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2. The Jacobian for the change of variables z = = g(u, v), y = h(u, v) is given by Әх ду ди дv J(u, v) = where S is the quarter ellipse Ə(x, y) d(u, v) (a) Calculate J(r, 0) if x = r cos(0) and y = r sin(0) (b) The Jacobian is used to implement a change of variables when integrating. If the region of integration is D and is given by z and y, then the new region D* will be given by u and v according to a = g(u, v) and y=h(u, v). The integration would change as follows [[ f(x, y)dydx = [[ f(g(u, v), h(u, v))|J (u, v)|dudv Now, use the change of variable z = ar cos 0, y = br sin to evaluate Je--d₁ 1일 ду дл du Əv + ≤ 1, z ≥ 0, y ≥ 0

2. The Jacobian for the change of variables z = = g(u, v), y = h(u, v) is given by Әх ду ди дv J(u, v) = where S is the quarter ellipse Ə(x, y) d(u, v) (a) Calculate J(r, 0) if x = r cos(0) and y = r sin(0) (b) The Jacobian is used to implement a change of variables when integrating. If the region of integration is D and is given by z and y, then the new region D* will be given by u and v according to a = g(u, v) and y=h(u, v). The integration would change as follows [[ f(x, y)dydx = [[ f(g(u, v), h(u, v))|J (u, v)|dudv Now, use the change of variable z = ar cos 0, y = br sin to evaluate Je--d₁ 1일 ду дл du Əv + ≤ 1, z ≥ 0, y ≥ 0

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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2. The Jacobian for the change of variables x = g(u, v), y = h(u, v) is given by ?х ду ду дх əu əv du əv J(u, v) = ə(x, y) d(u, v) (a) Calculate J(r, 0) if x = r cos(0) and y = r sin(0) (b) The Jacobian is used to implement a change of variables when integrating. If the region of integration is D and is given by z and y, then the new region D* will be given by u and v according to x = g(u, v) and y=h(u, v). The integration would change as follows where S is the quarter ellipse f(x, y)dydx = = = ſ. f(g(u, v), h(u, v))\J(u, v)|dudv Now, use the change of variable x = ar cos 0, y = br sin 0 to evaluate Je --d₁ е ≤1, x ≥ 0, y ≥ 0 1 3. Use question 2 to answer the following question. Find the volume of the region bounded by x² + 3y² = z and y² + z = 4.
Transcribed Image Text:2. The Jacobian for the change of variables x = g(u, v), y = h(u, v) is given by ?х ду ду дх əu əv du əv J(u, v) = ə(x, y) d(u, v) (a) Calculate J(r, 0) if x = r cos(0) and y = r sin(0) (b) The Jacobian is used to implement a change of variables when integrating. If the region of integration is D and is given by z and y, then the new region D* will be given by u and v according to x = g(u, v) and y=h(u, v). The integration would change as follows where S is the quarter ellipse f(x, y)dydx = = = ſ. f(g(u, v), h(u, v))\J(u, v)|dudv Now, use the change of variable x = ar cos 0, y = br sin 0 to evaluate Je --d₁ е ≤1, x ≥ 0, y ≥ 0 1 3. Use question 2 to answer the following question. Find the volume of the region bounded by x² + 3y² = z and y² + z = 4.
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