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What is a tensor?

I think groups representations are the most natural way to understand tensors. The following parts can be read essentially independently. The 2nd is the most relevant. Why tensors in physics can be ...
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1 vote
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Are tensors constructed such that one forms "act" on some complex vector field?

Is this supposed to be interpreted as $\omega$ 'acting on' some vector field, such that we get a number? Yes. [...] I am not sure if this object is a vector field at all. It is. I'm not entirely ...
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Expressing Maxwell's equations in tensor form using Electromagnetic field strength tensor

Since $F_{ij} = \epsilon_{ijk}B^k$ one has $\epsilon^{lij} F_{ij} = \epsilon^{lij}\epsilon_{ijk}B^k = 2\delta_{lk} B^k$, hence $B^l = \frac{1}{2} \epsilon^{lij} F_{ij}$. The third Maxwell's equation ...
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Expressing Maxwell's equations in tensor notation

The equation $$\partial_\mu F^{\nu\mu} = J^\nu$$ is to hold for all $\nu$, i.e. for $\nu=0$ and for all $\nu=i$. For $\nu=0$ this reads $$\partial_\mu F^{0\mu} = J^0 = \partial_i F^{0 i}$$ ...
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What is the idea behind 2-spinor calculus?

May I suggest a somewhat intuitive way to try to answer the question, inspired by the late Sir Michael Atiyah's view that "spinors are the square root of geometry". Atiyah argues in his ...
• 928
1 vote
Accepted

A particular contraction of Levi-Civita symbols and tetrads

It is zero. In $\epsilon_{\alpha\beta ij}$, all indices have to be distinct. So one of them must be $0$, and therefore $e^\alpha\wedge e^\beta\wedge e^0\wedge e^k$ vanishes, since $0$ appears twice. ...

Kerr Solution metric ansatz for EFEs

Pedagogical derivations (the ones I found by googling for a few minutes) seem to prefer the use of "oblate spheroidal coordinates" (OSC) as a good starting point to describe static, axially-...
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1 vote

Finding the proportionality constant in $\varepsilon^{\mu\nu}A_\mu^{\ \lambda} A_\nu^{\ \rho}\propto \varepsilon^{\lambda\rho}$

The properties of the epsilon tensor (as exploited by Cosmas Zachos) give a very elegant answer here. But, if you didn't know about those identities, you can still get the answer in a straightforward ...
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Accepted

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The foundations of geometric formulation of Newton's axioms

I think I sort of understand the $\nabla (dt)$ condition. Suppose we have a room and consider any point in the room, in the Newtonian case, it must be that if you kept a clock at each point, then , ...
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Argument of a scalar function to be invariant under Lorentz transformations

A scalar Lorentz invariant function satisfies $$f(k) = f(\Lambda k).$$ for all $\Lambda$ satisfying $\Lambda^T \eta \Lambda = \eta$. Let us look at the infinitesimal version of this equation. ...
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Preservation of symmetries of Tensors under lowering and raising indices

If $$T^{ij} = T^{ji},$$ Then $$T^{ij}g_{jk} = T^{ji}g_{jk}$$ $$T^{i}_{\ \ k} = T_{k}^{\ i} = (T^{i}_{\ \ k})^T$$ And $$T^{i}_{\ k}g_{li} = T_{k}^{\ i}g_{li}$$ $$T_{lk} = T_{kl},$$
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Metric tensor from hyperbolic PDE

Suppose that we have an equation of the form $$g^{\rho \sigma} \frac{\partial^2 \psi}{\partial x^\rho \partial x^\sigma} + A^\tau \frac{\partial \psi}{\partial x^\tau} = 0 \tag{1}$$ and we suspect ...

Why is a Lorentzian metric still Lorentzian after a general coordinate transformation?

Under general coordinate transformations $$g' = M g M^T , \qquad M_\mu{}^\alpha = \frac{\partial x^\alpha}{\partial x'^\mu} \in GL(n,{\mathbb R}) .$$ It's clear that $\det g' = \det g ( \det M )^2$...
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Software recommendation for a tensor calculation

The free Mathematica package xTensor can perform calculations such as this. You will also need to companion package xCoba, ...
It seems like you're curious as to what the three different components of the stress tensor mean. Roughly speaking, $T_{xx}$, $T_{yy}$, and $T_{zz}$ tell you how much force per area is being exerted ...