Tensors & their Transformation – An Introduction

Here (GR - 09), Tensors (or rather Dyads) are formally introduced for the first time. The approach taken is via the ‘Tensor Product’ in order to gain a basic understanding, although other simple explanations will be given later (though not exhaustive).

Having discussed the Transformation of Vectors (or rank one tensors) in the last video (GR - 08), that same idea is here applied (GR - 09) to what we generally call ‘Tensors’ (rank two tensors).

Furthermore, having earlier introduced the idea of the Covariant Derivative for Vectors (rank one tensors) in an earlier video (GR - 07), it is here shown how it can (and must) be applied to Tensors (rank two tensors). As in earlier videos, there is some repeated elaboration of the ‘Einstein Summation Convention’ for those coming at it for the first time.

This video is part of a series of videos on General Relativity (GR-01 to GR-20), which has been created to help someone who knows a little bit about “Newtonian Gravity” and “Special Relativity” to appreciate both the need for “General Relativity”, and for the way in which the ‘modelling’ of General Relativity helps to satisfy that need – in the physics sense.

The production of these videos has been very much a ‘one man band’ from start to finish (‘blank paper’ to ‘final videos’), and so there are bound to be a number of errors which have slipped through. It has not been possible, for example, to have them “proof-watched” by a second person. In that sense, I would be glad of any comments for corrections ……. though it may be some time before I get around to making any changes.

By ‘corrections and changes’ I clearly do not mean changes of approach. The approach is fixed – though some mistakes in formulae may have been missed in my reviewing of the final videos, or indeed some ‘approximate explanations’ may have been made which were not given sufficient ‘qualification’. Such changes (in formulae, equations and ‘qualifying statements’) could be made at some later date if they were felt to be necessary.

REPORTED CORRECTION
"Correction: 54:26 The three covariant derivatives of the V3 component (at the bottom of the screen) ought to show all nine of the gammas with an upper component of 3 rather than 2"

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