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[QUOTE="highflyyer, post: 5899846, member: 521514"] It looks all right to me! Let's try to intuitively understand what you did. ##T## is the generalization of a vector, in the sense that, ##T## is simply the sum of a bunch of components ##{T^{\mu\nu}}_{\sigma\rho}## multiplied by basis vectors ##e_{\mu} \otimes e_{\nu} \otimes \theta^{\sigma}\otimes \theta^{\rho}##. This is the interpretation of equation ##1## in your post. Therefore, in order to get the component ##{T^{\mu\nu}}_{\sigma\rho}##, you would [I]naively[/I] want to multiply the basis vector ##e_{\mu} \otimes e_{\nu} \otimes \theta^{\sigma}\otimes \theta^{\rho}## with itself. But then, you realize that ##T## exists not in flat space, but on a curved manifold. Therefore, you multiply the basis vector ##e_{\mu} \otimes e_{\nu} \otimes \theta^{\sigma}\otimes \theta^{\rho}## not with itself, but by its dual basis vector ##\theta^{\mu} \otimes \theta^{\nu} \otimes e_{\sigma} \otimes e_{\rho}##. That's exactly the interpretation of equation ##2## in your post. Your check of the consistency of equations ##1## and ##2## is simply a mathematical way of rewriting my above two paragraphs. [/QUOTE]
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