Let the function f(z) = u(x,y) + iv(x,y) be analytic in D, and consider the families of level curves u(x.y)=c1 and v(x,y)=c2 where c1 and c2 are arbitrary constants. Prove that these families are orthogonal. More precisely, show that if zo=(xo,yo) (o is a subscript) is a point in D which is common to two particular curves u(x,y)=c1 and v(x,y)=c2 and if f '(zo) is not equal to zero, then the lines tangent to those curves at (xo,yo) are perpendicular.(adsbygoogle = window.adsbygoogle || []).push({});

I really have absolutely no idea how to show this. It gives the suggestion that

[tex] \frac{\partial u}{\partial x} + \frac{\partial u}{\partial y}\frac{dy}{dx} = 0 [/tex]

and

[tex] \frac{\partial v}{\partial x} + \frac{\partial v}{\partial y}\frac{dy}{dx} = 0 [/tex]

So the total derivatives with respect to x of u and v are both zero. Should I equate these and look for some relationship between the partials? Since the function is analytic we know

[tex] u_x = v_y [/tex]

[tex] u_y = -v_x [/tex]

So this can be rewritten in several different ways, but I really just don't know what I am looking for.

Can anyone please offer some advice?

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# Showing two families of curves are orthogonal.

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