},{\frac {1}{4\pi }}{\text{Pot }}$ and $\nabla \times \nabla \times$ are inverse operators; i.e.,

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

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

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

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

$-{\frac {1}{4\pi }}{\text{Pot }}\nabla .\nabla u=-\nabla {\frac {1}{4\pi }}{\text{Pot }}\nabla u=-\nabla .\nabla {\frac {1}{4\pi }}{\text{Pot }}u=u,$ $-{\frac {1}{4\pi }}{\text{Pot }}\nabla .\nabla [\omega _{1}+\omega _{2}]=-\nabla .\nabla {\frac {1}{4\pi }}{\text{Pot }}[\omega _{1}+\omega _{2}]=\omega _{1}+\omega _{2}.$

With respect to the solenoidal function $\omega _{1},-\nabla .\nabla$ and $\nabla \times \nabla \times$ are equivalent; with respect to the irrotational function $\omega _{2},\nabla .\nabla$ and $\nabla \nabla .$ are equivalent; i.e.,

$-\nabla .\nabla \omega _{1}=\nabla \times \nabla \times \omega _{1},$⁠$\nabla .\nabla \omega _{2}=\nabla \nabla .\omega _{2}.$