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Graph, Wall, Tome: Difference between revisions
→$$R_{\mu v}$$
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* If $$dX^{1}$$ and $$dX^{2}$$ are perpendicular, then $$g_{12}$$ and $$g_{21}$$ would be 0, and we would get $$Length^{squared} = g_{11}(dX^{1})^{2} + g_{22}(dX^{2})^{2}$$ | * 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] | * See: [https://www.youtube.com/watch?v=UfThVvBWZxM&t=14m27s the video @ 14m27s] | ||
===== Computing vector rotation due to parallel transport ===== | |||
Then, they show parallel transport when following a parallelogram, but over a curved 3D manifold. To compute the vector rotation by components, they show: | |||
$$dV^{1} = dX^{1}dX^{2} (V^{1}R^{1}_{112} + V^{2}R^{1}_{212} + V^{3}R^{1}_{312})$$ | |||
$$dV^{2} = dX^{1}dX^{2} (V^{1}R^{2}_{112} + V^{2}R^{2}_{212} + V^{3}R^{2}_{312})$$ | |||
$$dV^{3} = dX^{1}dX^{2} (V^{1}R^{3}_{112} + V^{2}R^{3}_{212} + V^{3}R^{3}_{312})$$ | |||
or, using $$i$$ to summarize across all 3 components (difference vectors): | |||
$$dV^{i} = dX^{1}dX^{2} (V^{1}R^{i}_{112} + V^{2}R^{i}_{212} + V^{3}R^{i}_{312})$$ | |||
or , using $$j$$ to index over all 3 components (original vector): | |||
$$dV^{i} = dX^{1}dX^{2} \Sigma_{j} [(V^{j}R^{i}_{j12}]$$ | |||
* See: [https://www.youtube.com/watch?v=UfThVvBWZxM&t=19m33s the video @ 19m33s] | |||
===== Putting it all together ===== | |||
=== How do they relate? === | === How do they relate? === | ||