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Functions of Several Variables A function of two variables is a rule that assigns to each ordered pair of real numbers (x, y) in a subset D of the plane a unique real number denoted by f (x, y). The set D is the domain of f and its range is the set of values that f takes on, that is, {f (x, y) : (x, y) ∈ D}. We often write z = f (x, y) to make explicit the values taken on by f at the general point (x, y). The variables x and y are independent variables and z is the dependent variable. Functions of more than two variables can be defined similarly. 1 Example. Let √ f (x, y) = x+y+1 . x−1 (1) Evaluate f (3, 2), f (2, −5), f (1, −1), and f (−1, 4). (2) Find the domain of f . √ Solution. (1) f (3, 2) = 6/2; f (2, −5) is not defined; f (1, −1) is not defined; f (−1, 4) = −1. (2) The domain of f is D = {(x, y) : x + y + 1 ≥ 0, x 6= 1}. 2 Limits of Functions Let f be a function of two variables whose domain D includes points arbitrarily close to (a, b). Then we say that the limit of f (x, y) as (x, y) approaches (a, b) is L and we write lim f (x, y) = L (x,y)→(a,b) if for every number ε > 0 there is a corresponding number δ > 0 such that |f (x, y) − L| < ε whenever (x, y) ∈ D and 0 < p 3 (x − a)2 + (y − b)2 < δ. Continuous Functions A function f of two variables is called continuous at (a, b) if lim(x,y)→(a,b) f (x, y) = f (a, b). We say f is continuous on D if f is continuous at every point (a, b) in D. Let f and g be two functions defined on D, and let (a, b) be a point in D. If f and g are continuous at (a, b), then both f + g and f g are continuous at (a, b). Moreover, f /g is continuous at (a, b), provided g(a, b) 6= 0. A polynomial of two variables is a sum of terms of the form cxm y n , where c is a real number and m and n are nonnegative integers. Any polynomial is continuous on the whole plane IR2 . 4 Graphs If f is a function of two variables with domain D, then the graph of f is the set of all points (x, y, z) in IR3 such that z = f (x, y) and (x, y) is in D. Example. The graph of the function z = x2 + y 2 is called a paraboloid. z y O x 5 Example. Let us consider the function p z = 9 − x2 − y 2 . Its domain is the disk with radius 3 and center at the origin: {(x, y) : x2 + y 2 ≤ 9}. Since z ≥ 0 and x2 + y 2 + z 2 = 32 , its graph is an upper hemisphere. z y O x 6