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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
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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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 coordinates 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) 6.38. uv plane into which R is mapped under the transformation x = u + v, y = u – v. (b) Compute Compare 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 = 6 y = 0 (b) uv plane (a) xy plane Figure 6 9
Transcribed Image Text:Transformations, curvilinear coordinates 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) 6.38. uv plane into which R is mapped under the transformation x = u + v, y = u – v. (b) Compute Compare 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 = 6 y = 0 (b) uv plane (a) xy plane Figure 6 9
Figure 6.9 Under the given transformation, the line x + y = 6 is transformed into (u + v) + (u – v) = 6; i.e., 2u = 6 or u = 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 required region is bounded by u = 3, v = 1, and u = v, and is shown shaded in Figure 6.9(b). Ox dx a(x, y) du dv du (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 of any regions R in the xy plane are twice the areas of corresponding mapped regions R' in the uv plane.
Transcribed Image Text:Figure 6.9 Under the given transformation, the line x + y = 6 is transformed into (u + v) + (u – v) = 6; i.e., 2u = 6 or u = 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 required region is bounded by u = 3, v = 1, and u = v, and is shown shaded in Figure 6.9(b). Ox dx a(x, y) du dv du (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 of any regions R in the xy plane are twice the areas of corresponding mapped regions R' in the uv plane.
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