Unit Normal Vector Homework: Definition & Solution

[t_n_p]

Homework Statement

http://img171.imageshack.us/img171/5997/76283103.png

The Attempt at a Solution

I think a good place to start is with the definition of regular.
My definition is: σ is regular if ∂σ/∂s and ∂σ/∂v are linearly independent.

Want to confirm I'm on the right track before going further!

 

[t_n_p]

bump?
help anyone!?

 

[MathematicalPhysicist]

regularity means that:
$$d\sigma =/=0$$ for any s and v which is the same as what you wrote, i.e equivalent statements.
And you know how to find the normal to sigma, just multiply by scalar product N with the tangent of sigma, which is the derivative, and then divide by its norm.

 

[t_n_p]

ok, so after showing that sigma is regular, I want to find the unit vector normal (as you said above).

Found my equation in the text, just wondering if the notation is correct
http://img3.imageshack.us/img3/8527/11604435.png

 

[t_n_p]

to the top again

 

[t_n_p]

Well, I'm assuming the definition posted 2 posts above is correct, so then I proceed to evaluate ∂σ/∂s and ∂σ/∂v..

∂σ/∂s = ∂γ/∂s + r[(dn/ds)(cos v) + (db/ds)(sinv)]

and

∂σ/∂v = r[-n(s)sin v + b(s)cosv]

But now I'm wondering, how do I cross product these two if they are not vectors!