# 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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### Are there physical quantities constitute of magnitude, direction and rotation along that direction?

There are scalar quantities(magnitude) and vector quantities(magnitude and direction), but are there fundamental quantities that also depends on how it's oriented/rotated along the direction(magnitude,...
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### Yang-Mills Bianchi identity in tensor notation vs form notation

I've seen the Yang-Mills Bianchi identity written as both $$0 = dF^a + f^{abc} A^b \wedge F^c$$ and, in tensor notation, as $$\epsilon^{\mu\nu\lambda\sigma}D_{\nu} F^a_{\lambda\sigma} = 0.$$ Here ...
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### Trace of second-order tensor and its invariance under coordinate transformation

Let's consider an arbitrary scalar field. If I act twice on the scalar field with a gradient operator, I will obtain second-order tensor. If I will take a trace of this tensor, I will obtain another ...
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### Vector Calculus Recommendations [duplicate]

Is there any book which teaches multivariable calculus from a physics perspective? I understand math a lot better when it is applied to physics, and I was wondering if there is a "physical" approach ...
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### Can we generalize matrix model theory?

As in the question, can matrix model theory be generalized to a tensor model theory? Will the results be different or useful in describing real world phenomena? Details: in matrix model theory we ...
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### Is the inertia tensor a tensor field?

The inertia tensor seems like it cannot depend in any way on position, but every other tensor in physics is a tensor field (stress tensor, electromagnetic tensor...) so, which is it?
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### Looking for physical intuition into the Electromagnetic Tensor:

I have done some work with the electromagnetic tensor and I'm fairly good at manipulating it and using it to transform the Maxwell Equations into tensored forms. Admittedly though, I have no physical ...
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### Do the Christoffel symbols $\Gamma_{rn}^w\partial_sV_w = \Gamma_{sn}^w\partial_rV_w$?

In lecture 3 (about 97 min into the lecture) of Leonard Susskind's general relativity course, he suggests finding the Riemann curvature tensor in terms of the Christoffel symbols as an exercise. I ...
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### Using symmetry of Riemann tensor to vanish components

The Riemann tensor is skew-symmetric in its first and last pair of indices, i.e., \begin{align} R_{abcd} = -R_{abdc} = -R_{bacd} \end{align} Can I simply use this to say that, for example, the ...
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### Dirac Notation Tensor product

We can write a Singlet state of two $\frac{1}{2}$ spin particles like this: $$|S\rangle = \frac{1}{\sqrt{2}}\left( |+ \rangle ⊗ |-\rangle - |-\rangle ⊗|+\rangle \right)$$ is this the same as ...
I need to expresse the electromagnetic energy-momentum tensor in a vacuum $$T^\nu_{\ \ \ \mu} = F_{\mu\alpha}F^{\nu\alpha} - \frac14 F_{\alpha\beta}F^{\alpha\beta}\delta^\nu_{\ \ \mu}$$ in terms of ...