Engineering Calculus Notes 306

Engineering Calculus Notes 306 - First, let us nd the plane...

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294 CHAPTER 3. REAL-VALUED FUNCTIONS: DIFFERENTIATION Proposition 3.5.5. Suppose P ( x 0 ,y 0 ,z 0 ) is a regular point of the real-valued function f ( x,y,z ) and f ( x 0 ,y 0 ,z 0 ) = c . Then the level set of f through P L ( f,c ) := { ( x,y,z ) | f ( x,y,z ) = c } has a tangent plane P at P , which can be characterized in any of the following ways: • P is the plane through P with normal vector −→ f ( P ) ; • P is the set of all points P + −→ v where d P f ( −→ v ) = 0; • P is the level set L ( T P f,f ( P )) through P of the linearization of f at P . Let us see how this works out in practice for a few examples.
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Unformatted text preview: First, let us nd the plane tangent to the ellipsoid x 2 + 3 y 2 + 4 z 2 = 20 at the point P (2 , 2 , 1) (Figure 3.16 ). b (2 , 2 , 1) x y z Figure 3.16: The surface x 2 +3 y 2 +4 z 2 = 20 with tangent plane at (2 , 2 , 1) This can be regarded as the level set L ( f, 20) of the function f ( x,y,z ) = x 2 + 3 y 2 + 4 z 2 ....
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