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Engineering Calculus Notes 302

# Engineering Calculus Notes 302 - Theorem for real-valued...

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290 CHAPTER 3. REAL-VALUED FUNCTIONS: DIFFERENTIATION derivative of the restriction (which is nothing other than a partial of φ ( x,y )) is the appropriate ratio of partials of f , as given in Equation ( 3.24 ). But then, rather than trying to prove directly that φ is differentiable as a function of two variables, we can appeal to Theorem 3.3.4 to conclude that, since its partials are continuous, the function is differentiable. This concludes the proof of the Implicit Function
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Unformatted text preview: Theorem for real-valued functions of three variables. As an example, consider the level surface (±igure 3.15 ) L ( f, 1), where x y z b (1 , − 1 , 2) b (0 , 1 , 0) ±igure 3.15: The Surface L ( 4 x 2 + y 2 − z 2 , 1 ) f ( x,y,z ) = 4 x 2 + y 2 − z 2 : The partial derivatives of f ( x,y,z ) are ∂f ∂x ( x,y,z ) = 8 x ∂f ∂y ( x,y,z ) = 2 y ∂f ∂z ( x,y,z ) = − 2 z ;...
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