A relatively easy differential geometry question concerning principle curvatures

[Jxs63J]

Homework Statement

Show the principal curvatures on x sin z - y cos z = 0 are +-1/(1 + x^2 + y^2)

 

[nonequilibrium]

no attempt?

 

[nonequilibrium]

Got a reply from the OP:

Actually I have devoted hours and hours to this problem. It is in Chapter 9 of the old Schaum's book. I have tried to put this in a form u*e1 + v*e2 + f(u,v)*e3 with little success. I have used other hints from this chapter such as determinant [dx, fx, Dfx] = 0 and I can get some simplifications but then I get lost in overly extensive equations. I have tried graphing this function as z = arctan(y/x) but without Mathematica or Maple the graph gets hazy in my mind. I know it would have several sheets but try to stick with the "main sheet". So I feel I am missing something crucial, but at the same time feel it should have been easy and that it is a really cool problem.

Decided to post that here along with my help for future reference, in case other people google his question and arrive at this page.

We can rewrite (as noted by the OP) the equation to x tan(z) = y.
Using this, we can parametrize the surface as follows:
$$x(u,v) = (u, u \tan v, v)$$
Note: x is now not the first coordinate, but rather the position vector

The only economical way I know of calculating the principal values, is using the formulae $k_{1,2} = H \pm \sqrt{H^2 - K}$.

So we need to calculate H and K. This can be easily done using the parametrization of the surface and the following:
$$H \propto En - 2Fm + Gl \qquad \qquad K = \frac{ln-m^2}{EG-F^2}$$
where $E:= x_u \cdot x_u \qquad F:= x_u \cdot x_v \qquad G = x_v \cdot x_v$
and $l := x_{uu} \cdot \xi \qquad m := x_{uv} \cdot \xi \qquad n := x_{vv} \cdot \xi$
where xi is the normal $\xi = x_u \times x_v$.

I've only given H up to something proportional to it, because you'll only have to prove that H = 0, which you can expect by comparing the formulae for k_1 and k_2 given above with what you know will be the results.

 

[Jxs63J]

Mr Vodka:

What you are showing me is what I finally figured out last night. You have basically set this out as a Monge Patch (using the nomenclature of the text) which then allows calculation of the fundamental forms prior to solving for k1 and k2. This requires some plowing through, which I am in the middle - end of doing, albeit with much greater confidence as a result of now reading your note. I consider my finally figuring this out a result of thinking about my reply to you. I am very grateful. While I am not Erdos and thus can not offer a financial reward, a caramel apple appears in order. Are you near Albuquerque NM?

Again, grateful thanks for allowing my life to continue (though I will still need that final push).

Regards,

Jeff

 

[nonequilibrium]

No caramel apple for me, I'm a European citizen ;) But you're welcome!

If you get stuck with the calculations, let me know.