Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/91

CHAPTER IV.

(Supplementary to Chapter II.)

concerning the differential and integral calculus of vectors.

169. If $\omega$ is a vector having continuously varying values in space, and $\rho$ the vector determining the position of a point, we may set

${\begin{aligned}\rho &=xi+yj+zk,\\d\rho &=dx\,i+dy\,j+dz\,k,\end{aligned}}$

and regard $\omega$ as a function of $\rho$ or of $x,y,$ and $z.$ Then,

$d\omega =dx{\frac {d\omega }{dx}}+dy{\frac {d\omega }{dy}}+dz{\frac {d\omega }{dz}},$

that is,

$d\omega =d\rho .\left\lbrace i{\frac {d\omega }{dx}}+j{\frac {d\omega }{dy}}+k{\frac {d\omega }{dz}}\right\rbrace \cdot$

If we set

$\nabla \omega =i{\frac {d\omega }{dx}}+j{\frac {d\omega }{dy}}+k{\frac {d\omega }{dz}},$ $d\omega =d\rho .\nabla \omega .$

Here $\nabla$ stands for

$i{\frac {d}{dx}}+j{\frac {d}{dy}}+k{\frac {d}{dz}},$

exactly as in No. 52, except that it is here applied to a vector and produces a dyadic, while in the former case it was applied to a scalar and produced a vector. The dyadic $\nabla \omega$ represents the nine differential coefficients of the three components of w with respect to $x,y,$ and $z,$ just as the vector $\nabla u$ (where $u$ is a scalar function of $\rho$) represents the three differential coefficients of the scalar $u$ with respect to $x,y,$ and $z.$

It is evident that the expressions $\nabla .\omega$ and $\nabla \times \omega$ already defined (No. 54) are equivalent to $\{\nabla \omega \}_{\text{S}}$ and $\{\nabla \omega \}_{\text{X}}.$

160. An important case is that in which the vector operated on is of the form $\nabla u.$ We have then

$d\nabla u=d\rho .\nabla \nabla u,$

where

$\nabla \nabla u={\begin{Bmatrix}{\frac {d^{2}u}{dx^{2}}}ii&+{\frac {d^{2}u}{dx\,dy}}ij&+{\frac {d^{2}u}{dx\,dz}}ik\\+{\frac {d^{2}u}{dy\,dx}}ji&+{\frac {d^{2}u}{dy^{2}}}jj&+{\frac {d^{2}u}{dy\,dz}}jk\\+{\frac {d^{2}u}{dz\,dx}}ki&+{\frac {d^{2}u}{dz\,dy}}kj&+{\frac {d^{2}u}{dz^{2}}}kk.\end{Bmatrix}}$

This dyadic, which is evidently self-conjugate, represents the six