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Math Help - parameterisation of a curve

  1. #1
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    parameterisation of a curve

    This is from some online notes. let x(s) be a curve in R^3 parameterised by s. Then f(x(s) represents the values of f along the curve.

    My question is this. Is the graph of f in R^4 ? I though it would be as f is a function of 3 variables but in the notes its looks like it defines a surface in R^3
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    MHF Contributor FernandoRevilla's Avatar
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    Re: parameterisation of a curve

    Quote Originally Posted by Duke View Post
    This is from some online notes. let x(s) be a curve in R^3 parameterised by s. Then f(x(s) represents the values of f along the curve. My question is this. Is the graph of f in R^4 ? I though it would be as f is a function of 3 variables but in the notes its looks like it defines a surface in R^3
    We have

    \mathbb{R}\mathop  \to \limits^{x } \mathbb{R}^3\mathop  \to \limits^{f } \mathbb{R}^m\Rightarrow f\circ x:\mathbb{R}\to \mathbb{R}^m

    so, f(x(s)) represents the vectorial equation of a curve in \mathbb{R}^m .
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    Re: parameterisation of a curve

    what is m?
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    MHF Contributor FernandoRevilla's Avatar
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    Re: parameterisation of a curve

    Quote Originally Posted by Duke View Post
    what is m?
    You didn't define the final set of f , so any m positive integer will do .
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    Re: parameterisation of a curve

    if m=1, how would you represent f(x(s)) graphically. It next talks about x(s) lying on a flat surface of f and tangent vectors just to give you some context.
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    MHF Contributor FernandoRevilla's Avatar
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    Re: parameterisation of a curve

    Quote Originally Posted by Duke View Post
    if m=1, how would you represent f(x(s)) graphically.
    In the same way that for m=2 , f(x(s)) represents a movement in the plane, interpret it for m=1 as a movement of a particle in a straight line .

    It next talks about x(s) lying on a flat surface of f and tangent vectors just to give you some context.
    Could you transcribe the exact formulation of the question?
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    Re: parameterisation of a curve

    Its the 2 and 3 pages of this : https://vlebb.leeds.ac.uk/bbcswebdav...65/2.3-2.7.pdf. I don't understand what the function f is. Thanks.
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