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.

1. The study of the physics, chemistry, and dynamics of the earth's atmosphere, including the related effects at the air–earth boundary over both land and the oceans. Fundamental topics include the composition, structure, and motion of the atmosphere. The goals ascribed to meteorology are the complete understanding and accurate prediction of atmospheric phenomena.