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# 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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### Why do we need invariants to represent real life quantities?

Often it is said that one of the most useful properties of eigenvalues of a matrix is that they are invariant under change of basis. This in turn is said to be useful in physics because real, physical ...
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### How to prove a 4D vector is a 4-Vector?

This is a fairly open ended question. Given a set of 4 Components, that is, a 4D Vector, what is the process for determining rather or not it is a "4-Vector" as defined in special relativity? I want ...
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### Indices of the Riemann Tensor of the first kind

When establishing the identity $V^i_{,kl}-V^i_{,lk}=-R^i_{tkl}V^t$ (, denotes covariant differentiation), one of the steps involves raising one of the indices of the Riemann Tensor of the first kind . ...
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### Why Electrical conductivity tensor is symmetric? Or is it not always symmetric?

How to show that the electrical conductivity tensor is symmetric? (or it's not always symmetric?)
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### Einstein notation: can a free index be upper in one term and lower in another term?

Consider a linear combination of terms written using Einstein notation. Consider one free index in the linear combination: is it necessary that the index is upper in all terms or lower in all terms, ...
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### Doubt about the history of the tensor object in physics

The word "tensor" (or, maybe the whole physical idea of this object) could be traced date to 1898 on the work on Crystals due to Voigt . It seems that the references  and , introduce the idea ...
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### What physical quantity remains equal in different frames? [closed]

I recently came across a problem which involved going into rotating frames. And it was pretty tiring. (And difficult..). So I wondered if there was some quantity that you could measure from the ground ...
In a curved spacetime with a general Minkowskian metric $g_{\mu\nu}$, I understand the difference between the Levi-Civita symbol $\pmb{\in}^{\mu\nu\rho\sigma}$ (which is a tensor-density of weight $W=+... 3answers 66 views ### Can I contract index in this expression? I'm reading Carrol text on general relativity, on page 96 they arrive to the term \begin{equation} \frac{\partial x^{\mu}}{\partial x^{\mu '}}\frac{\partial x^{\lambda}}{\partial x^{\lambda '}}\frac{\... 0answers 34 views ### Order of positions of tensor/vector components in an inner/outer product Show that if$T_i$are the components of covariant vector T, then$S_{ij}=T_iT_j-T_jT_i$are the components of a skew-symmetric covariant tensor S. The question is whenever working with equations of ... 1answer 64 views ### Covariant derivative in a basis Reading through this paper, I saw that the energy momentum conservation: $$\nabla_\mu T^{\mu\nu}=0$$ can be evaluated as: $$\partial_t(\sqrt{-g}T^{t}_\nu)=-\partial_i(\sqrt{-g}T^{i}_\nu)+\sqrt{-g}T^... 0answers 63 views ### A specific derivation of Yang-Mills equations of motion I am not happy about the derivation of Yang-Mills equations of motion (YM eom) given here @Prahar https://physics.stackexchange.com/a/312681/42982: @Prahar said: Yang-Mills action is$$ S = \int ... 0answers 59 views ### Variations of tensors are tensors? Recently I posted a question about variation of metric. I thought I understood it and talked with my friend about it. After that he said he's not convinced because he can't prove variation of metric ... 3answers 118 views ### Functional derivative of metric To do functional derivative of some actions, we need to know a functional differential of metrics$g_{\mu \nu}(x)\$. One of the formulae about that is: $$g_{\mu\nu}\delta g^{\mu\nu} = - g^{\mu\nu} \... 0answers 35 views ### Derivative with respect to a coordiante differential (geodesic equation) If the arc length is chosen to be the action integral, that is$$ S=\int \sqrt {g_{kn}\frac{dx^k}{ds} \frac{dx^n}{ds}} dx \tag{11.13} $$Then Lagrangian is given by$$L=\sqrt {g_{kn}\frac{dx^k}{ds}...
A rank 2-tensor transforms under Lorentz transformation in the following way: $$T^{\mu \nu} \rightarrow T^{' \mu \nu} = \Lambda^\mu _\sigma\Lambda^\nu _\rho T^{\sigma \rho}$$ The field strength ...