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15.1 Partial Derivatives
Functions of Several Variables Deﬁnition 15.1.1 A function of 2 variables is a rule that assigns a unique real number f (x, y ) to each ordered pair (x, y ) in the domain D. The domain of f is the set of ordered pairs (x, y ) for which f is deﬁned. The range of f is the set {f (x, y )  (x, y ) ∈ Dom(f )}. The graph of f is the set of points {(x, y, f (x, y ))  (x, y ) ∈ Dom(f )}. NOTE: In a realvalued function of 2 variables x and y are independent variables, and z = f (x, y ) is the dependent variable. Example 15.1.1 Deﬁne f (x, y ) = 2 y − x2 xy − 5 . Find the domain as a set and sketch it. Example 15.1.2 Let f (x, y ) = 1 − x2 − y 2 . Find the domain as a set and sketch it. Find the range of f . What does the graph of z = f (x, y ) look like? Deﬁnition 15.1.2 The level curves of a function f of 2 variables are the (plane) curves with equations k = f (x, y ), where k is a constant in the range of f . NOTE: Level curves are drawn in the xy plane; traces are drawn in the plane z = k.) Example 15.1.3 Let f (x, y ) = y 2 − x2 . Draw the level curves for k = −4, 0, 9. Example 15.1.4 Let f (x, y ) = 4x2 + y 2 . Draw the level curves for k = 0, 4, 9, 16. These ideas can be extended to functions of 3 or more variables. Deﬁnition 15.1.3 The level surfaces of a function f of 3 variables are the surfaces with equations k = f (x, y, z ), where k is a constant in the range of f . Example 15.1.5 Let f (x, y, z ) = z − x2 + y 2 . Draw level surfaces for k = −1, 0, 1, 2. Example 15.1.6 Let f (x, y, z ) = x2 − y 2 + z 2 . Describe the level surfaces for this function. 17 ...
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This note was uploaded on 10/11/2010 for the course MTHSC 206 taught by Professor Chung during the Fall '07 term at Clemson.
 Fall '07
 Chung
 Derivative, Multivariable Calculus

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