Ch14Notes - MATH 214: PARTIAL DERIVATIVES 1. Functions of...

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Unformatted text preview: MATH 214: PARTIAL DERIVATIVES 1. Functions of Several Variables Definition. Suppose D is a set of n-tuples ( x 1 ,x 2 ,...,x n ) (usually n = 2 or 3 for us), where x i ∈ R . A real-valued function on D is a rule f that assigns a number w = f ( x 1 ,...,x n ) to every ( x 1 ,...,x n ) ∈ D . The set D is called the domain of f and the set { f ( x 1 ,...,x n ) | ( x 1 ,...,x n ) ∈ D } ⊂ R is called the range of f . The variable w is called the dependent variable of f and the variables x i are the independent variables of f . Example 1. The domain of f ( x,y,z ) = z √ xy is { ( x,y,z ) | xy 6 = 0 } = { ( x,y,z ) | x 6 = 0 } ∩ { ( x,y,z ) | y 6 = 0 } and the range is R . Example 2. The domain of g ( x,y ) = arctan x y is { ( x,y ) | - π 2 < x < π 2 ,y 6 = 0 } Definition. (1) Given a function f ( x,y ) , the set { ( x,y,z ) | z = f ( x,y ) } ⊂ R 3 is called the graph of f . For the functions f that we will consider, this will usually be a surface. (2) The set { ( x,y ) | f ( x,y ) = c } ⊂ R 2 is called the level curve of f at c . For the functions that we will consider, this will usually be a curve in the xy-plane. Example 3. The level curves of f ( x,y ) = x 2 + y 2 at c are circles of radius √ c if c > and are empty (i.e. they have no points) if c < . Example 4. The level curves of the function f ( x,y ) = xy at c are hyperbolas if c 6 = 0 and the level curve of f ( x,y ) = xy at c = 0 is the set { ( x,y ) | xy = 0 } = { ( x,y ) | x = 0 } ∪ { ( x,y ) | y = 0 } , i.e. the union of the x-axis and the y-axis. 2. Limits Definition. We say that f ( x,y ) approaches the limit L ∈ R as ( x,y ) approaches ( x ,y ) if for every > , there exists a δ > such that for every ( x,y ) with < p ( x- x ) 2 + ( y- y ) 2 < δ we have | f ( x,y )- L | < . In this case we write: lim ( x,y ) → ( x ,y ) f ( x,y ) = L Example 5. lim ( x,y ) → ( x ,y ) x = x , lim ( x,y ) → ( x ,y ) y = y , lim ( x,y ) → ( x ,y ) k = k for any k ∈ R . Rather than calculate limits directly, we’ll usually refer to the following theorem: Theorem 2.1. If lim ( x,y ) → ( x ,y ) f ( x,y ) = L and lim ( x,y ) → ( x ,y ) g ( x,y ) = M , then: (1) lim ( x,y ) → ( x ,y ) c 1 f ( x,y ) + c 2 g ( x,y ) = c 1 L + c 2 M (i.e. limits are linear ) 1 2 MATH 214: PARTIAL DERIVATIVES (2) lim ( x,y ) → ( x ,y ) f ( x,y ) g ( x,y ) = LM (this is sometimes called the product rule ) (3) lim ( x,y ) → ( x ,y ) f ( x,y ) g ( x,y ) = L M if M 6 = 0 (the quotient rule ) (4) If the greatest common denominator of r and s is 1 and s 6 = 0 , then lim ( x,y ) → ( x ,y ) f ( x,y ) r s = L r s (called the power rule ). For the power rule, we assume that L > if s is even (i.e. we don’t take even roots of negative numbers)....
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This note was uploaded on 07/31/2011 for the course MATH 214 taught by Professor Alexondrus during the Spring '11 term at University of Alberta.

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Ch14Notes - MATH 214: PARTIAL DERIVATIVES 1. Functions of...

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