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Math Help - Re: More exam revision!

  1. #1
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    Re: More exam revision!

    Revolving the unit speed curve c(u)=(g(u),h(u),0) about the x-axis
    produces a surface which can be covered by the coordinate patch

    PHI(u,v)=(g(u),h(u)cosv,h(u)sinv)

    Assume the h(u)>0 for all u. Show that the u-parameter curves are all
    geodesics

    [Difficulty]
    I'm revising for an exam and I don't know what to do!!

    [Thoughts]
    Ok first of all, should I be able to know how they got the coordinate
    patch? Once I have it I know how to work with it but I don't know how
    they got it!

    I have loads of theory to work off, I'm just not sure how to apply it.
    U-parameter curves can only be geodesic if unit speed so for starters
    do I need to check that this is unit speed? Or make it unit speed?

    Why is it only the u-parameter curves that will be geodesics?

    A regular curve a(t) on a surface is a geodesic of the surface is
    a''_tan=0

    Looking at my notes I have the following formula
    a''.(T X n) as the geodesic curvature. How is this, have the t's been
    suppressed? Like is it a''(t).(T(t) X n(t))?

    This doesn't really help me though does it? Since I don't have the
    original curve except in terms of functions?

    Sorry I'm really confused!
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  2. #2
    Super Member Rebesques's Avatar
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    How is this, have the t's been
    suppressed? Like is it a''(t).(T(t) X n(t))?

    Yes. Calculate this and show it is is zero on the curves v=constant.
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