Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/92

differential coefficients of the second order of $u$ with respect to $x,y,$ and $z.$^[1]

161. The operators $\nabla \times$ and $\nabla .$ may be applied to dyadics in a manner entirely analogous to their use with scalars. Thus we may define $\nabla \times \Phi$ and $\nabla .\Phi$ by the equations

${\begin{aligned}\nabla \times \Phi &=i\times {\frac {d\Phi }{dx}}+j\times {\frac {d\Phi }{dy}}+k\times {\frac {d\Phi }{dz}}\\\nabla .\Phi &=i.\,{\frac {d\Phi }{dx}}+j.\,{\frac {d\Phi }{dy}}+k.\,{\frac {d\Phi }{dz}}\end{aligned}}$

Then, if

${\begin{aligned}\Phi &=\alpha i+\beta j+\gamma k,\\\nabla \times \Phi &=\nabla \times \alpha i+\nabla \times \beta j+\nabla \times \gamma k.\\\nabla .\Phi &=\nabla .\alpha i+\nabla .\beta j+\nabla .\gamma k.\end{aligned}}$

Or, if

${\begin{aligned}\Phi &=i\alpha +j\beta +k\gamma ,\\\nabla \times \Phi &=i\left[{\frac {d\gamma }{dy}}-{\frac {d\beta }{dz}}\right]+j\left[{\frac {d\alpha }{dz}}-{\frac {d\gamma }{dx}}\right]+k\left[{\frac {d\beta }{dx}}-{\frac {d\alpha }{dy}}\right],\\\nabla .\Phi &={\frac {d\alpha }{dx}}+{\frac {d\beta }{dy}}+{\frac {d\gamma }{dz}}\cdot \end{aligned}}$

162. We may now regard $\nabla .\nabla$ in expressions like $\nabla .\nabla \omega$ as representing two successive operations, the result of which will be

${\frac {d^{2}\omega }{dx^{2}}}+{\frac {d^{2}\omega }{dy^{2}}}+{\frac {d^{2}\omega }{dz^{2}}}$

in accordance with the definition of No. 70. We may also write $\nabla .\nabla \Phi$ for

${\frac {d^{2}\Phi }{dx^{2}}}+{\frac {d^{2}\Phi }{dy^{2}}}+{\frac {d^{2}\Phi }{dz^{2}}},$

although in this case we cannot regard $\nabla .\nabla$ as representing two successive operations until we have defined $\nabla \Phi .$^[2]

That $\nabla .\nabla \Phi =\nabla \nabla .\Phi -\nabla \times \nabla \times \Phi$ will be evident if we suppose $\Phi$ to be expressed in the form $\alpha i+\beta j+\gamma k.$ (See No. 71.)

163. We have already seen that

$u''-u'=\int d\rho .\nabla u,$

where $u'$ and $u''$ denote the values of $u$ at the beginning and the end of the line to which the integral relates. The same relation will hold for a vector; i.e.,

$\omega ''-\omega '=\int d\rho .\nabla \omega .$

1. We might proceed to higher steps in differentiation by means of the triadics $\nabla \nabla \omega ,\nabla \nabla \nabla \omega ,$ the tetradios $\nabla \nabla \nabla \omega ,\nabla \nabla \nabla \nabla \omega ,$ etc. See note on page 74. In like manner a dyadic function of position in space ($\Phi$) might be differentiated by means of the triadic $\nabla \Phi ,$ the tetradic $\nabla \nabla \Phi ,$ etc.
2. See footnote to No. 160.