A differential (in )is said to be a locally exact differential in a region if it is an exact differential in some neighbourhood of each .
Show that every rectangle with sides parallel to axes satifies if it is a locally exact differential..
A differential (in )is said to be a locally exact differential in a region if it is an exact differential in some neighbourhood of each .
Show that every rectangle with sides parallel to axes satifies if it is a locally exact differential..
If pdx+ qdy is locally exact, then it is, as you say, an exact differential. That means that there exist some function F(x,y) such that dF= pdx+ qdy, at least inside .
But then . Let t be any parameter for the curve and we have where a and b are the beginning and ending values for t. But since this is a closed curve, t= a and t= b give the same point so x(b)= x(a), y(b)= y(a) and F(x(b),y(b))- F(x(a),y(a))= 0.