# Questions tagged [tensor-calculus]

Tensor calculus (tensor analysis) is a systematic extension of vector calculus to multivector and tensor fields in a form that is independent of the choice of coordinates on the relevant manifold, but which accounts for respective sub-spaces, their symmetries, and their connections.

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### Some formula on anti-symmetrization: general formula for anti-symmetrization $A^{i_1 \cdots i_n} X_{[a} \bar{A}_{i_1 \cdots i_n]}$ [closed]

Let $A, \bar{A}$ be the totally-anti-symmetric tensor. Here my convention for $A_{[a_1, \cdots, a_n]}= \frac{1}{n!} (A_{a_1 \cdots a_n} + \cdots)$ I want to find some general formula for \begin{...
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### How to use the definition of a rank-$2$ tensor for this kind of examples?

Suppose that, a rank-$2$ tensor transforms as \begin{align} T'^{ij}=\frac{\partial x'^i}{\partial x^k}\frac{\partial x'^k}{\partial x^l}T^{kl}. \end{align} How to use this criterion to investigate if ...
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### Graded cyclic properties in tensor calculus formalism of supergravity

I am trying to understand the chapter 4 of https://arxiv.org/abs/hep-th/0204035. I want to obtain equation 4.19 in this article. First let me summarized some equations we need Denoting the gauge ...
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### Independent Components of the Riemann Curvature Tensor

I am struggling to understand a general method to calculate the independent components of the Riemann Curvature Tensor (RCT). Firstly, as far as I am aware the number of independent components of the ...
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Given, $$T^{\mu\nu} = F^{\mu\lambda} F^\nu{}_{\lambda} - \frac{1}{4} \eta^{\mu\nu} F^{\lambda\sigma} F_{\lambda\sigma}.$$ Here $(T^{\mu\nu})$ is the energy-momentum ...