Example 1

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Derive the transformation rule for second order tensors ( ). Express this relation in matrix notation.

Solution

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A second-order tensor   transforms a vector   into another vector  . Thus,

 

In index and matrix notation,

 

Let us determine the change in the components of   with change the basis from ( ) to ( ). The vectors   and  , and the tensor   remain the same. What changes are the components with respect to a given basis. Therefore, we can write

 

Now, using the vector transformation rule,

 

Plugging the first of equation (3) into equation (2) we get,

 

Substituting for   in equation~(4) using equation~(1),

 

Substituting for   in equation (5) using equation (3),

 

Therefore, if   is an arbitrary vector,

 

which is the transformation rule for second order tensors.

Therefore, in matrix notation, the transformation rule can be written as