Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/68

The Maxwellian product of a vector and a scalar function of position in space is defined by the equation

${\text{Max }}(\omega ,u)=\iiint \iiint u{\frac {\rho '-\rho }{r^{3}}}.\omega '\,dv\,dv'=\iiint {\text{Max }}\omega \,dv=-{\text{New}}(u,\omega ).$

It is of course supposed that $u,w,\phi ,\omega ,$ are such functions of position that the above expressions have definite values.

104. By No. 97,

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

The volume-integral of this equation gives

$4\pi {\text{Pot}}(u,w)=\iiint {\text{New }}u.{\text{New }}w\,dv,$

if the integral

$\iint d\sigma .{\text{New }}u{\text{Pot }}w,$

for a closed surface, vanishes when the space included by the surface is indefinitely extended in all directions. This will be the case when everywhere outside of certain assignable limits the values of $u$ and $w$ are zero.

Again, by No. 102,

$4\pi \omega .{\text{Pot }}\phi$	$=\nabla \times {\text{Lap }}\omega .{\text{Pot }}\phi -\nabla {\text{Max }}\omega .{\text{Pot }}\phi$
	$=\nabla .[{\text{Lap }}\omega \times {\text{Pot }}\phi ]+{\text{Lap }}\omega .{\text{Lap }}\phi$
	$-\nabla .[{\text{Max }}\omega {\text{ Pot }}\phi ]+{\text{Max }}\omega {\text{ Max }}\phi .$

The volume-integral of this equation gives

$4\pi {\text{ Pot}}(\phi ,\omega )=\iiint {\text{Lap }}\phi .{\text{Lap }}\omega \,dv+\iiint {\text{Max }}\phi {\text{Max }}\omega \,dv,$

if the integrals

$\iint d\sigma .{\text{Lap }}\omega \times {\text{Pot }}\phi ,$⁠$\iint d\sigma .{\text{Pot }}\phi {\text{ Max }}\omega ,$

for a closed surface vanish when the space included by the surface is indefinitely extended in all directions. This will be the case if everywhere outside of certain assignable limits the values of $\phi$ and $\omega$ are zero.

CHAPTER III.

concerning linear vector functions.

105. Def.—A vector function of a vector is said to be linear, when the function of the sum of any two vectors is equal to the sum of the functions of the vectors. That is, if

${\text{func.}}[\rho +\rho ']={\text{func.}}[\rho ]+{\text{func.}}[\rho ']$

for all values of $\rho$ and $\rho ',$ the function is linear. In such cases it is easily shown that

${\text{func.}}[a\rho +b\rho '+c\rho ''+{\text{etc.}}]=a{\text{ func.}}[\rho ]+b{\text{ func.}}[\rho ']+c{\text{ func.}}[\rho '']+{\text{etc.}}$