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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
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