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Thread: Parametrization of a smooth surface

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    Parametrization of a smooth surface

    The reason that I am posting this is because I'm having a hard time figuring out how to pose the question in a manner that makes any sense. But I thought I'd post it here and see if anyone can help guide me.

    Suppose we have a smooth surface defined in 3-space that is given by X^3 = X^3 ( X^1 , ~X^2 ). (The superscripts indicate a contravariant index, not a power.)

    Suppose we have a parametrization of the surface, X^1 = v^1, X^2 = v^2, and thus X^3 = X^3(v^1,~ v^2 ). My question is: "Is there a transformation of coordinates of this surface such that the transformed coordinates are not allowable, but still have the same domain as the original parametrization?"

    I haven't been able to do much in the way of a proof but geometrically speaking I don't think such a transformation can be done. The surface is, of course, independent of the parametrization so it has the same number of critical points, the same curvature at a given point, etc. But I can't see how to make a proof out of it.

    I'm probably making a mountain out of a molehill but it bothers me that I can't come up with a way to write this thing down in clear terms.

    Thanks!

    -Dan
    Last edited by topsquark; Nov 7th 2017 at 10:01 PM.
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    Re: Parametrization of a smooth surface

    can you expand on what you mean by "not allowable" ?
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    Re: Parametrization of a smooth surface

    Quote Originally Posted by romsek View Post
    can you expand on what you mean by "not allowable" ?
    Sorry. I don't know what terms are standard. This is from "Introduction to Vector and Tensor Analysis," Wrede, pg. 223.

    An allowable transformation is of the form X^{ \mu } = X^{ \mu } \left ( \overline{X} ^{ \nu } \right ), where the \overline{X} ^{ \nu } are the new coordinates obeying the following:
    1) Each of the functions X^{ \mu } \left ( \overline{X} ^{ \nu } \right ) has continuous partial derivatives at least of first order

    2) The determinant  | X^{ \mu } / \overline{X} ^{ \nu } | \neq 0 for every coordinate triple of the domain determined by the function X^{ \mu }

    -Dan
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