PartialDifferentiation - Partial Differentiation 1-...

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Partial Differentiation 1- Functions of two variables Let D be a set of points (x,y) in the xy-plane Definition : A real-valued function of the variables x and y is a rule or correspondence that assigns a unique real number , denoted by z = f(x,y), to each point (x,y) in D. The set D is called the domain of the function. The number z = f(x,y) is called the value of the function at (x,y). The coordinates x and y of the point (x,y) are called the independent variables, while z is called the dependent variable. Example : Let f(x,y) = x x ! y , then f(1,0) = 1 1 ! 0 = 1, f(2, ! 2) = 2 2 ! ( ! 2) = 2 4 = 1 2 f(1,1) = 1 1 ! 1 is undefined D . (x,y) x y D . (x,y) x y R - z = f(x,y) >
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This function is undefined when x – y = 0 or x = y, this means that the domain of f is ! 2 \ {(x,y):x = y} . The graph of a function z = f(x,y) is a surface in the space R 3 2. Partial Derivatives Let z = f(x,y) a function of the variables x and y. If we hold the variable y fixed, say y = y 0 and view x as the only variable then we have f(x,y
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This note was uploaded on 07/23/2011 for the course MAP 2302 taught by Professor Staff during the Fall '08 term at FIU.

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PartialDifferentiation - Partial Differentiation 1-...

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