Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/63

where $r^{2}dq$ is the element of a spherical surface having center at $\rho$ and radius $r.$ We may also set

$\nabla u'.{\frac {\rho '-\rho }{r}}={\frac {du'}{dr}}\cdot$

We thus obtain

$\nabla .{\text{New }}u=\iiint {\frac {du'}{dr}}dq\,dr=4\pi \int {\frac {d{\bar {u}}'}{dr}}dr=4\pi {\bar {u}}'_{r=\infty }-4\pi {\bar {u}}'_{r=0},$

where ${\bar {u}}$ denotes the average value of $u$ in a spherical surface of radius $r$ about the point $\rho$ as center.

Now if ${\text{Pot }}u$ has in general a definite value, we must have ${\bar {u}}'=0$ for $r=\infty .$ Also, $\nabla .{\text{New }}u$ will have in general a definite value. For $r=0,$ the value of ${\bar {u}}'$ is evidently $u.$ We have, therefore,

$\nabla .{\text{New }}u$	$=-4\pi u,$
$\nabla .\nabla {\text{Pot }}u$	$=-4\pi u.$[1]

98. If ${\text{Pot }}\omega$ has in general a definite value,

${\begin{aligned}\nabla .\nabla {\text{Pot }}\omega &=\nabla .\nabla {\text{Pot }}[ui+vj+wk]\\&=\nabla .\nabla {\text{Pot }}ui+\nabla .\nabla {\text{Pot }}vj+\nabla .\nabla {\text{Pot }}wk\\&=-4\pi ui-4\pi vj-4\pi wk\\&=-4\pi \omega .\end{aligned}}$

Hence, by No. 71,

$\nabla \times \nabla \times {\text{Pot }}\omega -\nabla \nabla .{\text{Pot }}\omega =4\pi \omega .$

If we set

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

where $\omega _{1}$ and $\omega _{2}$ are such functions of position that $\nabla .\omega _{1}=0,$ and $\nabla \times \omega _{2}=0.$ This is expressed by saying that $\omega _{1}$ is solenoidal, and $\omega _{2}$ irrotational. ${\text{Pot }}\omega _{1}$ and ${\text{Pot }}\omega _{2},$ like ${\text{Pot }}\omega ,$ will have in general definite values.

It is worth while to notice that there is only one way in which a vector function of position in space having a definite potential can be thus divided into solenoidal and irrotational parts having definite potentials. For if $\omega _{1}+\epsilon ,\,\omega _{2}-\epsilon$ are two other such parts,

$\nabla .\epsilon =0$⁠and⁠$\nabla \times \epsilon =0.$

Moreover, ${\text{Pot }}\epsilon$ has in general a definite value, and therefore

1. Better thus: ${\begin{aligned}\nabla .\nabla {\text{Pot }}u=\iiint {\frac {1}{r}}\nabla .\nabla u\,dv=\iiint \nabla \left({\frac {1}{r}}\nabla u\right)dv-\iiint \nabla .\left(u\nabla {\frac {1}{r}}\right)dv\\+\iiint u\nabla .\nabla {\frac {1}{r}}dv=-\iint u\nabla {\frac {1}{r}}.d\sigma =-4\pi u.\end{aligned}}$ [MS. note by author.]