After listening to Pavel Grinfeld tensor calculus series of lectures on YT I thought I kind of understood the notation until I read this question: Relationship between covariant/contravariant basis vectors, where the expression
$$V^i = M^{ij} V_j$$
triggered the following comments from knowledgeable contributors:
Unless I am missing something, this doesn't seem to be standard math notation, but rather some physics convention.
Yes, this is Einstein summation notation. The OP should be aware that notation and conventions differ here between mathematics and physics; the physics convention may be somewhat more convenient for computations but IMO physicists don't do a good job of explaining tensors.
This is not intended to be a facetious question, but I wish someone would spell out clearly what is accepted practice in Math, and what is Physics. Pavel Grindfeld certainly has glowing credentials as a mathematician, but it is possible that he used Physics notation if he was teaching engineers or physicists. However, this contraction of the covariant coefficients of a vector $V$ with the metric tensor to raise the index seems in line with his notation, except perhaps that instead of $g^{ij}$ there is $M^{ij}.$ Certainly it is understood that only the coefficients are included in the expression.
How would this expression be more "mathematical sounding"? If the Einstein notation wasn't used at all, and there was a $\Sigma,$ or if the Einstein notation was preserved, but no super-script indices were used, as in $V_i = g_{ij} V_j,$ or something entirely different?
To wrap up the Math/Physics tensor notation possible differences, is there also some distinguishing element in usage regarding the index spacing to denote order, as in $T^{\nu\mu}{}_{\gamma}$ versus $T^{\nu\mu}_{\gamma}$? Clearly this matters in something like this: $T_{i}{}^{a}{}_{j}{}^{b}{}_{k} = T_{imj}{}^{bd}g^{am}g_{kd} \ne T_{ijk}{}^{ab},$ as explained by Jackozee Hakkiuz here.
Thank you!
