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### Constructing a basis for a tensor product

If the pair $(V\otimes W,\otimes)$ is a tensor product of vector spaces $V$ and $W$ with tensor product map $\otimes:V\times W\to V\otimes W$, then the pair satisfies the universal property: For any ...
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The notation you are using partly is not really well defined. There are (at least) two possible definitions of what $tr[g\otimes X\otimes Y]$ should mean, namely either $g(X,\ )\otimes Y$ or $g(\ ,Y)\... • 21k 1 vote Accepted ### Mechanics of a contraction on the Kronecker product matrix First a remark on the side: the distinction between upper and lower indices does not mean much in this context where we are not dealing with a metric; only the order of the indices matters. The result ... • 1,953 1 vote Accepted ### Abstract index notation - can't understand identity$(dx^{\mu})_aT^b\partial_bv^a=T^b\partial_b[(dx^{\mu})_av^a]$Unfortunately the version of that book in google stops at p. 66. So I start from some principles that are tersely explained, say, in Wikipedia. If$\boldsymbol{v}=v^\nu\partial_\nu$is a vector field ... • 15.3k 1 vote Accepted ### trace of a matrix squared:a formula We use the metric tensor to calculate the trace of a$2$-tensor: If$A=[a_{ij}]$is a$2$-tensor, then $$\text{tr}(A) = \sum g^{ij}a_{ij}.$$ What we are doing is using the metric tensor to raise one ... • 117k 1 vote ### Geometric meaning of second Covariant Derivative If you consider the "second covariant derivative" of a scalar function$f$instead of a vector field$w$, you get the nice symmetry result that$\nabla _{u,v}^{2}f=\nabla _{v,u}^{2}f\$, ...
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