1.
Show that the symmetric or antisymmetric property of a tensor is conserved under a transformation of coordinates.
Exercises 3.5 Exercise
2.
Show that the quantities \(\delta_{q}^{p}A_{p}\) transform like the components of a tensor.
3.
Prove that \(\delta_{ik} u_{k} =u_{i}\text{.}\)
4.
If \(A_{ij}\) is an antisymmetric tensor, Show that \(\delta_{j}^{i}A_{ij} =0\text{.}\)
5.
If \(A_{ij}\) is a skewsymmetric tensor of rank two, then prove that \((\delta_{ij}\delta_{lk} +\delta_{il}\delta_{jk})T_{ik} = 0\text{.}\)
6.
Show that if a physical quantity has no component in one coordinate system, then it does not have a component in any other coordinate systems.
7.
How many independent commponents are there in an antisymmetric tensor of rank two?
8.
Show that the direct product of a \(0^{th}\) rank tensor and a \(2^{nd}\) rank tensor is a tensor of rank two.
9.
Prove that the sum of diagonal components of a second-rank tensor is an invariant.
10.
State the transformation properties of tensors \(A_{i}\) and \(B^{ij}\text{.}\) Obtain the transformation properties of \(A_{i}B^{ij}\text{.}\) Explain your result.
11.
Contract and provide direct product of two tensors \(A_{\sigma}^{\mu \nu}\) and \(B_{l}^{m}\text{.}\) What will be the rank of the contracted tensor and the direct product?
12.
Show that any tensor of rank 2 can be expressed as the sum of a symmetric and an antisymmetric tensors of rank 2.
Hint.
Any tensor \(A^{ij}\) of rank 2, may be expressed as
\begin{equation*}
A^{ij} = \frac{1}{2}\left(A^{ij}+A^{ji}\right)+\frac{1}{2}\left(A^{ij}-A^{ji}\right)
\end{equation*}
13.
\(\vec{\omega}\) is any arbitrary contravarient vector. It is known that \(A_{ij}\vec{\omega}\) is a covariant vector. Show that \(A_{ij}\) is a covariant tensor of rank 2.
14.
Using the inner product of a tensor and applying contraction principle obtain the length L of a tensor \(A^{i}\text{.}\)
Hint.
take direct product of \(A^{i}\) and \(A_{j}\text{.}\)
