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**oblixps** Define a surface by

$\displaystyle x(u, v) = (u - \frac{u^3}{3} + uv^2, v - \frac{v^3}{3} + vu^2, u^2 - v^2) $

Show that, for $\displaystyle u^2 + v^2 < 3 $ Enneper's surface has no self-intersections. Hint: use polar coordinates $\displaystyle u = rcos(\theta), v = rsin(\theta)$ and show $\displaystyle x^2 + y^2 + \frac{4}{3}z^2 = \frac{1}{9}r^2(3 + r^2)^2 $, and then show that the equality implies that points in the (u, v) plane on different circles about (0, 0) cannot be mapped to the same point.

So what i did was plug in $\displaystyle u = rcos(\theta), v = rsin(\theta)$ into the parametrization of the surface and like the problem suggested i took the x, y, and z, and plugged them into the expression as they wanted. after some tedious calculations i finally reduced the long expression down to $\displaystyle r^2 + \frac{2}{3}r^4 + \frac{1}{9}r^6(sin^4(\theta) + cos^4(\theta)) $. when i match it up to the right side i see that everything is correct except the sines and cosines to the 4th power in the last term. however i done the problem many times and checked my work slowly along the way but i always get that the 4th power of sin and cos at the end that does not cancel out. can someone perform the calculation and see what one gets? i want to know whether it is the problem at fault or me at fault for making the same overlooked mistake over and over again.