Tuesday November 6 Changing coordinates 1 Consider the region R in R2 shown below at right In this problem you will do a change of coordinates to evaluate x 2y dA R y v 3 4 T 1 3 R 1 1 S 2 1 x u a Find a linear transformation T R2 R2 which takes the unit square S to R Write you answer both as a matrix ac db and as T u v au bv cu d v and check your answer with the instructor b Compute R x 2y dA by relating it to an integral over S and evaluating that Check your answer with the instructor 2 Another simple type of transformation T R2 R2 is a translation which has the general form T u v u a v b for a fixed a and b a If T is a translation what is its Jacobian matrix How does it distort area b Consider the region S u 2 v 2 1 in R2 with coordinates u v and the region R x 2 2 y 1 2 1 in R2 with coordinates x y Make separate sketches of S and R c Find a translation T where T S R d Use T to reduce x dA R to an integral over S and then evaluate that new integral using polar coordinates e Check your answer in d with the instructor Problems 3 and 4 on the back 3 Consider the region R shown below Here the curved left side is given by x y y 2 In this problem you will find a transformation T R2 R2 which takes the unit square S 0 1 0 1 to R y 2 1 0 1 R x 2 0 a As a warm up find a transformation that takes S to the rectangle 0 2 0 1 which contains R b Returning to the problem of finding T taking S to R come up with formulas for T u 0 T u 1 T 0 v and T 1 v Hint For three of these use your answer in part a c Now extend your answer in b to the needed transformation T Hint Try filling in between T 0 v and T 1 v with a straight line d Compute the area of R in two ways once using T to change coordinates and once directly 4 If you get this far evaluate the integrals in Problems 1 and 2 directly without doing a change of coordinates It s a fun filled task
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