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Calculating the line element after a change of coordinates

  1. Nov 14, 2015 #1
    1. The problem statement, all variables and given/known data

    Consider ##\mathbb{R}^3## in standard Cartesian co-ordinates, and the surface ##S^2## embedded within it defined by ##(x^2+y^2+z^2)|_{S^2}=1##. A particular set of co-ords on ##S^2## are defined by

    ##\zeta = \frac{x}{z-1}##,
    ##\eta = \frac{y}{z-1}##.

    Express ##1+\zeta^2+\eta^2## in terms of ##z##. By evaluating ##d\zeta## and ##d\eta##, show that the line element on ##S^2## is given by

    ##ds^2|_{S^2}=(dx^2+dy^2+dz^2)|_{S^2} = \frac{d\zeta^2+d\eta^2}{f(\zeta,\eta)}## (1),

    where you should give the form of ##f(\zeta,\eta)##

    2. Relevant equations


    ##1+\zeta^2+\eta^2=\frac{z^2+y^2}{(z-1)^2}+1=\frac{-2}{z-1}##
    ##d\zeta=\frac{dx}{z-1}-\frac{xdz}{(z-1)^2}## , ##d\eta=\frac{dy}{z-1}-\frac{ydz}{(z-1)^2}##

    ##d\eta^2 + d\zeta^2= \frac{dx^2}{(z-1)^2} +\frac{x^2dz^2}{(z-1)^4}-\frac{2xdxdz}{(z-1)^3}+\frac{dy^2}{(z-1)^2} +\frac{y^2dz^2}{(z-1)^4}-\frac{2ydydz}{(z-1)^3}## (2)

    3. The attempt at a solution

    So far I have been able to do the first two parts fine (the first two equations under 'Relevant equations'), the part I'm struggling with is trying to prove equation (1). So far I have tried computing ##d\eta^2+d\zeta^2##, which is equation 2 above, as well as rearranging the differentials and trying ##dx^2+dy^2##, but I feel like this is the wrong approach.

    I'm just wondering if anyone has any tips or can see what to do? I feel like I'm over thinking and over complicating the problem, or they may be something I'm missing.

    This is my first post, hopefully I've put it in the right place. And thanks in advance for any help!
     
  2. jcsd
  3. Nov 14, 2015 #2
    I just noticed a typo, it should be ##1+\zeta^2+\eta^2=\frac{x^2+y^2}{(z-1)^2}+1=\frac{-2}{z-1}##. This does not affect the question.
     
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