- #1

Shackleford

- 1,656

- 2

## 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 = ([itex]\sqrt{2}/2, \sqrt{2}/2, 0)[/itex] 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 sin

^{2}u, 3 sin

^{2}u sin v, sin u cos u)

||σ

_{u}x σ

_{v}|| = sqrt{9 sin

^{4}u + sin

^{2}u cos

^{2}u}

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?

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