or Minkowski Area. The values of the Levi-Civita symbol are unbiased of any metric tensor and coordinate program. Also, the specific time period "symbol" emphasizes that it is not a tensor because of how it transforms between coordinate units; on the other hand it may be interpreted as a tensor density.
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ε i j ε i n = δ j n displaystyle varepsilon _ ij varepsilon ^ in = delta _ j ^ n
The Levi-Civita image makes it possible for the determinant of a square matrix, as well as cross product or service of two vectors in 3-dimensional Euclidean Area, to be expressed in Einstein index notation.
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In any arbitrary curvilinear coordinate method and also inside the absence of a metric around the manifold, the Levi-Civita symbol as described higher than may very well be considered to be a tensor density industry in two various ways.
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then left aspect is zero, and ideal side is likewise zero because two of its rows are equal. In the same way for j c = j c + one displaystyle j_ c =j_ c+1