anonanon 152k1212 gold badges233233 silver badges398398 bronze badges $endgroup$ four $begingroup$ That is fantastic many thanks a bunch! I also posted an analogous query right here: math.stackexchange.com/inquiries/2135276/… that You may have some insight on? $endgroup$
The Levi-Civita image is most frequently Utilized in 3 and 4 Proportions, also to some extent in two Proportions, so these are definitely offered here before defining the general circumstance. Two Proportions[edit]
RexcirusRexcirus 43544 silver badges1515 bronze badges $endgroup$ 2 $begingroup$ Your remaining expression won't make A great deal feeling: The LHS has free of charge indices $i,j,k$ whereas your RHS has no cost indices $i,j,l$. $endgroup$
I am utilizing the Einstein notation, so multiple indices are summed) or they give two three-indices symbols with unique signal
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In any arbitrary curvilinear coordinate technique and in some cases inside the absence of a metric about the manifold, the Levi-Civita symbol as defined higher than may be considered to read this be a tensor density industry in two alternative ways.
but observe that If your metric signature consists of an odd amount of detrimental eigenvalues q, then the indication of the components of this tensor differ in the normal Levi-Civita symbol:
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The important thing defining house in the image is full antisymmetry while in the indices. When any two indices are interchanged, equivalent or not, the symbol is negated:
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