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

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Oliver Knill, Harvard Summer school, Summer 2009 Homework for Chapter 2. Curves and Surfaces Section 2.1: Functions, level surfaces, quadrics 1) (functions of two variables) Let f ( x, y ) = y 2 sin( x ). Find the equations for the three traces of the surface g ( x, y, z ) = z f ( x, y ) = 0, the graph z = f ( x, y ) of f . Sketch the surface. Solution: The contour map consists of circular regions for c ( 1 , 0) or y = radicalbig c + sin( x ) for c > 0. For c = 0, we have the xy -trace, which consists of y = ± radicalbig | sin( x ) | . The yz -trace: z = y 2 is a ”parabola”. The xz -trace: z = sin( x ) is the graph of the sin function. Minus 2 Minus 1 0 1 2 Minus 5 0 5 2) (level surfaces) Consider the surface z 2 4 z + x 2 2 x y = 0. Draw the three traces and What surface is it? Solution: Completion of the sphere gives ( z 2) 2 + ( x 1) 2 y = 5. The surface is a paraboloid rotational symmetric parallel to the y axis. To see this, it is helpful to draw the generalized traces obtained by intersecting with y = c which gives circles. Especially the intersection with the xy-plane is a circle. The other two traces are parabola. 3) (level surfaces) Surfaces satisfying the implicit equation x k + y k = z k with integer k are called Fermat surfaces. a) Sketch the Fermat surface for k = 2 with traces. b) Sketch the Fermat surface for k = 4 with traces. Remark: You have found integer points ( x, y, z ) lying on the Fermat surface x 2 + y 2 = z 2 in a previous homework. It was Fermat, who conjectured first, that there are no nontrivial lattice points on the Fermat surfaces for k > 2. This claim is now a theorem. Solution: a) The surface x 2 + y 2 = z 2 is a cone. b) The surface x 4 + y 4 = z 4 has on each height z a trace which has the shape when you deform a circle to a square. 1 4) a) Sketch the graph of the function f ( x, y ) = cos( x 2 + y 2 ) / (1 + x 2 + y 2 ). b Sketch the graph of the function g ( x, y ) = | x | − | y | . c) Sketch some contour curves f ( x, y ) = c of f . d) Sketch some contour curves g ( x, y ) = c of g . Solution: a) The graph is rotational symmetric because the function depends only on x 2 + y 2 = r 2 . It is a ripple in the pond. b) In each quadrant, the graph is linear and a plane. These four pieces come together above the coordinate axis. c) These are ellipses. d) The level curves are lines which break on the coordinate axes.
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