A tensor is a mathematical object that describes the relationships between different coordinate systems. In physics, tensors are used to describe physical quantities such as velocity, force, and stress.
Commutation is the act of switching the order of two mathematical operations. In the context of tensors, it refers to the order in which indices are written.
Commutation is important in proving tensor commutation because it allows us to test whether two tensors are equal. If the order of indices can be switched without changing the result, then the tensors are said to commute.
T^abc S_b vs S_b T^abc refers to the commutation of two tensors, T and S. The notation T^abc indicates a tensor with three indices (a, b, and c), while S_b indicates a tensor with one index (b). The vs indicates the order in which the tensors are written.
To prove tensor commutation in this case, we can use the properties of tensors to rearrange the indices and show that the two tensors are equal. This can involve using the commutative property of addition and multiplication, as well as the fact that repeated indices can be summed over. Additionally, we can use the Einstein summation convention to simplify notation.