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

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Unformatted text preview: 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 ....
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This note was uploaded on 11/11/2011 for the course MATH 254.01 taught by Professor Kwa during the Fall '10 term at Ohio State.

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

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