Schwere, Elektricität und Magnetismus:391

Bernhard Riemann: Schwere, Elektricität und Magnetismus
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VECTOR ANALYSIS.


gral of the curl of that function for any surface bounded by the line.

     To 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,



But


Therefore, since for a closed line,



Now


and


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



or by No. 30,



But represents an element of the surface generated by the motion of the element , and the last member of the equation is the surface-integral of 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 for that curve will be equal to the differential of the surface integral of 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 for a closed line bounding a plane surface infinitely small in all its dimensions is therefore



     This principle affords a definition of 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 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