On the transformation of integrals

§ 1. It is known that, ifx =x (u ,v ),y =y (u ,v ), be one-valued continuous functions of (u ,v ), possessing continuous differential coefficients, whilef (x ,y ) is any continuous function of (x ,y ), ∬c f (x ,y )dx dy = ∫c a ∫d b f {x (u ,v ),y (u ,v )}∂(x ,y )/∂(u ,v )du dv , where the integration on the left-hand side is taken over the area of the plane curve, C, which is the image in the (x ,y )-plane of the rectangle (a ,b ;c ,d ) in the {u ,v )-plane. Here it is tacitly assumed that C divides the plane into two distinct parts, a limitation which, however, disappears when we employ the definition of integration over an area which I have found it necessary to introduce into analysis. 2. If we denote by F (x,y ) one of the indefinite integrals off (x, y ) with respect tox , and byx = x (t) ,y= y (t),