Calculus of several variables

Abstract

The purpose of this chapter is to develop certain aspects of differential calculus from the case of functions of one variable, familiar from school mathematics, to functions of several variables. Together with linear algebra, the subject of Chapter 5, differential calculus is a standard tool for the construction of models in economics and calculus methods are used in a variety of different applications in economics. A particularly large group of problems in which calculus methods manifest themselves is that class of problems involving the maximisation of some function. Edgeworth (1881) took the view that:

The principal inquiries in Social Science may be viewed as maximum problems. For Economics investigates the arrangements between agents each tending to his own maximum utility, and Politics and (Utilitarian) Ethics investigates the arrangements which conduce to the maximum sum total of utility. Since, then, Social Science, as compared with the Calculus of Variations, starts from similar data — loose quantitative relations — and travels to a similar conclusion — determination of maximum — why should it not pursue the same method, Mathematics?