Axes_transform_and_tensor_notation

Axes_transform_and_tensor_notation - Axes Transformation...

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Axes Transformation Rules and Tensor Notation Rules for Cartesian Tensors Author: John M. Cimbala, Penn State University Latest revision: 20 September 2007 Axes Transformation Rules Let unprimed coordinates x 1 , x 2 , and x 3 denote the original axes. x 1 x 2 x 1 x 2 α 11 22 21 12 Let primed coordinates x 1 , x 2 , and x 3 denote the rotated axes. Let C ij be the cosine matrix , defined as cos ij ij C , where ij is 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 = A 1 1 3 first-order tensor vector A m = C im A i 2 2 9 second-order tensor “tensor” A mn = C im C jn A ij 3 3 27 third-order tensor - A mnp = C im C jn C kp A ijk 4 4 81 fourth-order tensor - A mnpq = C im
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This note was uploaded on 07/23/2008 for the course ME 521 taught by Professor Cimbala during the Fall '07 term at Pennsylvania State University, University Park.

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