# 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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### Is every Lorentz invariant a Lorentz scalar?

All examples of lorentz invariant quantities that I have come across seem to be scalars: rest mass, proper time, spacetime interval,dot product of two 4 vectors etc. Another thing is that these are ...
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### Does completely antisymmetric tensor act on a tensor always produce a tensor or not?

So completely antisymmetric tensor $\epsilon$ act on a tensor can produce a new object. i.e. $G_{\alpha\beta}=\frac{1}{2}\epsilon_{\alpha\beta\mu\nu}F^{\mu \nu}$. However, According to Landau's ...
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### What is the physical difference between the mixed and fully contravariant Cauchy stress tensor components?

One traditional representation of the stress tensor among relativists is a rank-2 fully contravariant tensor, associating a contravariant force per unit area $t^i$ to a unit normal $n_j$ defined on a ...
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### Action in Electromagnetism expressed in differential geometry and tensor notation

$$S = -\frac{1}{4} \int F_{\mu\nu}F^{\mu\nu} = -\frac{1}{2} \int F \wedge *F$$ Trying to figure out why this identity holds true and getting stuck.
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One way to motivate the Christoffel symbol is to consider the partial derivative of a tensor, $T_\alpha$ $\frac{\partial T_\alpha}{\partial x^\gamma}=\frac{\partial^2 x^\beta}{\partial x^\alpha \... 1answer 65 views ### Tensor analysis: confusion about notation, and contra/co-variance I'm learning about tensors in the context of special relativity, and I'm a bit confused some notation. I understand a four-vector is a four dimensional vector, which is written in the form$(ct, x, y,...
On page 22 of Sean Carroll's Spacetime and Geometry, he says that tensors can act on other tensors and gives the following example: $$U^{\mu}_{\nu} = T^{\mu \rho}_{\sigma} S^{\sigma}_{\rho \nu}$$ ...