% Bernd Berg Nov 4 2018.
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\centerline{\bf Special and General Relativity (PHZ 4601/5606) Fall 2018 
Solutions} \medskip

\centerline{\bf Set 11} 

\begin{description} % After Rindler 10.2.
\item {27. Some relations between covariant and ordinary differentiation}
\end{description}

\noindent For vectors:
$$ A_{i:j} = A_{i,j} + A_a\,\Gamma^a_{ij}~~{\rm and}~~
   \Gamma^a_{ij}=\Gamma^a_{ji} $$
holds. Therefore, 
$$ A_{i:j}-A_{j:i} = A_{i,j}-A_{j,i}\,. $$
For antisymmetric tensors: Adding the equations
\begin{eqnarray} \nonumber
  T_{ij:k} &=& T_{ij,k} - T_{aj}\,\Gamma^a_{ik} - T_{ia}\,\Gamma^a_{jk}\,,
  \\ \nonumber
  T_{jk:i} &=& T_{jk,i} - T_{ak}\,\Gamma^a_{ji} - T_{ja}\,\Gamma^a_{ik}\,,
\end{eqnarray} 
gives
\begin{eqnarray} \nonumber
  \qquad\qquad\qquad T_{ij:k} + T_{jk:i} &=& T_{ij,k} + T_{jk,i} 
  - T_{ia}\,\Gamma^a_{jk} - T_{ak}\,\Gamma^a_{ji}\,, \qquad\qquad (1)
\end{eqnarray} 
where two terms cancelled one another due to antisymmetry of $T_{ij}$.
Adding now to (1)
\begin{eqnarray} \nonumber
  T_{ki:j}&=&T_{ki,j}-T_{ai}\,\Gamma^a_{kj}-T_{ka}\,\Gamma^a_{ji}\,,
\end{eqnarray} 
using symmetry and antisymmery, gives the desired result
\begin{eqnarray} \nonumber
  T_{ij:k} + T_{jk:i} + T_{ki:j} &=& T_{ij,k} + T_{jk,i} + T_{ki,j}\,.
\end{eqnarray} 

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