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In almost any arbitrary curvilinear coordinate program as well as during the absence of a metric over the manifold, the Levi-Civita image as outlined over may very well be thought of as a tensor density discipline in two different ways.

The phrase "n-dimensional Levi-Civita image" refers to The reality that the amount of indices over the symbol n matches the dimensionality of the vector Room in query, which can be Euclidean or non-Euclidean, for instance, R three displaystyle mathbb R ^ 3

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In summary, the dialogue is regarding the properties in the Levi-Civita image. The primary element discusses a method that does not seem to match the predicted answer, and the second element will involve proving Yet another formula. The discussion finishes that has a final Alternative that satisfies the Qualities.

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= δ i l ( δ j m δ k n − δ j n δ k m ) − δ i m ( δ j l δ k n − δ j n δ k l ) + δ i n ( δ j l δ k m − δ j m δ k l ) .

ε i j ε i n = δ j n displaystyle varepsilon _ ij varepsilon ^ in = delta _ j ^ n