Lately I've been trying to teach myself GR and it's been going fairly well, but yesterday for practice I decided to compute the curvature tensor of a paraboloid and it's not working. I've tried using three different coordinate systems, starting with what I thought would be the most obvious one,(adsbygoogle = window.adsbygoogle || []).push({});

x= r cos(theta)

y= r sin(theta)

z= r^2

but in every case the Christoffel symbols have failed. For example, using the above, the basis vectors are

e sub r = < cos(theta), sin<theta>, 2r >

e sub theta= <-r sin(theta), r cos(theta), 0 >

and the partial derivative of e sub r (for example) with respect to r = <0,0,2>. Yet there is no linear combination of these two basis vectors that will make <0,0,2>. The other two coordinate systems I've tried have had similar difficulties.

A paraboloid strikes me as a legitimate 2-dimensional manifold. What am I doing wrong?

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# Paraboloid problem

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