Second Rank Tensor
A Second Rank Tensor is a tensor with rank of 2.
- See: Matrix.
References
2012
- Are matrices and second rank tensors the same thing?
- QUOTE: Matrices are often first introduced to students to represent linear transformations taking vectors from [math]\displaystyle{ ℝ^n }[/math] and mapping them to vectors in [math]\displaystyle{ ℝ^m }[/math]. A given linear transformation may be represented by infinitely many different matrices depending on the basis vectors chosen for [math]\displaystyle{ ℝ^n }[/math] and [math]\displaystyle{ ℝ^m }[/math], and a well-defined transformation law allows one to rewrite the linear operation for each choice of basis vectors.
Second rank tensors are quite similar, but there is one important difference that comes up for applications in which non-Euclidean (non-flat) distance metrics are considered, such as general relativity. 2nd rank tensors may map not just ℝn to ℝm, but may also map between the dual spaces of either [math]\displaystyle{ ℝ^n }[/math] or [math]\displaystyle{ ℝ^m }[/math]. The transformation law for tensors is similar to the one first learned for linear operators, but allows for the added flexibility of allowing the tensor to switch between acting on dual spaces or not.
Note that for Euclidean distance metrics, the dual space and the original vector space are the same, so this distinction doesn't matter in that case.
- QUOTE: Matrices are often first introduced to students to represent linear transformations taking vectors from [math]\displaystyle{ ℝ^n }[/math] and mapping them to vectors in [math]\displaystyle{ ℝ^m }[/math]. A given linear transformation may be represented by infinitely many different matrices depending on the basis vectors chosen for [math]\displaystyle{ ℝ^n }[/math] and [math]\displaystyle{ ℝ^m }[/math], and a well-defined transformation law allows one to rewrite the linear operation for each choice of basis vectors.