Unit Normal Vector Homework: Definition & Solution

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Homework Help Overview

The discussion revolves around the concept of unit normal vectors in the context of regularity in differential geometry. Participants are exploring definitions and methods related to the normal vector of a curve or surface.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the definition of regularity and its implications for finding normal vectors. There are attempts to confirm understanding of the definitions and to clarify notation related to the normal vector. Questions arise about the mathematical operations involved, particularly regarding the cross product of derivatives that are not vectors.

Discussion Status

The discussion is ongoing, with some participants confirming their understanding of regularity and others seeking clarification on notation and mathematical operations. There is no explicit consensus, but guidance is being offered regarding definitions and approaches to the problem.

Contextual Notes

Participants are working with specific definitions and equations from their texts, and there is an emphasis on ensuring the correctness of notation and mathematical expressions. Some posts indicate a need for further assistance and clarification on certain points.

t_n_p
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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!
 
Last edited by a moderator:
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bump?
help anyone!?
 
regularity means that:
[tex]d\sigma =/=0[/tex] 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.
 
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
 
Last edited by a moderator:
to the top again
 
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!
 

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