Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/50

Now⁠	$\delta \omega =$	$\textstyle \sum \displaystyle \left[{\frac {d\omega }{dx}}\delta x\right]=$	$\textstyle \sum \displaystyle \left[{\frac {d\omega }{dx}}\delta (i.\delta \rho )\right],$
and⁠	$d\omega =$	$\textstyle \sum \displaystyle \left[{\frac {d\omega }{dx}}dx\right]=$	$\textstyle \sum \displaystyle \left[{\frac {d\omega }{dx}}\delta (i.d\rho )\right],$

where the summation relates to the coordinate axes and connected quantities. Substituting these values in the preceding equation,

$\delta \int \omega .d\rho =\int \textstyle \sum \displaystyle \left((i.\delta \rho )\left({\frac {d\omega }{dx}}.d\rho \right)-(i.d\rho )\left({\frac {d\omega }{dx}}.\delta \rho \right)\right),$

or by No. 30,

$\delta \int \omega .d\rho =\int \textstyle \sum \displaystyle \left[i\times {\frac {d\omega }{dx}}\right].[\delta \rho \times d\rho ]=\int \nabla \times \omega .[\delta \rho \times d\rho ].$

But $\delta \rho \times d\rho$ represents an element of the surface generated by the motion of the element $d\rho ,$ and the last member of the equation is the surface-integral of $\nabla \times \omega$ for the infinitesimal surface generated by the motion of the whole line. Hence, if we conceive of a closed curve passing gradually from an infinitesimal loop to any finite form, the differential of the line-integral of $\omega$ for that curve will be equal to the differential of the surface integral of $\nabla \times \omega$ for the surface generated: therefore, since both integrals commence with the value zero, they must always be equal to each other. Such a mode of generation will evidently apply to any surface closing any loop.

61. The line-integral of $\omega$ for a closed line bounding a plane surface $d\sigma$ infinitely small in all its dimensions is therefore

$\nabla \times \omega .d\sigma .$

This principle affords a definition of $\nabla \times \omega$ which is independent of any reference to coordinate axes. If we imagine a circle described about a fixed point to vary its orientation while keeping the same size, there will be a certain position of the circle for which the line-integral of $\omega$ will be a maximum, unless the line-integral vanishes for all positions of the circle. The axis of the circle in this position, drawn toward the side on which a positive motion in the circle appears counter-clockwise, gives the direction of $\nabla \times \omega ,$ and the quotient of the integral divided by the area of the circle gives the magnitude of $\nabla \times \omega .$

$\nabla ,\nabla .,$ and $\nabla \times$ applied to Functions of Functions of Position.

62. A constant scalar factor after $\nabla ,\nabla .,$ or $\nabla \times$ may be placed before the symbol.

63. If $f(u)$ denotes any scalar function of $u$, and $f'(u)$ the derived function,

$\nabla f(u)=f'(u)\nabla u.$