Tuesday, November 6 ∗∗ Changing coordinates1. Consider the region R in R2shown below at right. In this problem, you will do a change ofcoordinates to evaluate:ÏRx − 2y dASuvTRxy(1,1)(2,1)(3,4)(1,3)(a) Find a linear transformation T : R2→ R2which takes the unit square S to R.Write you answer both as a matrix¡a bc d¢and as T (u,v) = (au+bv, cu+d v), and check youranswer with the instructor.(b) ComputeÎRx − 2y dA by relating it to an integral over S and evaluating that. Check youranswer with the instructor.2. Another simple type of transformation T : R2→ R2is a translation, which has the general formT (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 =©u2+ v2≤ 1ªin R2with coordinates (u, v), and the region R =©(x − 2)2+ (y − 1)2≤ 1ªin R2with coordinates (x, y).Make separate sketches of S and R.(c) Find a translation T where T (S) = R.(d) Use T to reduceÏRx dAto 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 − y2. In thisproblem, you will find a transformation T : R2→ R2which takes the unit square S = [0, 1]× [0, 1]to R.(2,1)Rxy(2,0)(0,1)(a) As a warm up, find a transformation that takes S to the rectangle [0, 2] × [0, 1] which con-tains 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” be-tween 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 ofcoordinates. It’s a fun-filled task. .
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