After we create the Levi-Civita image like a determinant like we did above, the thoroughly antisymmetric assets that it have gets obvious: swapping any two indices corresponds to interchanging their corresponding rows from the matrix because of which the determinant, that's the Levi-Civita by itself, changes sign.
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RexcirusRexcirus 43544 silver badges1515 bronze badges $endgroup$ 2 $begingroup$ Your final expression won't make Considerably sense: The LHS has absolutely free indices $i,j,k$ whereas your RHS has free indices $i,j,l$. $endgroup$
ε i j ε i n = δ j n displaystyle varepsilon _ ij varepsilon ^ in = delta _ j ^ n
Samples of Levi-Civita properties in 4 Proportions contain the symmetric and antisymmetric mother nature from the tensor, its relation into the metric tensor, and its transformation Qualities underneath coordinate transformations.
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This tensor really should not be bewildered Along with the tensor density subject pointed out earlier mentioned. The presentation Within this section closely follows Carroll 2004.
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In almost any arbitrary curvilinear coordinate procedure and perhaps during the absence of the metric about the manifold, the Levi-Civita symbol as described over might be considered to be a tensor density industry in two alternative ways.
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exactly where gab would be the illustration with the metric in that coordinate procedure. We can similarly take into account a contravariant Levi-Civita tensor by elevating the indices Along with the metric as standard,
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