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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 α 22 α 21 α 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 α , where ij α 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 C jn C kp C lq A ijkl . . . .
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