# Alternating unit tensor

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## 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*:**A**×

**B**= ε

_{ijk}

*A*

_{i}

*B*

_{j}δ

_{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*= . . . +

*f*ε

_{ij3}

*U*

_{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.