HELMHOLTZS THEOREM WHEN THE DOMAIN IS INFINITE AND WHEN THE FIELD HASSINGULAR POINTS

A direct proof, based on classical analysis, is given to extend Helmho ltz's fundamental theorem of vector analysis to cases in which the dom ain is infinite and/or the field u(r) has isolated singular points. No restriction whatsoever is placed upon the growth of u(r) as \r\-->inf inity, or as the singular points are approached, Consequently all repr esentation theorems dependent upon Helmholtz's theorem are also freed from such restrictions.