Functions of Several Variables

The rest of the course is devoted to calculus of several variables in which we study continuity,
di®erentiability and integration of functions from Rn to R, and their applications.
In calculus of single variable, we had seen that the concept of convergence of sequence played
an important role, especially, in de¯ning limit and continuity of a function, and deriving some
properties of R and properties of continuous functions. This motivates us to start with the notion
of convergence of a sequence in Rn. For simplicity, we consider only R2 or R3. General case is
entirely analogous.
Convergence of a sequence : Let Xn = (x1;n; x2;n; x3;n) 2 R3. We say that the sequence (Xn)
is convergent if there exists X0 2 R3 such that kXn ?X0k ! 0 as n ! 1. In this case we say that
Xn converges to X0 and we write Xn ! X0.
Note that corresponding to a sequence (Xn), Xn = (x1;n; x2;n; x3;n), there are three sequences
(x1;n)(x2;n) and (x3;n) in R, and vice-versa. Thus the properties of (Xn) can be completely under-
stood in terms of the properties of the corresponding sequences (x1;n)(x2;n) and (x3;n) in R. For
example,
(i) Xn ! X0 in R3 , the coordinates xi;n ! xi;0 for every i = 1; 2; 3 in R.
(ii) (Xn) is bounded (i.e., 9 M such that kXnk · M 8 n) , each sequence (xi;n); i = 1; 2; 3,