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A.1.2 Cartesian Components of a Tensor
Let form an orthonormal basis in a Cartesian coordinate system . Then the Cartesian components of are or equivalently, (Recall that , thus .)
The Cartesian components of a tensor are obtained as follows. Let . The components of are given by . But , so . Note that is the component along of the vector . By convention, we denote this component as
components of tensor .
Thus, . Taking the dot product on both sides with yields , or in indicial form. In matrix form, The matrix of tensor with respect to can also be denoted by or . The columns of are given by , e.g., This result, when generalized, leads to the useful identity
A scaling transformation with different scale factors along should satisfy the following relations by definition: Verify that is a tensor. Also find the matrix of in .
Solution. Is a tensor? Let any and , then and
Now that we have demonstrated that is a tensor, its components are given by , thus Then, the matrix of is given by