CHAPTER VII. 



GENERAL ANALYTICAL THEOREMS. 



GREEN'S THEOREM. 



175. A THEOREM, first given by Green, and commonly called after him, 

 enables us to express an integral taken over the surfaces of a number of 

 bodies as an integral taken through the space between them. This theorem 

 naturally has many applications to Electrostatic Theory. It supplies a means 

 of handling analytically the problems which Faraday treated geometrically 

 with the help of his conception of tubes of force. 



176. THEOREM. If u, v, w are continuous functions of the Cartesian 

 coordinates a, y, z, then 



2 jj(lu + mv + nw) dS = - fff(j~ + ~ + 1?) dxdydz (94). 



Here 2 denotes that the surface integrals are summed over any number of 

 closed surfaces, which may include as special cases either 



(i) one of finite size which encloses all the others, or 

 (ii) an imaginary sphere of infinite radius, 



and I, m, n are the direction-cosines of the normal drawn in every case from 

 the element dS into the space between the surfaces. The volume integral is 

 taken throughout the space between the surfaces. 



Consider first the value of 1 1 / dxdydz. Take any small prism with its 



axis parallel to that of x, and of cross-section dydz. Let it meet the surfaces 

 at P, Q, R,S,T,U,... t (fig. 53), cutting off areas dS P , dS Q> dS*, .... 



The contribution of this prism to I M^ dxdydz is dydz\^- doc, where the 

 integral is taken over those parts of the prism which are between the surfaces. 



fdw, [ Q faj [ s du 



Thus \ dx=\ dx+\ 5- 



] OX } pOX ] R OX 



R "bx 

 UP+UQ U R 



