hw3-difftop - Math 343 Homework 3 Due Wednesday February...

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Unformatted text preview: Math 343 Homework 3 Due Wednesday February 17, 2010. 1 To be handed in for grading 1. Is the cuspidal cubic { ( x,y ) R 2 | y 2 = x 3 } a submanifold of R 2 ? Why or why not? Hint: Review the definition of a submanifold from Lecture 1. 2. Take a smooth real valued function f : R 2 R , the graph of f graph( f ) := { ( x,y,z ) R 3 | z = f ( x,y ) } . The tangent plane to this surface at the point ( a,b,f ( a,b )) is given by the equation z = f ( a,b ) + f x ( a,b )( x- a ) + f y ( a,b )( y- b ) . The aim of this question is for you to recover this equation (first seen in multivariable calculus) using the more general setting of manifolds. (a) Let M = graph( f ). Carefully explain why M is a 2-dimensional manifold. (b) Given ( a,b,c ) M then T ( a,b,c ) M is defined to be T ( a,b,c ) M = im d ( a,b ) , where is a parametrization in a neighborhood of the point ( a,b ). From (a) explicitly compute d ( a,b ) and find a basis for im d ( a,b ) . (c) The best linear approximation of M at ( a,b,c ) is given by ( a,b,c )+ T ( a,b,c ) M . In our case, this can be rewritten as the equation of the plane at ( a,b,c ) parallel to T ( a,b,c ) M . Find the equation of this plane and show that it matches the equation given above. Hint 1: The equation of a plane at a point ( a,b,c ) R 3 perpendicular to the vector ~v = v 1 v 2 v 3 is given by v 1 ( x- a ) + v 2 ( y- b ) + v 3 ( z- c ) = 0. Hint 2: Review the cross product of two vectors and its geometric interpretation. 3. Find the equation of the tangent plane to x 2 + y 2- z 2 =- 1 at the point (2 , 2 ,- 3)....
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hw3-difftop - Math 343 Homework 3 Due Wednesday February...

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