ABSTRACT
This chapter discusses a functional with multiple independent variables. The simplest case is that of two independent variables, and this will be the vehicle to introduce the process. Minimal surfaces occur in intriguing applications. For example, soap films spanned over various types of wire loops intrinsically attain such shapes, no matter how difficult the boundary curve is. Various biological cell interactions also manifest similar phenomena. The boundary condition represents a three-dimensional curve defined over the perimeter of the domain. The curve may be piecewise differentiable, but continuous and forms a closed loop, a Jordan curve. The problem has obvious relevance in mechanical engineering and computer aided manufacturing. An even more practical three-variable case, important in engineering dynamics, is when the Euclidean spatial coordinates are extended with time. Generalization to more spatial coordinates is not very frequent, although in some manufacturing applications five-dimensional hyper-spaces do occur.
