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9.21. Question. A function u(x, y) is said to be a harmonic function if Uxx + Uyy = 0. (a) Show if u is harmonic, -u₁ dx + u dy = 0 is an exact equation. So there exists (at least locally) the so-called harmonic conjugate function v(x, y) such that vx = -uy and Vy = Ux. (b) Verify that the following u are harmonic and find the corresponding harmonic conjugates v: 1. u = 2xy 2. uecos y

9.21. Question. A function u(x, y) is said to be a harmonic function if Uxx + Uyy = 0. (a) Show if u is harmonic, -u₁ dx + u dy = 0 is an exact equation. So there exists (at least locally) the so-called harmonic conjugate function v(x, y) such that vx = -uy and Vy = Ux. (b) Verify that the following u are harmonic and find the corresponding harmonic conjugates v: 1. u = 2xy 2. uecos y

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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9.21. Question. A function u(x, y) is said to be a harmonic function if Uxx + Uyy = 0. (a) Show if u is harmonic, -u₁ dx + u dy = 0 is an exact equation. So there exists (at least locally) the so-called harmonic conjugate function v(x, y) such that vx = -uy and Vy = Ux. (b) Verify that the following u are harmonic and find the corresponding harmonic conjugates v: 1. u = 2xy 2. uecos y
Transcribed Image Text:9.21. Question. A function u(x, y) is said to be a harmonic function if Uxx + Uyy = 0. (a) Show if u is harmonic, -u₁ dx + u dy = 0 is an exact equation. So there exists (at least locally) the so-called harmonic conjugate function v(x, y) such that vx = -uy and Vy = Ux. (b) Verify that the following u are harmonic and find the corresponding harmonic conjugates v: 1. u = 2xy 2. uecos y
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