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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 differentiable function f : R 2 R can be expressed near each of its regular points as the graph of a function. Specifically, suppose f ( x 0 ,y 0 ) = c and ∂f ∂y ( x 0 ,y 0 ) negationslash = 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 ) , defined 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) Furthermore, at any point x ( x 0 δ 1 ,x 0 + δ 1 ) , the derivative of φ ( x ) is dx = bracketleftbigg ∂f ∂x ( x,φ ( x )) bracketrightbiggslashbiggbracketleftbigg ∂f ∂y ( x,φ ( x )) bracketrightbigg
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Unformatted text preview: First we show that Equation ( 3.17 ) determines a well-de±ned function φ ( 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 ) su²ciently near −→ x , say for | x − x | ≤ δ and | y − y | ≤ δ 2 . For any a ∈ [ x − δ,x + δ ], 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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