After we compose the Levi-Civita symbol like a determinant like we did above, the totally antisymmetric property that it have results in being obvious: swapping any two indices corresponds to interchanging their corresponding rows while in the matrix as a result of which the determinant, that is the Levi-Civita itself, variations indicator.
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RexcirusRexcirus 43544 silver badges1515 bronze badges $endgroup$ 2 $begingroup$ Your ultimate expression doesn't make much perception: The LHS has free of charge indices $i,j,k$ While your RHS has totally free indices $i,j,l$. $endgroup$
ε i j ε i n = δ j n displaystyle varepsilon _ ij varepsilon ^ in = delta _ j ^ n
These properties are essential for comprehension the habits with the tensor in different coordinate units.
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The covariant Levi-Civita tensor (also referred to as the Riemannian quantity form) in any coordinate technique that matches the chosen orientation is
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