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100note3 - Math100 Partial Derivatives 3.1 Functions of Two...

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Unformatted text preview: Math100, Partial Derivatives 3.1 Functions of Two or More Variables Definition 3.1 f is a function in n variables if f is a function in which a vector in the n-dimensional space is plugged in and a number is returned. Definition 3.2 Let f be a function in n variables. The domain of f is the collection of all points x in the n-dimensional space such that f ( x ) is defined. Example 3.3 f ( x, y ) = p x 2 + y 2- 1 is a function in two variables whose domain is the exterior of the unit circle. Example 3.4 g ( x, y ) = y/x is a function in two variables whose domain is the plane excluding the y- axis. Definition 3.5 Let f be a function in n variables. The graph of f is the collection of points ( x 1 , ..., x n +1 ) in the ( n + 1)-dimensional space satisfying x n +1 = f ( x 1 , x 2 , ..., x n ) . Example 3.6 The graph of the function f ( x, y ) = p x 2 + y 2 is a cone in the three-dimensional space. Definition 3.7 Let f be a function in two variables. A level curve of f is a collection of points ( x, y ) in the plane satisfying f ( x, y ) = k for a fixed number k . Example 3.8 If f ( x, y ) = p x 2 + y 2- 1 , the level curves of f are the circles with the origin as their common centers and radius at least 1 . Example 3.9 If g ( x, y ) = y/x , the level curves of g are straight lines through the origin with the vertical line excluded. Remark 3.10 If x does not belong to the domain of f , then, for all numbers a , the point ( x, a ) does not belong to the graph of f . Definition 3.11 Let f be a function in n variables. x is an n-dimensional vector and L is a number. We say that f ( x ) approaches L as x approaches x if | f ( x )- L | is as small as one wish provided that || x- x || is sufficiently small. Symbollically, we write lim x → x f ( x ) = L. 1 Example 3.12 Show that lim ( x,y ) → (0 , 0) xy x 2 + y 2 does not exist. proof: Let f ( x, y ) = xy/ ( x 2 + y 2 ). Then, for each t 6 = 0, f ( t, 0) = 0 f ( t, t ) = 1 2 so that lim t → f ( t, 0) = 0 and lim t → f ( t, t ) = 1 2 . In other words, f ( x, y ) approaches 0 as ( x, y ) “approaches the origin along the x-axis”, f ( x, y ) approaches 1 2 as ( x, y ) “approaches the origin along the straight line with slope 1”. Therefore, the limit of f ( x, y ) as ( x, y ) approaches the origin does not exist. 3.2 Partial Derivatives Definition 3.13 Let f be a function in two variables. The derivative of f with respect to x (or the first variable) at the point ( x , y ) is the derivative of f ( x, y ) at x . Its symbol is either f x ( x , y ) or ∂f ∂x ( x , y ) . Similarly, the derivative of f with respect to y (or the second variable) at the point ( x , y ) is the derivative of f ( x , y ) at y . Its symbol is either f y ( x , y ) or ∂f ∂y ( x , y ) ....
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This note was uploaded on 12/25/2011 for the course MATH 101 taught by Professor Ching during the Spring '11 term at HKU.

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100note3 - Math100 Partial Derivatives 3.1 Functions of Two...

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