The principal theme of this book is “the existence and differentiability of the solutions of variational problems involving multiple integrals.” We shall discuss the corresponding questions for single integrals only very briefly since these have been discussed adequately in every other book on the calculus of variations.Moreover, applications to engineering,physics,etc.,are not discussed at all;however,we do discuss mathematical applications to such subjects as the theory of harmonic integrals and the so-called “d-Neumann” problem (see Chapters 7 and 8).Since the plan of the book is described in Section 1.2 below we shall merely make a few observations here.