ABSTRACT
This chapter provides ideas and results for the basic problem of several independent variables, or an immediate extension of this problem. It presents miscellaneous results which include the Hamilton-Jacobi theory, the problems of several independent variables, sufficiency theory, and, finally, the several dependent variables case. These ideas and results are included in most modern day texts in the calculus of variations but are not really necessary for the understanding of the material. The chapter presents the problems whose Euler-Lagrange equations are partial differential equations and involves the Hamiltonian function and the resulting Hamilton-Jacobi equations. Historically, sufficiency theory came much later than necessity theory for several reasons. It has simply been much more difficult to obtain a sufficiency theory. There is an important point to make which is that necessary conditions stand individually while sufficient conditions go together. Thus, for example for necessary conditions, it is seen that the Legendre, Weierstrass or Jacobi conditions can be considered separately.
