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Find the Jacobian J of the transformations • x=2u - 1, y = 3v-4, z = (w-4)/2. • r = u cos(v), y = usin(v), z = w. What coordinate transformation does this change of variables correspond to? • r = usin(v) cos(w), y = u sin(v) sin(w), z = u cos(v). What coordinate transforma- tion does this change of variables correspond to?

Find the Jacobian J of the transformations • x=2u - 1, y = 3v-4, z = (w-4)/2. • r = u cos(v), y = usin(v), z = w. What coordinate transformation does this change of variables correspond to? • r = usin(v) cos(w), y = u sin(v) sin(w), z = u cos(v). What coordinate transforma- tion does this change of variables correspond to?

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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Problem 8.1 Integrals: General coordinate transformations a) Find the Jacobian J of the transformations • x=2u - 1, y = 3v-4, z = (w-4)/2. • x = u cos(v), y = usin(v), z = w. What coordinate transformation does this change of variables correspond to? • x = u sin(v) cos(w), y = usin(v) sin(w), z = u cos(v). What coordinate transforma- tion does this change of variables correspond to? b) Use the transformation x = u²-v², y = 2uv to evaluate the integral 1-z ² √² + y²³ dy dz Hint: Show that the image of the triangular region G with vertices (0,0), (1,0), (1, 1) in the uv-plane is the region of integration R in the xy-plane defined by the limits of integration.
Transcribed Image Text:Problem 8.1 Integrals: General coordinate transformations a) Find the Jacobian J of the transformations • x=2u - 1, y = 3v-4, z = (w-4)/2. • x = u cos(v), y = usin(v), z = w. What coordinate transformation does this change of variables correspond to? • x = u sin(v) cos(w), y = usin(v) sin(w), z = u cos(v). What coordinate transforma- tion does this change of variables correspond to? b) Use the transformation x = u²-v², y = 2uv to evaluate the integral 1-z ² √² + y²³ dy dz Hint: Show that the image of the triangular region G with vertices (0,0), (1,0), (1, 1) in the uv-plane is the region of integration R in the xy-plane defined by the limits of integration.
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