The Calculus of Variations has assumed an increasingly important role in modern developments in analysis, geometry, and physics. Originating as a study of certain maximum and minimum problems not treatable by the methods of elementary calculus, variational calculus in its present form provides powerful methods for the treatment of differential equations, the theory of invariants, existence theorems in geometric function theory, variational principles in mechanics. Also important are the applications to boundary value problems in partial differential equations and in the numerical calculation of many types of problems which can be stated in variational form. No literature representing these diverging viewpoints is to be found among standard texts on calculus of variations, and in this course an attempt will be made to do justice to this variety of problems. The subject matter with which calculus of variations is concerned is a class of extremum (i.e. maximum or minimum) problems which can be considered an extension of the familiar class of extremum problems dealt with by elementary differential calculus. In the elementary problems one seeks extremal values of a function of one or more (but in any case a finite number) real variables. In the more general problems considered by calculus of variations, the functions to be extremized, sometimes called functionals,have functions as independent variables. The area A(f) below a curve y = f (x), for example, is a functional since its value depends upon a whole function f. (It is possible to treat a functional as a function of an enumerable set of Fourier coefficients, but this attack usually leads to almost insuperable difficulties.)