6.38) my professor says I have to explain the steps in the solved problems in the picture. Not just copy eveything down from the text. Transformations, curvilinear coordinatesA region R in the xy plane is bounded by x+ y = 6, x – y = 2, and y = 0. (a) Determine the region R' in thea(x, y)6.38.uv plane into which R is mapped under the transformation x = u + v, y = u – v. (b) ComputeCompare the result of (b) with the ratio of the areas of R and R'.(c)a(u,v)(a) The region R shown shaded in Figure 6.9 (a) is a triangle bounded by the lines x + y = 6, x – y = 2, and y= 0, which for distinguishing purposes are shown dotted, dashed, and heavy, respectively.R'x+y = 6y = 0(b) uv plane(a) xy planeFigure 6 9 Figure 6.9Under the given transformation, the line x + y = 6 is transformed into (u + v) + (u – v) = 6; i.e., 2u = 6 oru = 3, which is a line (shown dotted) in the uv plane of Figure 6.9(b).Similarly, x- y = 2 becomes (u + v) – (u – v) = 2 or v = 1, which is a line (shown dashed) in the uv plane.In like manner, y =0 becomes u – v=0 or u=v, which is a line shown heavy in the uv plane. Then the requiredregion is bounded by u = 3, v = 1, and u = v, and is shown shaded in Figure 6.9(b).Ox dxa(x, y) du dvdu(u +v)ди(b)a(u,v):2|ду ду-(и — 0)-(и — v)|du dul|du(c) The area of triangular region R is 4, whereas the area of triangular region R’is 2. Hence, the ratio is 4/2= 2, agreeing with the value of the Jacobian in (b). Since the Jacobian is constant in this case, the areas ofany regions R in the xy plane are twice the areas of corresponding mapped regions R' in the uv plane.

Answered: Image /qna-images/answer/3d635de1-e7df-4bbe-91e8-f651c10b20e4.jpg

A region R in the xy plane is bounded by x+y = 6, x- y = 2, and y = 0. (a) Determine the region R' in the a(x, y) a(u,v) 6.38. uv plane into which R is mapped under the transformation x = u + v, y = u – v. (b) Compute (a) The region R shown shaded in Figure 6.9 (a) is a triangle bounded by the lines x + y = 6, x - y = 2, and y = 0, which for distinguishing purposes are shown dotted, dashed, and heavy, respectively. Compare the result of (b) with the ratio of the areas of R and R'. R' x+y = 6 y = 0 (b) uv plane (a) xy plane E = n

A region R in the xy plane is bounded by x+y = 6, x- y = 2, and y = 0. (a) Determine the region R' in the a(x, y) a(u,v) 6.38. uv plane into which R is mapped under the transformation x = u + v, y = u – v. (b) Compute (a) The region R shown shaded in Figure 6.9 (a) is a triangle bounded by the lines x + y = 6, x - y = 2, and y = 0, which for distinguishing purposes are shown dotted, dashed, and heavy, respectively. Compare the result of (b) with the ratio of the areas of R and R'. R' x+y = 6 y = 0 (b) uv plane (a) xy plane E = n

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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