Engineering Calculus Notes 272

Engineering Calculus Notes 272 - First we show that...

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260 CHAPTER 3. REAL-VALUED FUNCTIONS: DIFFERENTIATION Theorem 3.4.2 (Implicit Function Theorem for R 2 R ) . The level set of a continuously diFerentiable function f : R 2 R can be expressed near each of its regular points as the graph of a function. Speci±cally, suppose f ( x 0 ,y 0 ) = c and ∂f ∂y ( x 0 ,y 0 ) n = 0 . Then there exists a rectangle R = [ x 0 δ 1 ,x 0 + δ 1 ] × [ y 0 δ 2 ,y 0 + δ 2 ] centered at −→ x 0 = ( x 0 ,y 0 ) (where δ 1 2 > 0 ), such that the intersection of L ( f,c ) with R is the graph of a C 1 function φ ( x ) , de±ned on [ x 0 δ 1 ,x 0 + δ 1 ] and taking values in [ y 0 δ 2 ,y 0 + δ 2 ] . In other words, if ( x,y ) R , (i.e., | x x 0 | ≤ δ 1 and | y y 0 | ≤ δ 2 ), then f ( x,y ) = c ⇐⇒ φ ( x ) = y. (3.17) ²urthermore, at any point x ( x 0 δ 1 ,x 0 + δ 1 ) , the derivative of φ ( x ) is dx = b ∂f ∂x ( x,φ ( x )) Bsb ∂f ∂y ( x,φ ( x )) B . (3.18) Proof. The proof will be in two parts.
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Unformatted text preview: First we show that Equation ( 3.17 ) determines a well-dened function ( x ): For notational convenience, we assume without loss of generality that f ( x ,y ) = 0 (that is, c = 0), and f y ( x ,y ) > . Since f ( x,y ) is continuous, we know that f y ( x ) > 0 at all points x = ( x,y ) suciently near x , say for | x x | and | y y | 2 . For any a [ x ,x + ], consider the function of y obtained by xing the value of x at x = a : g a ( y ) = f ( a,y ) ;...
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This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.

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