Inverse and Implicit Functions

Abstract

This chapter uses the results of the previous three chapters to prove the inverse function theorem, that an invertible derivative connotes a locally invertible mapping. Equivalently, the implicit function theorem states that under some conditions, a set of constraints on a set of variables locally specifies some of the variables as functions of the others. The Lagrange multiplier condition follows, giving a method to solve optimization problems with constraints, i.e., to begin doing calculus in curved spaces.