Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/149

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100.]
GREEN'S THEOREM.
109

from the surface . The quantities and correspond to superficial densities, but at present we must consider them as defined by the above equations.

Green's Theorem is obtained by integrating by parts the expression

(5)


throughout the space within the surface .

If we consider as a component of a force whose potential is , and as a component of a flux, the expression will give the work done by the force on the flux.

If we apply the method of integration by parts, we find




;

(6)


or

.

(7)


In precisely the same manner by exchanging and , we should find


(8)


The statement of Green's Theorem is that these three expressions for are identical, or that




(9)



Correction of Green's Theorem for Cyclosis.

There are cases in which the resultant force at any point of a certain region fulfils the ordinary condition of having a potential, while the potential itself is a many-valued function of the coordinates. For instance, if

we find , a many-valued function of and , the values of forming an arithmetical series whose common difference