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7. Let u = u(x, y) and let x = r cos and y = r sin 0. (a) Express u and uy in terms of u, and up. 1 (b) Show that || Vu||² = u² + √zª³, (c) Let u(r, 0) = r² cos² 0. Use part (b) to compute ||Vu||2. Then compute ||Vu||² directly by observing that u(x, y) = x².

7. Let u = u(x, y) and let x = r cos and y = r sin 0. (a) Express u and uy in terms of u, and up. 1 (b) Show that || Vu||² = u² + √zª³, (c) Let u(r, 0) = r² cos² 0. Use part (b) to compute ||Vu||2. Then compute ||Vu||² directly by observing that u(x, y) = x².

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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Question
### Problem 7 Let \( u = u(x, y) \) and let \( x = r \cos \theta \) and \( y = r \sin \theta \). #### Tasks: (a) **Express \( u_x \) and \( u_y \) in terms of \( u_r \) and \( u_\theta \).** (b) **Show that \( ||\nabla u||^2 = u_r^2 + \frac{1}{r^2} u_\theta^2 \).** (c) **Let \( u(r, \theta) = r^2 \cos^2 \theta \). Use part (b) to compute \( ||\nabla u||^2 \). Then compute \( ||\nabla u||^2 \) directly by observing that \( u(x, y) = x^2 \).** **Explanation:** - Part (a) involves finding the expressions for the partial derivatives of \( u \) with respect to \( x \) and \( y \) in terms of the derivatives with respect to polar coordinates \( r \) and \( \theta \). - Part (b) requires showing the equivalence of the expression for the square of the gradient norm in terms of derivatives with respect to polar coordinates. - Part (c) involves applying the results from part (b) to a specific function and verifying it through a different method.
Transcribed Image Text:### Problem 7 Let \( u = u(x, y) \) and let \( x = r \cos \theta \) and \( y = r \sin \theta \). #### Tasks: (a) **Express \( u_x \) and \( u_y \) in terms of \( u_r \) and \( u_\theta \).** (b) **Show that \( ||\nabla u||^2 = u_r^2 + \frac{1}{r^2} u_\theta^2 \).** (c) **Let \( u(r, \theta) = r^2 \cos^2 \theta \). Use part (b) to compute \( ||\nabla u||^2 \). Then compute \( ||\nabla u||^2 \) directly by observing that \( u(x, y) = x^2 \).** **Explanation:** - Part (a) involves finding the expressions for the partial derivatives of \( u \) with respect to \( x \) and \( y \) in terms of the derivatives with respect to polar coordinates \( r \) and \( \theta \). - Part (b) requires showing the equivalence of the expression for the square of the gradient norm in terms of derivatives with respect to polar coordinates. - Part (c) involves applying the results from part (b) to a specific function and verifying it through a different method.
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