2. Let the function f(z) u(x, y) + 1 u(x, y) be analytic in a domain D, and cons

Question

2. Let the function f(z) u(x, y) + 1 u(x, y) be analytic in a domain D, and consider the c and c2 are arbitrary families of level curves u(x, y) = ci and u(x, y) = c2, where 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) and u(x, y) c2 and iff'(zo)0, then the lines tangent to those curves at (xo, yo) are perpendicular. Suggestion: Notehow D(x, y)2 that follows from the pair of equations u(x, y) - ci and au au dy ax ay dx av av dyo x dy dx

Explanation / Answer

Solution:

2. Using partial derivative arguments,
u(x,y) = c_1 ==> du/dx * dx + du/dy * dy = 0
==> dy/dx = -(du/dx)/(du/dy).

Similarly, v(x,y) = c_2 ==> dy/dx = -(dv/dx)/(dv/dy).
(Note that du/dx, du/dy, dv/dx, and dv/dy are all partial derivatives.)

Now, we multiply u and v's slopes together:
[-(du/dx)/(du/dy)] * [-(dv/dx)/(dv/dy)]
= [(du/dx)/(du/dy)] * [(dv/dx)/(dv/dy)]
= [(dv/dy)/(-dv/dx)] * [(dv/dx)/(dv/dy)], by Cauchy-Riemann Eqns
= -1.

Since the slopes multiply to -1, the two families of level curves are orthogonal

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