Schwere, Elektricität und Magnetismus:391

VECTOR ANALYSIS.

<section begin=t1 /> gral of the curl of that function for any surface bounded by the line.

Vorlage:IdtTo prove this principle, we will consider the variation of the line-integral which is due to a variation in the closed line for which the integral is taken. We have, in the first place,

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Therefore, since $\int d(\omega \cdot \delta \rho )=0$ for a closed line,

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

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or by No. 30,

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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 $\mathrm{\nabla }\times \omega$ 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 $\mathrm{\nabla }\times \omega$ 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.

Vorlage:Idt61. The line-integral of $\omega$ for a closed line bounding a plane surface $d\sigma$ infinitely small in all its dimensions is therefore

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Vorlage:IdtThis principle affords a definition of $\mathrm{\nabla }\times \omega$ which is independent of any reference to coördinate 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 <section end=t1 />