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In this problem we will graphically investigate properties of conformal maps. For each of the maps (x, y) (u, v) below, complete parts i) through iii). (a) u(x, y) = x² + y², v(x, y) = 2xy (b) u(x, y) = x² - y², v(x, y) = 2xy (c) u(x, y) = = e cos x, v(x, y) = e sinx -y i) Plot the curves u(x, y) = 0, 1 and v(x, y) = 0, 1 in the x, y-plane. Do the curves intersect at right angles? ii) Based on your picture from part (a), is the map conformal? Why or why not? iii) Prove that the map f(x, y) = u(x, y) +iv(x, y) is/is not conformal using the Cauchy-Riemann conditions. If possible, represent the map by a function f(z) where z= x + iy.

In this problem we will graphically investigate properties of conformal maps. For each of the maps (x, y) (u, v) below, complete parts i) through iii). (a) u(x, y) = x² + y², v(x, y) = 2xy (b) u(x, y) = x² - y², v(x, y) = 2xy (c) u(x, y) = = e cos x, v(x, y) = e sinx -y i) Plot the curves u(x, y) = 0, 1 and v(x, y) = 0, 1 in the x, y-plane. Do the curves intersect at right angles? ii) Based on your picture from part (a), is the map conformal? Why or why not? iii) Prove that the map f(x, y) = u(x, y) +iv(x, y) is/is not conformal using the Cauchy-Riemann conditions. If possible, represent the map by a function f(z) where z= x + iy.

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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In this problem we will graphically investigate properties of conformal maps. For each of the maps (x, y) → (u, v) below, complete parts i) through iii). (a)_u(x,y)= x² + y², v(x, y) = 2xy (b) _u(x, y) = x² – y², v(x, y) = 2xy -Y (c)_u(x, y) = e cosx, v(2,y) = e sinx i) Plot the curves u(x, y) = 0, 1 and v(x, y) = 0,1 in the x, y-plane. Do the curves intersect at right angles? ii) Based on your picture from part (a), is the map conformal? Why or why not? iii) Prove that the map f(x, y) = u(x, y) + iv(x, y) is/is not conformal using the Cauchy-Riemann conditions. If possible, represent the map by a function f(z) where z = x + iy.
Transcribed Image Text:In this problem we will graphically investigate properties of conformal maps. For each of the maps (x, y) → (u, v) below, complete parts i) through iii). (a)_u(x,y)= x² + y², v(x, y) = 2xy (b) _u(x, y) = x² – y², v(x, y) = 2xy -Y (c)_u(x, y) = e cosx, v(2,y) = e sinx i) Plot the curves u(x, y) = 0, 1 and v(x, y) = 0,1 in the x, y-plane. Do the curves intersect at right angles? ii) Based on your picture from part (a), is the map conformal? Why or why not? iii) Prove that the map f(x, y) = u(x, y) + iv(x, y) is/is not conformal using the Cauchy-Riemann conditions. If possible, represent the map by a function f(z) where z = x + iy.
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