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ε i j ε i n = δ j n displaystyle varepsilon _ ij varepsilon ^ in = delta _ j ^ n
These Houses are essential for comprehending the habits on the tensor in numerous coordinate programs.
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This tensor shouldn't be perplexed Using the tensor density area described previously mentioned. The presentation In this particular portion intently follows Carroll 2004.
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Utilizing the funds pi notation Π for regular multiplication of numbers, an explicit expression for that image is:[citation needed]
but detect that When the metric signature includes an odd variety of adverse eigenvalues q, then the indicator with the parts of this tensor differ from your normal Levi-Civita symbol:
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On the pseudo-Riemannian manifold, just one may define a coordinate-invariant covariant tensor area whose coordinate representation agrees Along with the Levi-Civita symbol where ever the coordinate technique is these types of that The premise with the tangent House is orthonormal with regard to the metric and matches a selected orientation.
The real key defining property with the image is total antisymmetry inside the indices. When any two indices are interchanged, equivalent read here or not, the image is negated:
Up to a few sets of CTs for each stage is often paralleled for every meter with no will need for your meter multiplier