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Graph, Wall, Tome: Difference between revisions

711 bytes added ,  29 January 2020
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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? ===
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