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Engineering Calculus Notes 321

# Engineering Calculus Notes 321 - x,y and −→ p is the...

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3.5. SURFACES AND THEIR TANGENT PLANES 309 The parametrization T ( r 0 ,s 0 ) −→ p assigns to each vector −→ v R 2 a vector T ( r 0 ,s 0 ) −→ p ( −→ v ) in the tangent plane at ( r 0 ,s 0 ): namely if γ ( τ ) is a curve in the ( s,t )-plane going through ( r 0 ,s 0 ) with velocity −→ v then the corresponding curve −→ p ( γ ( τ )) in S goes through −→ p ( r 0 ,s 0 ) with velocity T ( r 0 ,s 0 ) −→ p ( −→ v ). T ( r 0 ,s 0 ) −→ p is sometimes called the tangent map at ( r 0 ,s 0 ) of the parametrization −→ p . We can also use the two partial derivative vectors −→ p ∂r and −→ p ∂s to find an equation for the tangent plane to S at P . Since they are direction vectors for the plane, their cross product gives a normal to the plane: −→ N = −→ p ∂r × −→ p ∂s and then the equation of the tangent plane is given by −→ N · [( x,y,z ) −→ p ( r 0 ,s 0 )] = 0 . You should check that in the special case when S is the graph of a function f
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Unformatted text preview: ( x,y ), and −→ p is the parametrization of S as −→ p ( x,y ) = ( x,y,f ( x,y )) then ∂ −→ p ∂x = −→ ı + ∂f ∂x −→ k ∂ −→ p ∂y = −→ + ∂f ∂y −→ k −→ N = − ∂f ∂x −→ ı − ∂f ∂y −→ + −→ k yielding the usual equation for the tangent plane. We summarize these observations in the following Remark 3.5.9. If −→ p : R 2 → R 3 is regular at ( r ,s ) , then 1. The linearization of −→ p ( r,s ) at r = r , s = s T ( r ,s ) −→ p ( r + △ r,s + △ s ) = −→ p ( r ,s ) + △ r ∂ −→ p ∂r + △ s ∂ −→ p ∂s has Frst-order contact with −→ p ( r,s ) at r = r , s = s ;...
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