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Geodesics of a Parabolic Cylinder

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(Please see the attached file for the complete problem description)
Show that the problem of finding geodesics on a surface g(x,y,z) = 0 joining points (x_1,y_1,z_1) and (x_2,y_2,z_2) can be found by obtaining the minimum of:
(please see the attached file)
Hence find the geodesics for the parabolic cylinder y = x^2.

Please show all of your work.
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Solution Summary

This solution derives the general form of the geodesic equation for a 2D surface and derive the geodesic equations for the case of a parabolic cylinder.

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Please see the attached file for the complete solution
We are given a surface S given by the equation (please see the attached file). We wish to show that the geodesics of S from (please see the attached file) to (please see the attached file) are the curves (please see the attached file) for which the quantity
(1) (please see the attached file)
is minimized, where (please see the attached file) for (please see the attached file) and (please see the attached file) We also wish to find the geodesics of the parabolic ...

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