ABSTRACT
In this chapter we will pursue some centuries-old questions: when does an equation F (x, y) = 0 determine a function y = f(x)? Is f differentiable, and if so, how do you calculate its derivative? These questions can be asked when the variables x, y take values in R, or in any Euclidean spaces. A path to the answers will lead us through a special case when x,y belong to spaces of the same dimension, say to Rn, and F(x,y) = x−g(y), for some function g : Rn → Rn. In other words, we will establish the existence and properties of the inverse function. Both of these can be best understood through the study of the derivative of g, and the Jacobian matrix Jg. All these results will then be used in the field of the so-called constrained optimization.
