8
votes
Accepted
What kind of tensor is the electromagnetic field tensor (Faraday tensor)
Every quantity that you are used to, in physics, are usually defined in terms of the upper indices. The nice property of this is that, because upper indices correspond to vector components, this means ...
- 11.2k
4
votes
Accepted
Understanding the definition of the covariant derivative
In the case of a scalar field, the covariant derivative equals the partial derivative.
In the case of a tensor field of higher rank, there are further terms involving connection coefficients: for rank ...
- 61.8k
3
votes
Understanding the definition of the covariant derivative
The covariant derivative of a covector field $V_\mu$ is:
$$ V_{\mu;\nu} = V_{\mu,\nu} - \Gamma^\lambda{}_{\mu\nu} V_{\lambda} $$
The covariant derivative of a scalar field is just its partial ...
- 3,313
3
votes
The definition of the Lie Derivative
It's just because you want to compare two objects at the same point. In differential geometry you can NOT compare objects at different points since they live in different spaces. The pull back allows ...
- 541
2
votes
When transfoming an intertia tensor from one set of principle axes to another, why does it not change the tensor?
The inertia tensor of a cube with edge length a is
$$\mathbf I_1=\frac M6\,a^2\,\begin{bmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 \\
\end{bmatrix}
$$
with arbitrary ...
- 12.9k
2
votes
What kind of tensor is the electromagnetic field tensor (Faraday tensor)
Fundamentally, as is the case with gauge fields in general, it's a second degree differential form, $F = ½ F_{μν} dx^μ ∧ dx^ν$, so that its components are those of an anti-symmetric rank two covariant ...
- 2,383
1
vote
Parallel transport of a vector on a $2d$ plane
Parallel vector field along a curve means that the "covariant differential of a vector along the curve vanishes" which implies
\begin{align}
\dot{A}^\mu+\Gamma^\mu_{\nu\lambda}A^\nu\dot{x}^\...
- 66
1
vote
Question about the the velocity and acceleration in tensor notation
This answer is based on Pavel Grinfeld's Youtube lecture.
The trajectory is parameterized with respect to time as a given. That is, the contravariant bases are expressed as a function of time $Z^i \...
- 11
1
vote
Derivative for the Maxwell field
$$\frac{\partial(\partial_{\mu}A^{\sigma})}{\partial(\partial^{\nu}A_{\lambda})}$$
I can't understand whether I must raise the lower index of the partial derivative, and lower the one of the vector ...
- 23.3k
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