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7. Find the Jacobian of the transformation T(x, y) = (w sin(v²),w cos(v²)). X=w Sin (v?) w = sin (v?) v =W COS (v²). 2v %3D y= w cos (v?) Ym= cos (v?) %3D gu=-W sin (v^). 2v J= (sin (v*)) (-wsin(v3).2v) - (wcos (v}.2v) (cos(v?))

7. Find the Jacobian of the transformation T(x, y) = (w sin(v²),w cos(v²)). X=w Sin (v?) w = sin (v?) v =W COS (v²). 2v %3D y= w cos (v?) Ym= cos (v?) %3D gu=-W sin (v^). 2v J= (sin (v*)) (-wsin(v3).2v) - (wcos (v}.2v) (cos(v?))

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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7. Find the Jacobian of the transformation T(x,y) = (w sin(v²),w cos(v²)). X= w sin (v?) Xw = sin (v?) Xv=W COS (v?). 2v y= w cos (v²) Ym= cos (v?) yu=-w sin (v")-2v J= (sin (v?)) (-wsin(u3).2v) - (wcos (v%.2v) (cos(v?) (Sin(v?))(2vw sin(v*)-(( 2vNcos (ve) Ccos (ve)) - Zvw sin (v) - (2vwcos (v?) - 2vw sin? (v?) -Zvw Cos? (v?) - Zvw (sin? (ve) + cos (v*)) = - 2vw (1) = -2vW 8. Use the transformations v = x + y and w = x - y to set up the transformed integral S,(x2 – 2xy + y“)e** dA if R is the region bounded by x – y = 1, x – y = 2, x = 0, and y = 0. You only need to set it up; you do not need to solve it 2 Ex
Transcribed Image Text:7. Find the Jacobian of the transformation T(x,y) = (w sin(v²),w cos(v²)). X= w sin (v?) Xw = sin (v?) Xv=W COS (v?). 2v y= w cos (v²) Ym= cos (v?) yu=-w sin (v")-2v J= (sin (v?)) (-wsin(u3).2v) - (wcos (v%.2v) (cos(v?) (Sin(v?))(2vw sin(v*)-(( 2vNcos (ve) Ccos (ve)) - Zvw sin (v) - (2vwcos (v?) - 2vw sin? (v?) -Zvw Cos? (v?) - Zvw (sin? (ve) + cos (v*)) = - 2vw (1) = -2vW 8. Use the transformations v = x + y and w = x - y to set up the transformed integral S,(x2 – 2xy + y“)e** dA if R is the region bounded by x – y = 1, x – y = 2, x = 0, and y = 0. You only need to set it up; you do not need to solve it 2 Ex
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