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Results tagged with tensor-calculus
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user 61966
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.
8
votes
3
answers
4k
views
Sign of the totally anti-symmetric Levi-Civita tensor $\varepsilon^{\mu_1 \ldots}$ when rais...
I am confused with the sign we get when we want to raise or lower all indices of the totally anti-symmetric tensor of any rank. Take the metric to be mostly plus ($-+\ldots+$). Then is it
$$\varepsilo …
- 2,218
0
votes
2
answers
63
views
Finding $M_{ij}$ from $J_{i} = -\frac{1}{2}\epsilon_{ijk}M_{jk}$ of the Lorentz group
I cannot understand why if
$$J_{i} = -\frac{1}{2}\epsilon_{ijk}M_{jk}$$
then
$$M_{ij}= -\epsilon_{ijk}J_{k}.$$
Here $M_{ij}$ is the generator of the 4 dimensional Lorentz algebra (although I have onl …
- 2,218
0
votes
1
answer
121
views
What is the relation between $\eta^{ab}x^2$ and $x^ax^b$?
What is the relation between $\eta^{ab}x^2$ and $x^ax^b$? Here $\eta$ is the Minkowski metric in $d=4$ and $x$ is a 4-vector.
In particular, a tensor like $$x^a x^b x^c (\eta_{ab}x^2 - x_ax_b) \tag{1 …
- 2,218
0
votes
1
answer
285
views
Prove $G^{+\rho(\mu}H^{+\nu)}{}_{\rho} = -\frac{1}{4}\eta^{\mu \nu}G^{+\rho \sigma}H^+_{\rho... [closed]
I want to prove the following fact for two antisymmetric tensors:
$$
G^{+\rho (\mu} H^{+\nu)}{}_{ \rho} = -\frac{1}{4}\eta^{\mu \nu} G^{+\rho \sigma}H_{\rho \sigma}^{+}. \tag{4.39}$$
(See e.g. …
- 2,218
4
votes
1
answer
499
views
Confusion about square bracket notation in the the Kronecker delta symbol
I am not sure I understand what the short-hand anti-symmetrization means. I.e. I know that
$$\delta_{cd}^{[ab]} ~=~ \frac{1}{2}(\delta_{c}^{a}\delta_{d}^{b} - \delta_{c}^{b}\delta_{d}^{a})$$
but how …
- 2,218
12
votes
1
answer
2k
views
Difference between Cartesian product $\times$ and tensor product $\otimes$ on groups
After a comment of John Baez to a question I asked on MathOverflow, I would like to ask what the difference between, for example, $SU(3)\times SU(2) \times U(1) $ and $SU(3) \otimes SU(2) \otimes U(1) …
- 2,218