Is the result a tensor tensor?
The result is not a tensor field, however, since it does not satisfy the trans-formation law (3.20). To obtain a derivative tensor field, we will need thecovariant derivative, as we saw in Chapter2and as we will see again later inthis chapter.
How do you prove a tensor is an atensor?
Proof. The left side of (2. 41)isadifference of tensors and is therefore atensor. Denoting this tensor temporarily as δAipq, wehavethat δAipq =RihqpAh, where bothδAipq andAhare tensors. In this equality, Ahis arbitrary,Rihqp is given, andδAipq is the resulting tensor. Now make a change-of-basistransformation.
What is the Riemann curvature tensor?
hqp is called theRiemann curvature tensor. The Riemann cur-vature tensor plays a central role in differential geometry and the theory ofmanifolds, as well as in applications of these theories. Here we do not pro-vide enough material on these subjects to really appreciate the importance ofthe Riemann tensor.
What are the elementary operations of a tensor?
The elementary operations on tensors include common vector operations (ad-dition and multiplication by a scalar) as well as operations that are uniqueto tensors. Among the latter, the most important is the tensor product. Thesecond is contraction.