to problems in the calculus of variations in more than one variable. Ten years after, in , Lamberto Cesari changed the continuity requirement in Tonelli's definition to a less restrictive integrability requirement, obtaining for the first time the class of functions of bounded variation of several variables in its full generality: as Jordan did before him, he applied the concept to resolve of a problem concerning the convergence of Fourier series, but for functions of two variables. After him, several authors applied BV functions to study Fourier series in several variables, geometric measure theory, calculus of variations, and mathematical physics. Renato Caccioppoli and Ennio de Giorgi used them to define measure of nonsmooth boundaries of sets (see the entry "Caccioppoli set" for further information). Olga Arsenievna Oleinik introduced her view of generalized solutions for nonlinear partial differential equations as functions from the space BV in the paper , and was able to construct a generalized solution of bounded variation of a first order partial differential equation in the paper : few years later, Edward D. Conway and Joel A. Smoller applied BV-functions to the study of a single nonlinear hyperbolic partial differential equation of first order in the paper , proving that the solution of the Cauchy problem for such equations is a function of bounded variation, provided the initial value belongs to the same class. Aizik Isaakovich Vol'pert developed extensively a calculus for BV functions: in the paper he proved the chain rule for BV functions and in the book he, jointly with his pupil Sergei Ivanovich Hudjaev, explored extensively the properties of BV functions and their application. His chain rule formula was later extended by Luigi Ambrosio and Gianni Dal Maso in the paper . Formal definition BV functions of one variable The total variation of a continuous real-valued (or more generally complex-valued) function f, defined on an interval [a, b] ⊂ ℝ is the quantity where the supremum is taken over the set of all partitions of the interval considered. If f is differentiable and its derivative is Riemann-integrable, its total variation is the vertical component of the arc-length of its graph, that is to say, A continuous real-valued function on the real line is said to be of bounded variation (BV function) on a chosen interval [a, b] ⊂ ℝ if its total variation is finite, i.e. It can be proved that a real function ƒ