On the Riemann–Stieltjes double integral

The article deals with the new properties of the Riemann–Stieltjes curvilinear integral. It is proved that the Riemann–Stieltjes curvilinear integral is independent of path of integration if an integrable and an integrating functions depend only on one variable. A new necessary condition of the functional dependence of functions of two variables is found. The author proposes a new approach to the definition of the Riemann–Stieltjes double integral, which contains not one but two integrating functions. General properties of the Riemann–Stieltjes double integral are discussed. Methods for calculating the double integral for the case of smooth or piecewise-smooth integrating functions are presented. A formula for the conversion of the Riemann–Stieltjes double integral into an iterated integral is obtained.