Engineering Calculus Notes 322

# Engineering Calculus Notes 322 - Equation 3.25 using R = 1...

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310 CHAPTER 3. REAL-VALUED FUNCTIONS: DIFFERENTIATION 2. it parametrizes a plane through P = −→ p ( r 0 ,s 0 ) = ( x 0 ,y 0 ,z 0 ) which contains the velocity vector of any curve passing through P in the surface S parametrized by −→ p ; 3. the equation of this plane is −→ N · ( x x 0 ,y y 0 ,z z 0 ) = 0 where −→ N = −→ p ∂r × −→ p ∂s . This plane is the tangent plane to S at P . Let us consider two quick examples. First, we consider the sphere parametrized using spherical coordinates in
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Unformatted text preview: Equation ( 3.25 ); using R = 1 we have −→ p ( θ,φ ) = (sin φ cos θ, sin φ sin θ, cos φ ) (see Figure 3.20 ). b c −→ N x y z Figure 3.20: Tangent Plane to Sphere at p √ 3 2 √ 2 , − √ 3 2 √ 2 , 1 2 P Let us fnd the tangent plane at P ± √ 3 2 √ 2 , − √ 3 2 √ 2 , 1 2 ²...
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