lecture08

lecture08 - 2006 Fall MATH 100 Lecture 8 1 ATH 100 Lecture...

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Unformatted text preview: 2006 Fall MATH 100 Lecture 8 1 ATH 100 Lecture 8 Tangent plane ( ) plane tangent a it call then we plane, common a on lying through pass on lies curves any of lines tangent if , on point a is , , If : plane tngent of Definition P S S z y x P ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) , , by given at plane tangent a has surface the , at able differenti is , then , , on point a is , , Let : Theorem = − − − + − ⇔ = z z y y y x f x x y x f P y x y x f y x f z z y x P y x 2006 Fall MATH 100 Lecture 8 2 ATH 100 Lecture 8 Tangent plane ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 , , , , and on lying curve any be , , : Assume 1 , , , , curve any of ector tangent v that the show want to We : Proof s z s y s x z y x S s z z s y y s x x C y x f y x f n y x = = = = − = ⊥ ( ) ( ) ( ) ( ) , , 1 , , i.e. or and , , So = ⋅ − ∂ ∂ ∂ ∂ = − ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ = = ds dz ds dy ds dx y f x f ds dz ds dy y f ds dx x f ds dy y f ds dx x f ds dz s y s x f s z 2006 Fall MATH 100 Lecture 8 3 ATH 100 Lecture 8 Tangent plane ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 , , , , by given is , at plane tangent the of vector normal the and along , , at ector tangent v the is , , − = y x f y x f n y x C z y x ds s dz ds s dy ds s dx y x ( ) ( ) ( ) Equation of normal line ( ) , ( ) , ( ) 1 x y x x f x y y y f x y z z θ θ θ θ θ θ = + = + = + − 2006 Fall MATH 100 Lecture 8...
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This note was uploaded on 09/17/2011 for the course MATH 100 taught by Professor Qt during the Fall '09 term at HKUST.

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lecture08 - 2006 Fall MATH 100 Lecture 8 1 ATH 100 Lecture...

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