Integrals Over Surfaces in Parametric Form

fff(x, y, z, X, Y, Z) du dv, where X, Y, Z are the three jacobians of y, z with respect to u, v. In the present paper a similar investigation is conducted for a wider class of surfaces, including for example surfaces for which the derivatives x, etc., are summable together with their squares, and also surfaces z = z(x, y) which are absolutely continuous in the sense of Tonelli. Along with theorems concerning semi-continuity, we find that for our class of surfaces the Lebesgue area is given by the classical double integral. The consideration of this wider class of surfaces is desirable not merely for the sake of generalization, but also because of the fact that the important class of saddle surfaces bounded by a Jordan curve and of finite area is not included in the class of rectifiable surfaces, but is included (as will be shown in a later paper) in the class of surfaces for which x2, etc., are summable. Since this paper is intended as a sequel to the preceding one' on double integrals, we retain the notation and definitions of that paper, and request the reader to read the first portion of it as an introduction. The first part of the present paper is concerned with the definition of the class of surfaces considered and with the statement of the problem. In the second part we obtain some lemmas on approximation by polyhedra which enable us to reduce the problem to a simpler one. The third part contains the statement of the principal theorems of the paper. The fourth part is devoted in part to the presentation of certain sets of conditions on the functions y, z which are sufficient to assure us that the surface x = x(u, v), y = y(u, v), z =z(u, v) belongs to our class of surfaces, and in part to the indication of certain fragmentary extensions of our theorems.