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,

$\delta \int \omega \cdot d\rho =\int \delta \omega \cdot d\rho +\int \omega \cdot \delta d\rho .\,$

But	$\omega \cdot \delta d\rho =d(\omega \cdot \delta \rho )-d\omega \cdot \delta \rho .\,$

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

$\delta \int \omega \cdot d\rho =\int \delta \omega \cdot d\rho -\int d\omega \cdot \delta \rho .\,$

Now	$\delta \omega =\sum \left\lbrack {\frac {d\omega }{dx}}\delta x\right\rbrack =\sum \left\lbrack {\frac {d\omega }{dx}}(i\cdot \delta \rho )\right\rbrack ,\,$

and	$d\omega =\sum \left\lbrack {\frac {d\omega }{dx}}dx\right\rbrack =\sum \left\lbrack {\frac {d\omega }{dx}}(i\cdot d\rho )\right\rbrack ,\,$

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

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

or by No. 30,

$\delta \int \omega \cdot d\rho =\int \sum \left\lbrack i\times {\frac {d\omega }{dx}}\right\rbrack \cdot \lbrack \delta \rho \times d\rho \rbrack =\int \nabla \times \omega \cdot \lbrack \delta \rho \times d\rho \rbrack .\,$

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

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

     This principle affords a definition of $\nabla \times \omega \,$ 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