ABSTRACT
This chapter discusses the concept of functions by allowing the members of the dependent set not being restricted to be numbers, but to be functions themselves. Despite the simplicity of the example, the connection of a geometric problem to a variational formulation of a functional is clearly visible. The chapter describes the fundamental problem of calculus of variations as finding extrema of functionals, most commonly formulated in the form of an integral. The topic of the calculus of variations evolves from the analysis of functions. Certain similarities to the extremum evaluation of regular functions by the teaching of classical calculus are obvious. The problem of the brachistochrone may be the first problem of variational calculus, already solved by Johann Bernoulli in the late 1600s. The name stands for the shortest time in Greek, indicating the origin of the problem.
