14.1 - 14.1 FUNCTIONS OF SEVERAL VARIABLES KIAM HEONG KWA 1...

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14.1: FUNCTIONS OF SEVERAL VARIABLES KIAM HEONG KWA 1. Functions of Two Variables A function f of two variables is a rule that assigns to each ordered pair of real numbers ( x,y ) in a set D R 2 a unique real number denoted by f ( x,y ). The domain of f is the set D . If f is given by a formula and no domain is specified, then D is understood to be the set of all pairs ( x,y ) for which the given formula is well-defined. The range of f is the set of values that f takes on, that is { f ( x,y ) R | ( x,y ) D } ; The graph of f is the set of all points ( x,y,z ) in R 3 such that z = f ( x,y ), ( x,y ) D ; A level curve , or contour curve, of f is the set of all points ( x,y ) in the domain of f such that f ( x,y ) = k for a given value k . It shows where the graph of f has height k . The collection of all level curves of f forms the contour map of f . When one writes z = f ( x,y ), it is customary to call x and y as the independent variables and z as the dependent variables . Remark 1. A revision on cylinders and quadric surfaces is useful. See section 12.6 of the text. Example 1 (Problem 6 in the text) . Let f ( x,y ) = ln( x + y - 1) . Find and sketch the domain of f . Find also the range of f . Part of solution. The domain is { ( x,y ) R 2 | y > 1 - x } . To find the range, first note that the range of f must be contained in R . Conversely, let z be an arbitrary real number. Then
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