M.Thamban Nair         Notes for the B.Tech. course: MA 1101
          Calculus of Several Variables
                 Chapter 1
                 Functions of Several Variables
                 1.1  Introduction
                    Functions of more than one variables come naturally in applications. For exam-
                 ple, in physics one comes across the relation
                                            PV =c, constant,
                                             T
                 where P,V,T represents the pressure, volume and temperature of an ideal gas. Since
                                      P = cT,   V = cT,  T = PV
                                           V        P         c
                 each of P,V,T can be thought of as a function of the remaining two variables.
                    In elementary geometry, we know that the area of a triangle of base length b
                 and altitude h, area of a rectangle of sides a and b are given by
                                          1bh,  ab  and πr2,
                                          2
                 respectively, and they are functions of the variables (b,h), (a,b) and r, respectively.
                 Also, distance of a point (x,y) in the plane from the origin (0,0) is given by
                                               p 2    2
                                                 x +y .
                 Recall that the above follows from Pythagoras theorem.
                    Weshall introduce some notations and basic definitions:
                    • N: set of all natural numbers.
                    • R: set of all real numbers.
                       n
                    • R : set of all n-tuple of real numbers, i.e., the set of all (x ,...,x ) with
                                                                         1      n
                     x ∈Rfor i=1,...,n.
                       i
                                          n
                    • For x = (x ,...,x ) ∈ R , we denote by |u| (absolute value or modulus of u),
                              1      n
                                             2    2      2
                     the positive square root of x1 + x2 + ...,x , i.e.,
                                                         k
                                                q 2    2       2
                                           |x| := x +x +...,x .
                                                   1   2       k
                                                                                     1
                      2                              Functions of Several Variables
                                                      2                        2
                      Note that, if u = (u ,u ) ∈ R and x = (x ,x ) ∈ R , then
                                            1  2                    1   2
                                                p            2             2                    2
                                     |x −u| =      (x −u ) +(x −u ) +...+(x −u ) .
                                                     1     1       2     2              k     k
                                                        2                                           2
                          Thus, for u = (u ,u ) ∈ R , the set of all points x = (x ,x ) ∈ R such that
                                             1  2                                        1   2
                                                                                                              2
                      |x − u| = r represents the circle with centre u and radius r, that is, {x ∈ R :
                                                                               2
                      |x − u| = r}. The region inside this circle is {x ∈ R : |x − u| < r} and the region
                                                                              2
                      inside the circle including the boundary is {x ∈ R : |x − u| ≤ r}. Note that the
                      above sets are same as
                                                     2            2              2    2
                                             {x ∈ R : (x −u ) +(x −u ) =r },
                                                           1    1        2     2
                                                     2             2             2     2
                                              {x ∈ R : (x −u ) +(x −u )  0.
                                            0     0  0
                                                             2
                     (1) The set of all points u := (x,y) ∈ R that satisfy |u−u | < r is called the open
                                                                                 0
                     disc with centre u := (x ,y ) and radius r.
                                        0      0  0
                                                                2
                     (2) The set of all points u := (x,y) ∈ R that satisfy |u − u | ≤ r is called the
                                                                                      0
                     closed disc with centre u := (x ,y ) and radius r.                                  ♦
                                                0      0  0
                                                                            2
                     Definition 1.4 Let D be a region, i.e., a subset of R .
                     (1) D is said to be a bounded set if it is contained in a disc (open or closed) for
                     some radius r > 0. Sets which are not bounded are called unbounded sets.
                     (2) A point (x ,y ) is called an interior point of D if it is the centre of an open
                                    0  0
                     disc contained in D.
                     (3) A point (x ,y ) is called a boundary point of D every open disc containing
                                    0  0
                     this point contains some point from D as well as some point not in D.
                     (4) The set of all interior points of D is called the interior of D, and it is denoted
                     by int(D).
                     (5) The set of all boundary points of D is called the boundary of D, denoted by
                     bd(D).
                     (6) The set D is called an open set if its interior is itself.
                     (7) The set D is called a closed set if it contains all its boundary points.        ♦
                                         2
                        Thus, for D ⊆ R ,
                                                                                             2
                        • u ∈bd(D) ⇐⇒ ∀r>0, B(u ,r)∩D6=∅ & B(u ,r)∩R \D6=∅;
                            0                               0                        0
                        • D is open ⇐⇒ ∃r >0,          B(u ,r) ⊆ D,
                                                           0