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Graph, Wall, Tome: Difference between revisions
→$$R_{\mu v}$$
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The same video then proceeds to explain $$R_{\mu v}$$. It progresses through some concepts. | The same video then proceeds to explain $$R_{\mu v}$$. It progresses through some concepts. | ||
===== Computing length in non-orthogonal bases ===== | |||
First, just describing the length of a vector on a curved space is hard. It is given by: | |||
$$Length^{squared} = g_{11}dX^{1}dX^{1} + g_{12}dX^{1}dX^{2} + g_{21}dX^{2}dX^{1} + g_{22}dX^{2}dX^{2}$$ | |||
Some notes: | |||
* This is not Pythagorean theorem, because $$dX^{1}$$ and $$dX^{2}$$ are not perpendicular. | |||
* Instead, looks like a formula to get the diagonal from two opposite vertices in a parallelogram. | |||
* If $$dX^{1}$$ and $$dX^{2}$$ are perpendicular, then $$g_{12}$$ and $$g_{21}$$ would be 0, and we would get $$Length^{squared} = g_{11}(dX^{1})^{2} + g_{22}(dX^{2})^{2}$$ | |||
* See: [https://www.youtube.com/watch?v=UfThVvBWZxM&t=14m27s the video @ 14m27s] | |||
=== How do they relate? === | === How do they relate? === | ||