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Differentiation of Multivariable Functions

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CHAPTER 3 Differentiation of Multivariable Functions 16. Functions of Several Variables The concept of a function of several variables can be qualitatively un- derstood from simple examples in everyday life. The temperature in a room may vary from point to point. A point in space can be defined by an or- dered triple of numbers that are coordinates of the point in some coordinate system, say, (x,y,z). Measurements of the temperature at every point from a set D in space assign a real number T (the temperature) to every point of D. The dependence of T on coordinates of the point is indicated by writing T = T (x,y,z). Similarly, the concentration of a chemical can depend on a point in space. In addition, if the chemical reacts with other chemicals, its concentration at a point may also change with time. In this case, the concen- tration C = C(x,y,z,t) depends on four variables, three spatial coordinates and the time t. In general, if the value of a quantity f depends on values of several other quantities, say, x1, x2,..., xm, this dependence is indicated by writing f = f(x1,x2,...,xm). In other words, f = f(x1,x2,...,xm) indicates a rule that assigns a unique real number f to each ordered m-tuple of real numbers (x1,x2,...,xm): f : (x ,x ,...,x ) f(x ,x ,...,x ) 1 2 m → 1 2 m Each number in the m-tuple may be of a different nature and measured in different units. In the above example, the concentration depends on ordered quadruples (x,y,z,t), where x, y, and z are the coordinates of a point in space (measured in units of length) and t is time (measured in units of time). To analyze properties of functions of several variables, a notion of a distance between two ordered m tuples is needed. For example, a rate of change of a function is naturally− defined as the difference of values of the function at two points divided by the distance between them. This allows us to determine that one function changes more rapidly than the other. In what follows, functions on Euclidean spaces will be studied. In other words, the distance between two ordered m tuples (or two arguments of a function) is assumed to be the Euclidean distance.− To simplify notations, the argument of a function of several variables will often be written in the vector form f(x ,x ,...,x ) = f(r) , r Rm . 1 2 m ∈ The value of a function at a particular point P of a Euclidean space will also be denoted by f(P ) to emphasize that this value is independent of the choice of a coordinate system in which coordinates of P are given. For 241 242 3. DIFFERENTIATION OF MULTIVARIABLE FUNCTIONS f D D f y P P f(P ) f(P ) x R f R f Figure 16.1. Left: A function f of two variables is a rule, f : P f(P ), that assigns a number f(P ) to every point P of a→ planar region D. The set R of all numbers f(P ) is the range of f. The region D is the domain of f. Right: A function f of three variables is a rule that assigns a number f(P ) to every point P of a solid region D. example, a temperature in a room T = T (P ) depends on a point P , but the coordinates of P (the ordered triples of numbers) are different in different coordinate systems. 16.1. Real-Valued Functions of Several Variables. Definition 16.1. (Real-Valued Function of Several Variables). Let D be a set of ordered m-tuples of real numbers (x1,x2,...,xm). A func- tion f of m variables is a rule that assigns to each m tuple in the set D − a unique real number denoted by f(x1,x2,...,xm). The set D is the do- main of f, and its range is the set of values that f takes on it, that is, f(x ,x ,...,x ) (x ,x ,...,x ) D . { 1 2 m | 1 2 m ∈ } This definition is illustrated in Fig. 16.1. The rule may be defined by different means. For example, if D is a collection of N points Pi, i = 1, 2,...,N, then a function f can be defined by a table (Pi, f(Pi)), where f(Pi) is the value of f at Pi. A function f can be defined geometrically. For example, the height of a mountain relative to sea level is a function of its position on the globe. So the height is a function of two variables, the longitude and latitude. A function can be defined by an algebraic rule that prescribes algebraic operations to be carried out with real numbers in any n-tuple to obtain the value of the function. For example, f(x,y,z) = x2 y + z3 . − The value of this function at (1, 2, 3) is f(1, 2, 3)= 12 2+33 = 26 . − Unless specified otherwise, the domain of a function defined by an algebraic rule is the set of m-tuples for which the rule makes sense. Example 16.1. Find the domain and the range of the function of two variables f(x, y) = ln(1 x2 y2). − − 16. FUNCTIONS OF SEVERAL VARIABLES 243 Solution: The logarithmis defined for any strictly positive number. There- fore, the doublets (x, y) must be such that 1 x2 y2 > 0 or x2 + y2 < 1. Hence, − − D = (x, y) x2 + y2 < 1 . { | } Since any doublet (x, y) can be uniquely associated with a point on a plane, the set D can be given a geometrical description as a disk of radius 1 whose boundary, the circle x2 + y2 = 1, is not included in D. For any point in the interior of the disk, the argument of the logarithm lies in the interval 0 < 1 x2 y2 1. So the range of f is the set of values of the logarithm in the− interval− (0≤, 1], which is < f 0. −∞ ≤ Example 16.2. Find the domain and the range of the function of three variables f(x,y,z) = x2 z x2 y2. − − Solution: The square rootp is defined only for nonnegative numbers. There- fore, ordered triples (x,y,z) must be such that z x2 y2 0, that is, − − ≥ D = (x,y,z) z x2 + y2 . { | ≥ } This set can be given a geometrical description as a point set in space because any triple can be associated with a unique point in space. The equation z = x2 + y2 describes a circular paraboloid. So the domain is the spatial (solid) region containing points that lie on or above the paraboloid. The function is non- negative. By fixing x and y and increasing z, one can see that the value of f can be any positive number. So the range is 0 f(x,y,z) < . The domain of a function of m variables is≤ viewed as a subset∞ of an m-dimensional Euclidean space. For example, the domain of the function f(r)=(1 x2 x2 x2 )1/2 = (1 r 2)1/2 , − 1 − 2 −···− m −k k where r = x1,x2,...,xm , is the set of points inthe m-dimensional Euclidean space whoseh distance fromi the origin (the zero vector) does not exceed 1, D = r Rm r 1 ; { ∈ | k k ≤ } that is, it is a closed m-dimensional ball of radius 1. So the domain of a multivariable function defined by an algebraic rule can be described by conditions on the components of the ordered m-tuple r under which the rule makes sense. 16.2. The Graph of a Function of Two Variables. The graph of a function of one variable f(x) is the set of points (x, y) of a plane such that y = f(x) , x D ∈ where the domain D is a collection of points on the x axis. The graph is obtained by moving a point of the domain parallel to the y axis by an amount determined by the value of the function y = f(x). The graph provides a 244 3. DIFFERENTIATION OF MULTIVARIABLE FUNCTIONS z (x,y,z) z z = f(x, y) 1 y D y 2 3 (x,y, 0) D x x Figure 16.2. Left: The graph of a function of two vari- ables is the surface defined by the equation z = f(x, y). It is obtained from the domain D of f by moving each point (x,y, 0) in D along the z axis to the point (x,y,f(x, y)). Right: The graph of the function studied in Example 16.3. useful picture of the behavior of the function. The idea can be extended to functions of two variables. Definition 16.2. (Graph of a Function of Two Variables). The graph of a function f(x, y) with domain D is a collection of points (x,y,z) in space such that z = f(x, y) , (x, y) D. ∈ The domain D is a set of points in the xy plane. The graph is then obtained by moving each point of D parallel to the z axis by an amount equal to the corresponding value of the function z = f(x, y). If D is a portion of the plane, then the graph of f is generally a surface (see Fig. 16.2 (left panel)). One can think of the graph as “mountains” of height f(x, y) on the xy plane.
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