On the transformation of integrals

Abstract

§ 1. It is known that, if x = x ( u , v ), y = y ( u , v ), be one-valued continuous functions of ( u , v ), possessing continuous differential coefficients, while f ( 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 of f ( x, y ) with respect to x , and by x = x (t) , y= y (t),