Alternating unit tensor

alternating unit tensor

A mathematical function with symbol ε_ijk defined to switch between the discrete values of +1, 0, and -1, depending on the values of the three indices i, j, and k:

It is one of the tools used in Einstein's summation notation to handle operations equivalent to cross products in vector notation. For example: A × B = ε_ijkA_iB_jδ_k, where A_i and B_j represent vectors in summation notation, and δ_k is the unit vector. Subscripts or indices i, j, and k must each take on the values of 1, 2, and 3, representing the Cartesian directions of x, y, and z. The alternating unit tensor is useful to describe the Coriolis term in the equations of motion: ∂U_i/ ∂t = . . . + fε_ij3U_j + . . ., where f is the Coriolis parameter, U_i and U_j are wind components, and t is time. Summation notation is commonly used in atmospheric turbulence studies.

Stull, R. B. 1988. An Introduction to Boundary Layer Meteorology. 666 pp.

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A mathematical function with symbol ε_ijk defined to switch between the discrete values of +1, 0, and -1, depending on the values of the three indices i, j, and k: