Engineering Calculus Notes 298

Engineering Calculus Notes 298 - R 3 For a real-valued...

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286 CHAPTER 3. REAL-VALUED FUNCTIONS: DIFFERENTIATION These represent the direction vectors of the lines tangent at P ( x 0 ,y 0 ,z 0 ) to the intersection of the planes y = y 0 and x = x 0 , respectively, with our graph. A parametrization of the tangent plane is x = x 0 + s y = y 0 + t z = z 0 + ∂f ∂x ( x 0 ,y 0 ) s + ∂f ∂y ( x 0 ,y 0 ) t and the two lines are parametrized by setting t (resp. s ) equal to zero. A normal vector to the tangent plane is given by the cross product −→ n = −→ v 1 × −→ v 2 = ∂f ∂x ( x 0 ,y 0 ) −→ ı ∂f ∂y ( x 0 ,y 0 ) −→ + −→ k . The adventurous reader is invited to think about how this extends to graphs of functions of more than two variables. Level Surfaces: The Implicit Function Theorem in
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Unformatted text preview: R 3 For a real-valued function f ( x,y,z ) of three variables, the level set L ( f,c ) is dened by an equation in three variables, and we expect it to be a surface . For example, the level sets L ( f,c ) of the function f ( x,y,z ) = x 2 + y 2 + z 2 are spheres (of radius c ) centered at the origin if c > 0; again for c = 0 we get a single point and for c < 0 the empty set: the origin is the one place where f ( x,y,z ) = 2 x + 2 y + 2 z k...
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