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Assume that F(z) $(x, y) + ¡b(x, y) is analytic in the domain D and that F'(z) ‡ () in D. Consider the families of level curves {p(x, y) = constant} and {$(x, y) = constant}. which are the equipotentials and streamlines for the fluid flow V(x, y) F'(). Prove that the two families of curves are orthogonal. Hint: Suppose that (x), yo) is a point common to the two curves $(x, y) = c, and $(x, y) = c₂. Take the gradient of þ and 4. and show that the normals to the curves are perpendicular. =

Assume that F(z) $(x, y) + ¡b(x, y) is analytic in the domain D and that F'(z) ‡ () in D. Consider the families of level curves {p(x, y) = constant} and {$(x, y) = constant}. which are the equipotentials and streamlines for the fluid flow V(x, y) F'(). Prove that the two families of curves are orthogonal. Hint: Suppose that (x), yo) is a point common to the two curves $(x, y) = c, and $(x, y) = c₂. Take the gradient of þ and 4. and show that the normals to the curves are perpendicular. =

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 25 of Chapter 3, Section 3.3: Analytic and Harmonic Functions from Complex Analysis for Mathematics and Engineering

25. Assume that F(z) Q(x, y) + ib(x, y) is analytic in the domain D and that F'(z) # () in D. Consider the families of level curves {p(x, y) constant) and {$(x, y) = constant}, which are the equipotentials and streamlines for the fluid flow V(x, y) F'(). Prove that the two families of curves are orthogonal. Hint: Suppose that (xo. Yo) is a point common to the two curves p(x, y) = c; and $(x, y) = c₂. Take the gradient of þ and v. and show that the normals to the curves are perpendicular. = -
Transcribed Image Text:25. Assume that F(z) Q(x, y) + ib(x, y) is analytic in the domain D and that F'(z) # () in D. Consider the families of level curves {p(x, y) constant) and {$(x, y) = constant}, which are the equipotentials and streamlines for the fluid flow V(x, y) F'(). Prove that the two families of curves are orthogonal. Hint: Suppose that (xo. Yo) is a point common to the two curves p(x, y) = c; and $(x, y) = c₂. Take the gradient of þ and v. and show that the normals to the curves are perpendicular. = -
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