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Math Help - A regularly parametrized surface

  1. #1
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    A regularly parametrized surface

    Let
    γ : R2 R3 be a unit speed curve of nonvanishing curvature and let its scalar curvature, torsion and Frenet frame be denoted κ, τ and [u, n, b], as usual. Consider the mapping M : R2 R3, M(s, t) = γ(s) + tb(s).
    1)Show that M is a regularly parametrized surface. (You may assume that M is one-to-one.)
    You may assume the frenet formulae.

    I got (u(s)-tτn(s)) x b(s) for the cross product of the partial derivatives (which I need to show is non-zero for all s,t), but how do I simplify further and is this correct?








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  2. #2
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    Re: A regularly parametrized surface

    b = u x n, so n x b = u, b x u = n,
    ( u - tτn) x b = uxb - tτ nxb = -n -tτ u
    It's length is 1+(tτ)^2, never vanishes.
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  3. #3
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    Re: A regularly parametrized surface

    Can you explain your first line please. That's surely not true in general is it?
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  4. #4
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    Re: A regularly parametrized surface

    The cross product a b is defined as a vector c that is perpendicular to both a and b, with a direction given by the right-hand rule and a magnitude equal to the area of the parallelogram that the vectors span.
    u, b, n are all unit vectors that are perpendicular to each other.
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  5. #5
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    Re: A regularly parametrized surface

    thanks mate
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