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RexcirusRexcirus 43544 silver badges1515 bronze badges $endgroup$ two $begingroup$ Your closing expression does not make Substantially sense: The LHS has cost-free indices $i,j,k$ Whilst your RHS has cost-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 important for comprehending the habits from the tensor in several coordinate methods.
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For every sequence of nonempty open up sets You will find there's disjoint sequence of nonempty open up sets "underneath" it
The subsequent are samples of the overall identity previously mentioned specialised to Minkowski Place (Together with the destructive indicator arising through the odd quantity of negatives while in the signature from the metric tensor in both signal Conference):
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Over a pseudo-Riemannian manifold, one may outline a coordinate-invariant covariant tensor field whose coordinate illustration agrees with the Levi-Civita symbol wherever the coordinate program is these that The idea in the tangent House is orthonormal with respect for the metric and matches a selected orientation.
The crucial element defining property in the image is full antisymmetry from the indices. When any two indices are interchanged, equivalent or not, the symbol is negated:
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