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Mar 19 at 22:58 comment added Vincent Thacker This is incorrect. The basis 1-forms and the basis vectors are not elements of the same vector space. They are elements of $V^*$ and $V$ respectively. So you cannot dot them. Instead it should be written as $e^\mu (e_\nu)$. Any book that uses the dot is mathematically wrong.
Mar 19 at 16:47 comment added Albertus Magnus Yes, it is good to get familiar with the idea that contraction is the result of a bilinear mapping acting on basis vectors and 1-forms respectively.
Mar 19 at 11:50 history edited Sancol. CC BY-SA 4.0
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Mar 19 at 11:29 history answered Sancol. CC BY-SA 4.0