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Natural basis and dual basis of a circular paraboloid
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[QUOTE="jambaugh, post: 5839494, member: 76054"] Now that my power's back on... (Irma's fault)... Yep, here are the full metric forms in 3-space if you add the W coordinate: [tex] \mathbf{g} \sim \left( \begin{array}{ccc} (1+\tfrac{1}{4U}) & 0 & -1\\ 0 & U & 0\\ -1 & 0 & 1\end{array}\right)[/tex] and dual metric is: [tex] \mathbf{g}^* = \left(\begin{array}{ccc}4U & 0 & 4U\\ 0 & 1/U & 0\\ 4U & 0 & 4U +1\end{array}\right)[/tex] The problem here, to get the 2-dimensional metric and dual metric is that the tangent space (spanned by ##e_U,e_V## [edited]) is of course different at every point and the dual vectors are not in that tangent space. One must define a tangent component of the 3-dim gradient. [To Be Continued] [/QUOTE]
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Natural basis and dual basis of a circular paraboloid
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