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Theory Manual Version 3.6
 Subsection A.1.11: Orthogonal Tensor Up Section A.1: Second-Order Tensors Subsection A.1.13: Symmetric and Antisymmetric Tensors 

A.1.12 Transformation Laws for Cartesian Components of Vectors and Tensors

figure ../Figures/FigOrthoBases.png
Figure A.1 Orthonormal bases and .
Let and be two orthogonal bases in a Cartesian coordinate system. could be made to coincide with through a rigid body rotation (i.e., a transformation that preserves vector length and angles), where . Since , the components of are direction cosines between and .
Rotation about
figure ../Figures/FigRotationAboutX3.png
Figure A.2 Rotation about .
Reflection about plane, .
For any vector , its components with respect to and are and , respectively. Using the above relation,
or In matrix form, or Here and are matrices of the same vector, expressed in two different coordinate systems. This is not the same as , where is the linear transformation of by .
Now consider a tensors . Its components with respect to and are given by and , respectively. Thus, , or In matrix form, , or Equivalently, we can show that or . As for vectors, we note that and are the matrices of the same tensor , with respect to two different coordinate systems. This is not the same as .
 Subsection A.1.11: Orthogonal Tensor Up Section A.1: Second-Order Tensors Subsection A.1.13: Symmetric and Antisymmetric Tensors