115] GREEN'S THEOREM. 343 



or, transposing , and denoting the symmetrical integral by J, 



9Cf\ T fCfl~ 8UdV i 3UdV l dUdV ~\J 



20) J =JJJ h^ + ^^ + ir^r* 



37008(110) 



This result is known as Green's Theorem. 1 } 



By the definition of differentiation in any direction (p. 333) the 

 parenthesis in the surface integral on the right is 



~==P v eos(P v ri), 



if P v is the parameter of V. In like manner the symmetrical func- 

 tion of U and V on the left, the integrand in J, is the geometric 

 product of the vector parameters of U and V. This symmetric 

 function, which we will denote by <d (U, F), 



dUdV . dU dV 



is often called the mixed differential parameter of U and F From 

 its geometrical properties, or by direct calculation, it is also a 

 differential invariant for a transformation of coordinates. 



Since the integral on the left is symmetrical in U and F, we 

 may interchange them on the right, so that 



21) J- 

 Writing this equal to the former value, and transposing, we obtain 



, , -, 



-f ^2 + ^r- 2 ) f dr. 

 ' ^2/ 2 dz*)} > 



which will be referred to as Green's theorem in its second form. 



We shall, unless the contrary is stated, always mean by n the 

 internal normal to a closed surface, but if necessary we shall 

 distinguish the normals drawn internally and externally as n { and n*. 

 If we do not care to distinguish the inside from the outside we shall 

 denote the normals toward opposite sides by % and w 2 . 



1) An Essay on the Application of Mathematical Analysis to the theories of 

 Electricity and Magnetism. Nottingham, 1828. Geo. Green, Reprint of papers, p. 25. 



