Regularity of Surface M and Matrix at Point P

[Shackleford]

Homework Statement

Consider the surface M parametrized by σ(u, v) = (sin u cos v, sin u sin v, 3 cos u), 0 < u < π, 0 < v < 2π.

(ii) Show that the surface is regular.

(iii) Let P = (\sqrt{2}/2, \sqrt{2}/2, 0) in M. Find the matrix S_p with respect to the basis {σ_u(P), σ_v(P)} in T_P(M).

Homework Equations

Surface is regular if σ_u x σ_v does not equal zero at any point in the domain.

The Attempt at a Solution

σ_u = (cos v cos u, sin v cos u, -3 sin u)

σ_v = (-sin u sin v, sin u cos v, 0)

σ_u x σ_v = (3 cos v sin²u, 3 sin²u sin v, sin u cos u)

||σ_u x σ_v|| = sqrt{9 sin⁴u + sin²u cos²u}

Of course, it looks like the norm of the vector product should simplify further.

I'm also unsure about the given basis for the matrix of the shape operator. By direction calculation, the matrix contains expressions of the first and second fundamental forms. Do I evaluate the partials at the point P and use those in the inner product calculations?

 

[HallsofIvy]

If u= 0 then \sigma_u\times \sigma_v= 0.

 

[Shackleford]

If u= 0 then \sigma_u\times \sigma_v= 0.

Yes. I'm a bit confused. It says to show that the surface is regular.

 

[Ray Vickson]

Yes. I'm a bit confused. It says to show that the surface is regular.

The original description excluded the boundaries $u =0$, $u = \pi$, $v = 0$, $v = 2 \pi$.

 

[Shackleford]

The original description excluded the boundaries $u =0$, $u = \pi$, $v = 0$, $v = 2 \pi$.

Okay. It doesn't look like any combination of u and v in the domain yields (0, 0, 0), so I can state that it is regular. However, it seems that the norm argument should simplify a bit more.

 

[Ray Vickson]

Okay. It doesn't look like any combination of u and v in the domain yields (0, 0, 0), so I can state that it is regular. However, it seems that the norm argument should simplify a bit more.

Who cares? You just want to know if it = 0 or not.

 

[Shackleford]

Who cares? You just want to know if it = 0 or not.

(iii) Let P = (\sqrt{2}/2, \sqrt{2}/2,0) in M. Find the matrix Sp with respect to the basis {σ_u(P), σ_v(P)} in T_P(M).