Sketch the families of level curves of the component functions u and v when z -...
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Sketch the families of level curves of the component functions u and v when z - 1 = (2)f z +1' and note how the result in Exercise 7 is illustrated here. Exercise 7 Let the function f (z) = u(x, y) + iv(x, y) be analytic in a domain D, and consider the families of level curves u(x, y) = c and v(x, y) = c2, where c and c2 are arbitrary real constants. Prove that these families are orthogonal. More precisely, show that if zo = (xo. Yo) is a point in D which is common to two particular curves u(x, y) = c and v(x, y) = c2 and if f (zo) = 0, then the lines tangent to those curves at (x0, yo) are %3D perpendicular. Suggestion: Note how it follows from the pair of equations u(x, y) = c, and v(x, y) = c2 that ди dy + ax θυ dy 0. dy dx du dv = 0 and ax %3D dy dx Sketch the families of level curves of the component functions u and v when z - 1 = (2)f z +1' and note how the result in Exercise 7 is illustrated here. Exercise 7 Let the function f (z) = u(x, y) + iv(x, y) be analytic in a domain D, and consider the families of level curves u(x, y) = c and v(x, y) = c2, where c and c2 are arbitrary real constants. Prove that these families are orthogonal. More precisely, show that if zo = (xo. Yo) is a point in D which is common to two particular curves u(x, y) = c and v(x, y) = c2 and if f (zo) = 0, then the lines tangent to those curves at (x0, yo) are %3D perpendicular. Suggestion: Note how it follows from the pair of equations u(x, y) = c, and v(x, y) = c2 that ди dy + ax θυ dy 0. dy dx du dv = 0 and ax %3D dy dx
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Related Book For
Complex Variables and Applications
ISBN: 978-0073051949
8th edition
Authors: James Brown, Ruel Churchill
Posted Date:
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