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InbredDummy
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Homework Statement
Show that the helicoid and the catenoid are conjugate minimal surfaces
Homework Equations
the helicoid is given by the parameterization
X(u,v) = (asinh(v)*cosu, asinh(v)*sinu, au) = (x1, x2, x3)
the catenoid is given by the parameterization
Y(u,v) = (acoshv*cosu, a coshv*sinu, av) = (y1, y2, y3)
The Attempt at a Solution
so i need that d(x1)/du = d(y1)/dv, etc, the component-wise functions must satisfy the Cauchy Riemann equations, but I'm not getting the right answers. clearly for x3 and y3, the C-R equations are satisfied.
i get:
d(x1)/du = -asinhv*sinu
d(y1)/dv = asinhv*cosu
and the C-R equations are not even close to satisfied. but i get that the off terms satisfy the C-R equations,
d(y2)/dv = asinhv*sinu
any help? i know this is a famous classical problem in minimal surfaces, but I am so stuck