Now |
![{\displaystyle \delta \omega =}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7322b5bbf519ad546d0184fe1770f73edab3bf07) |
![{\displaystyle \textstyle \sum \displaystyle \left[{\frac {d\omega }{dx}}\delta x\right]=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7a99f0ad682409fac12bfefb3dbf704f8fb5c4e9) |
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and |
![{\displaystyle d\omega =}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0be319feaeba5ca68402f9af0c6322c91e4f0c29) |
![{\displaystyle \textstyle \sum \displaystyle \left[{\frac {d\omega }{dx}}dx\right]=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/28db52bcdf229325c2969f6d876113eba34404db) |
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where the summation relates to the coordinate axes and connected quantities. Substituting these values in the preceding equation,
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or by No. 30,
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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
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This principle affords a definition of
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
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
and the quotient of the integral divided by the area of the circle gives the magnitude of
and
applied to Functions of Functions of Position.
62. A constant scalar factor after
or
may be placed before the symbol.
63. If
denotes any scalar function of
, and
the derived function,
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