# 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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### Question on covariant derivative calculations

I am trying to show by explicit calculation that $$\nabla_{\mu}T^{\nu}_{\phantom{\nu}\lambda}=g_{\lambda\xi}\nabla_{\mu}T^{\nu\xi}\tag{1}$$ where the covariant derivative commutes with the metric ...
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### Deriving the transformation law for the Christoffel symbols

I am a first year undergraduate teaching myself General Relativity from the book by Bernard Schutz. In one of the problems he asks to derive the transformation law for the Christoffel symbols from the ...
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### Functional derivatives of tensors in phase space in the context of General Relativity

In classical mechanics the functional derivative of a scalar function $f(x,\dot{x})$ respect to the trajectory $x^i(t)$ is \begin{equation} \frac{\delta f\big[x(t),\dot{x}(t)\big]}{\delta x^i(t')}=\...
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### How to verify the gauge invariance of the term $\mathcal{L}= -\frac{1}{2}tr[(F^{i}_{\mu \nu}\sigma^i)^2]$?

This is equation (15.38) from Peskin and Schroeder. I am unable to compute this term to verify if it is invariant (I know that it is but I'd like to verify that). I would appreciate it if someone can ...
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