Axes Transformation Rules and Tensor Notation Rules for Cartesian Tensors Author: John M. Cimbala, Penn State University Latest revision: 20 September 2007 Axes Transformation RulesLet unprimedcoordinates x1, x2, and x3denote the originalaxes. x1x2x1′x2′α11α22α21α12Let primedcoordinates x1′, x2′, and x3′denote the rotatedaxes. Let Cijbe the cosine matrix, defined as cosijijCα≡, where ijαis defined as the angle between the old(i) and new(j) axes. (Note: The figure to the right is in two dimensions only, but can be extended easily to three dimensions.) Tensor order Number of free indices Number of elements Mathematical name Common name Transformation rule 0 0 1 zero-order tensor scalar A′= A1 1 3 first-order tensor vector A′m= CimAi2 2 9 second-order tensor “tensor” A′mn= CimCjnAij3 3 27 third-order tensor - A′mnp= CimCjnCkpAijk4 4 81 fourth-order tensor - A′mnpq= CimCjnCkpClqAijkl. . . .
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