or Minkowski space. The values of the Levi-Civita image are independent of any metric tensor and coordinate system. Also, the precise phrase "symbol" emphasizes that it's not a tensor as a result of how it transforms concerning coordinate programs; however it can be interpreted to be 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 symbol enables the determinant of the sq. matrix, as well as the cross solution of two vectors in three-dimensional Euclidean House, being expressed in Einstein index notation.

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In almost any arbitrary curvilinear coordinate process and perhaps within the absence of a metric on the manifold, the Levi-Civita image as defined above may be thought to be a tensor density discipline in two other ways.

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then remaining side is zero, and suitable side is additionally zero since two of its rows are equal. In the same way for j c = j c + one displaystyle j_ c =j_ c+1