General relativity/The Tensor Product

<General relativity

If $𝐓$ and $𝐒$ are tensors of rank $n$ and $m$, then there exists a tensor $𝐓\otimes 𝐒$ of rank $n+m$. The components of the new tensor (pronounced "T tensor S") are obtained by multiplying the components of the old tensors. In other words, if $𝐓=T_{\beta }^{\alpha \gamma }{𝐞}_{\alpha }\otimes 𝐝x^{\beta }\otimes {𝐞}_{\gamma }$ and $𝐒=S_{\mu \nu }𝐝x^{\mu }\otimes 𝐝x^{\nu }$, then $𝐓\otimes 𝐒=T_{\beta }^{\alpha \gamma }{S}_{\mu \nu }{𝐞}_{\alpha }\otimes 𝐝x^{\beta }\otimes {𝐞}_{\gamma }\otimes 𝐝x^{\mu }\otimes 𝐝x^{\nu }$.

More to come...