homework2 - Oliver Knill Harvard Summer school Summer 2010...

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Oliver Knill, Harvard Summer school, Summer 2010 Homework for Chapter 2. Curves and Surfaces Section 2.1: Functions, level surfaces, quadrics 1) (functions of two variables) Let f ( x, y ) = ( x 2 1) / ( y 2 + 1). Draw the graph. The level surface S : g ( x, y, z ) = z f ( x, y ) = 0 of course is the graph of f . Find the equations for the three traces of the surface S . Sketch the contour map. 2) (level surfaces) Consider the surface z 2 4 z + x 2 2 x y = 0. Draw the three traces. What surface is it? 3) (level surfaces) Draw the Fermat surface x 4 + y 4 = z 4 and its traces. 4) a) Sketch the graph and contour map of the function f ( x, y ) = cos( x 2 + y 2 ) / (1 + x 2 + y 2 ). b) Sketch the graph and contour map of the function g ( x, y ) = | x | − | y | . 5) (level surfaces) Verify that the line vector r ( t ) = ( 1 , 3 , 2 ) + t ( 1 , 2 , 1 ) is contained in the surface z 2 x 2 y = 0. Section 2.2: Parametric surfaces 1) (parametrized surfaces) Plot the surface with the parametrization vector r ( u, v ) = ( v 3 cos( u ) , v 3 sin( u ) , v ) , where u [0 , 2 π ] and v R . 2) (parametrized surfaces) Find a parametrization for the plane which contains the three points P = (3 , 7 , 1), Q = (1 , 2 , 1) and R = (0 , 3 , 4).
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