Levi-Civita tensor


to see

\varepsilon^i\wedge\varepsilon^j\wedge\varepsilon^k\wedge\varepsilon^l(e_s,e_t,e_u,e_v)={\varepsilon^{ijkl}}_{stuv}

since we are requiring “canonical” duality, i.e.  covectors, \varepsilon^k:V\to R, do

\varepsilon^k(e_l)={\delta^k}_l.

one uses

\varepsilon^i\!\wedge\!\varepsilon^j\!\wedge\!\varepsilon^k\!\wedge\!\varepsilon^l\!=\!\!\sum_{\sigma\in S_4}\!(\!-1\!)^{\sigma}\!\varepsilon^{\sigma(i)}\!\otimes\!\varepsilon^{\sigma(j)}\!\otimes\!\varepsilon^{\sigma(k)}\!\otimes\!\varepsilon^{\sigma(l)}

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2 Comments

Filed under differential geometry, fiber bundle, geometry, multilinear algebra

2 responses to “Levi-Civita tensor

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