Tuesday November 13 Solutions Surface Parameterpalooza 1 Let S be the portion of the plane x y z 1 which lies in the positive octant a Draw a picture of S Solution The picture is shown below 1 0 0 0 0 5 0 5 0 0 0 0 0 5 1 0 1 0 b Find a parametrization r D S being sure to clearly indicate the domain D Check your answer with the instructor Solution One can use the parametrization r u v u v 1 u v with the domain D given by D u v 0 u 1 0 v 1 u c Use your answer in b to compute the area of S via an integral over D Solution Using the parametrization in b one gets ru 1 0 1 rv 0 1 1 p so ru rv 1 1 1 and kru rv k 3 Hence the area of S is p Z 1 Z 1 u 3 dS kru rv kd vd u 2 D 0 0 d Check your answer in c using only things you learned in the first few weeks of this class Solution The picture of S is a triangle with vertices A 1 0 0 B 0 1 0 and C 0 0 1 Thus AB 1 1 0 and AC 1 0 1 and the area is p 1 3 k AB AC k 2 2 2 Consider the surface S which is the part of z x 2 y 2 1 where z 0 a Draw a picture of S Solution The picture is shown below b Find a parametrization r D S Check your answer with the instructor Solution One can use the parametrization r r r cos r sin 1 r 2 with the domain 0 r 1 0 2 3 Let S be the surface given by the following parametrization Let D 1 1 0 2 and define r u v u cos v u sin v v a Consider the vertical line segment L u 0 in D Describe geometrically the image of L under r Solution The image of u 0 under r is a line segment 0 0 v where 0 v 2 b Repeat for the vertical segments where u 1 and u 1 Solution When u 1 the image r 1 v cos v sin v v is a helix with 0 v 2 and so is u 1 Thus the images of u 1 and u 1 form the double helix c Use your answers in a and b to make a sketch of S Solution The picture is shown below 4 Consider the ellipsoid E given by x2 y 2 z 2 1 9 4 a Draw a picture of E Solution The picture is shown below b Find a parametrization of E Hint Find a transformation T R3 R3 which takes the unit sphere S to E and combine that with our existing parametrization of the plain sphere S Solution One can use the following parametrization r 3 sin cos 2 sin sin cos with the domain 0 0 2
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