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Graph, Wall, Tome: Difference between revisions
→$$R_{\mu v}$$
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$$dV^{3} = dX^{1}dX^{2} (V^{1}R^{3}_{112} + V^{2}R^{3}_{212} + V^{3}R^{3}_{312})$$ | $$dV^{3} = dX^{1}dX^{2} (V^{1}R^{3}_{112} + V^{2}R^{3}_{212} + V^{3}R^{3}_{312})$$ | ||
or, using $$i$$ to summarize across all 3 components (difference vectors): | or, using $$i$$ to summarize across all 3 components (difference vectors): | ||
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$$dV^{i} = dX^{1}dX^{2} \Sigma_{j} [(V^{j}R^{i}_{j12}]$$ | $$dV^{i} = dX^{1}dX^{2} \Sigma_{j} [(V^{j}R^{i}_{j12}]$$ | ||
See: [https://www.youtube.com/watch?v=UfThVvBWZxM&t=19m33s the video @ 19m33s] | |||
Open questions: | |||
* Why a parallelogram? | |||
* How to properly overlay the parallelogram onto the 3d manifold, in order to do the parallel transport? | |||
* How does this relate to the length computation above? | |||
===== Putting it all together ===== | ===== Putting it all together ===== | ||