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Writing a tensor with respect to a particular basis
When defining tensors as multilinear maps, I am having trouble understanding why a tensor, let's say of type (2,1), can be written in the following way:
$$T = T^{\mu\nu}_{\rho} e_\mu \otimes e_\nu ...
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Conversion of motion equation from Cartesian to Polar coordinates: Is covariant differentiation necessary?
I have earlier posted the same question here on math stackexchange but without any answer. As the question concerns tensors, I guess that I have come to the right ...
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Need some basic help with notation and the Christoffel symbols
Apologies in advance if some of the questions below seem overly simple.
In an introductory GR book, I find the following expression for the autoparallel of the affine connection (the upper bound of ...
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Contraction of the metric tensor
This is perhaps a simple tensor calculus problem -- but I just can't see why...
I have notes (in GR) that contains a proof of the statement
In space of constant sectional curvature, $K$ is ...
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Zero-zero (lower indicies) term for affine connection ($\Gamma_{00}^\lambda$), why do some terms dissapear?
More simply a tensor algebra question, but in General relativity I have the following when I calculate $\Gamma_{00}^\lambda$:-
$$
\Gamma_{00}^\lambda = \frac{1}{2}g^{\nu\lambda}\left(
\frac{\partial ...
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Relativistic basic question - four vector, Lorentz matrix
I have heard relativistics only very compressed during my student time. Now I looked up the definitions again and a question comes into my mind:
A contravariant vector is transformed like this: ...
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covarient derivative of electromagnetic field tensor
I'm trying to prove the energy momentum tensor in curved spacetime for Electromagnetic field is Divergence-less directly(Without using general lie derivative method which can prove any energy momentum ...
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Vector identities equivalence under different coordinates
I've learned to represent curl, rot and Laplacian in the general form using scaling factors, Levi Civita symbol and delta.
I was asked to prove some general identities in vector calculus.
I was ...
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Positive Permutation Tensor
I have seen that one can make an operator with
$$
L^i=\epsilon^{ijk}x_{j}\partial_{k}
$$
But what if I want to make instead items that are sums, instead of differences. For instance
...
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How do I write the energy of a constant, uniform 2D charge distribution?
Let's consider a 2D electromagnetic field defined in a square domain $[0,\Lambda]^2$, with periodic boundary conditions, with a constant charge distribution, uniform all over the aforementioned ...
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Determine the tensor of contraint and deformation of a cube under compression
We have a cube under compression with dimension l1*l2*l3, is put between 2 rigid plates in the axis 1 (two plates block the deformation of the cube in thí axis), the cube is also put on a rigid plate, ...
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Invariants of a tensor [closed]
I asked (almost) the same question in the math exchange.
I'm teaching a course, and I need a simple and intuitive proof that the invariants of a matrix ($3\times3$, but it doesn't matter) can be ...
