The ideas Gauss developed to described the geometry of a curved two-dimensional surface is generalized to abstract N dimensional spaces. This is the most math intensive video in the series so far. This is unavoidable as we are introducing the fundamental mathematical objects that allowed Einstein to give his ideas of curved spacetime precise form. Only the broad strokes are shown here. The following appendix videos fill some of the details.
Relativity 7b1 - slope and curvature http://youtu.be/FqOH42zc9Es Relativity 7b2 - metric tensor details http://youtu.be/kdYzAFs4Mt0 Relativity 7b3 - Cristoffel symbols and the geodesic equation http://youtu.be/hmq7HuWXTM8 Relativity 7b4 - Riemann and Ricci tensors http://youtu.be/sBDGPIuJ-HA
Relativity playlist: http://www.youtube.com/playlist?list=PLF56602BAC693237E
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Playlist videos for General Theory of Relativity: