Differentiation in $$\mathbb {R}^{n}$$ R n

The analysis of the behavior of a function is efficiently carried out when we study the way in which the function varies. In this chapter, techniques used in studying functions of one real variable $$ f:I\rightarrow \mathbb {R}$$ f : I → R , defined in an open interval $$I\subset \mathbb {R}$$ I ⊂ R , are extended for functions of several real variables $$f:U\rightarrow \mathbb {R}^{m}$$ f : U → R m defined over an open subset $$U\subset \mathbb {R}^{n}$$ U ⊂ R n . Several real variables is understood to mean a finite number of variables $$(x_{1},\ldots ,x_{n})\in \mathbb {R}^{n}$$ ( x 1 , … , x n ) ∈ R n . The simple topological nature of $$\mathbb {R}^{n}$$ R n allows the theory to be more easily understood as all the concepts and techniques. The same concepts and techniques will be studied in the chapters ahead within the framework of Banach spaces.