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

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where the summation relates to the coordinate axes and connected quantities. Substituting these values in the preceding equation, or by No. 30,  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 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 $$\nabla \times \omega.$$

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,