# 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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### Second order tensors as linear operators

A tensor is formally defined as an object whose components obey some transformation rules. I, however, find it more intuitive to look at (second order) tensors as a linear operator/function between ...
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### If $S_{\mu\nu\sigma} V^{\mu}V^{\nu}V^{\sigma} = T_{\mu\nu\sigma} V^{\mu}V^{\nu}V^{\sigma}$, then is it true $S_{\mu\nu\sigma} = T_{\mu\nu\sigma}$?

For any vector $V$, suppose that the following equality holds $$S_{\mu\nu\sigma} V^{\mu}V^{\nu}V^{\sigma} = T_{\mu\nu\sigma} V^{\mu}V^{\nu}V^{\sigma}$$ for two tensors $S$ and $T$. Does it follow ...
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### Every symmetric second rank covariant tensor can be transformed into diagonal form with diagonal elements 0,±1

I started learning about tensors, and this theorem was mentioned: "Every symmetric second rank covariant tensor can be transformed into diagonal form in which the diagonal elements are either 1,0 or −...
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### I want to know why engineering strain is not a tensor

One of the most classical examples in the mechanics of materials is that engineering strain is not tensor. I want to know why the engineering strain doesn't meet the tensor definitation. ...
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### About variation of Ricci tensor

I have been doing some calculation on variation of Ricci's tensor with respect to the metric, that, according with S. Carroll (An Introduction to General Relativity: Spacetime and Geometry, equation 4....
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### How can I show the Einstein Tensor using second Identity of Bianchi? [closed]

The Einstein tensor given by: $$G_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}$$ Can be shown using Bianchi identity?
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### What is the implication if the Kretschmann scalar for two metrics are same? [closed]

Does it mean that they are describing the same spacetime?
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### How can I prove that $\Gamma_{kij}+\Gamma_{kji}=\partial_k g_{ij}$? [closed]

I want a simple proof of this identity: $$\Gamma_{kij}+\Gamma_{kji}=\partial_k g_{ij}$$ If there's no answer, give me a hint or something would help to prove it, and thanks!
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### MTW the contraction of a basis bi-vector with a basis two-form. Why am I getting this factor of 2?

My question pertains to the discussion of two-forms and bi-vectors in MTW, Chapters 3 and 4. I set out to understand how the expression for the contraction of a basis p-vector with a basis p-form ...
### Sign of components of $\vec{L}$ seems to change under parity; book says otherwise
Book: Gravitational Waves-Vol 1 by Maggiore; pg 147, note 48 The book says that the components of angular momentum $\vec{L}$ are unchanged under parity (reversing the orientation of the axes). Now, ...