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\begin{document}
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\begin{center}
{\large\bf Covariant Differentiation}
\end{center}
\bigskip

\large
In the following we use the notation of Rindler (7.1), p.132 
for partial derivatives:
\begin{eqnarray} \label{p}
  p^{i'}_i = \frac{\partial x^{i'}}{\partial x^i}\,,~~~
  p^i_{i'} = \frac{\partial x^i}{\partial x^{i'}}\,,~~~
  p^{i'}_{ij} = \frac{\partial^2 x^{i'}}{\partial x^i\,
  \partial x^j}\,.
\end{eqnarray}
Note that $p^{i'}_{ik}\,p^k_{k'}$ should be used for
$$ \frac{\partial~}{\partial x^{k'}}\, p^{i'}_i\,, $$
because in $p^{i'}_{ik'}$ {\bf the order} of the $x^i$ and $x^{k'}$ 
differentiations would matter, while $i$ and $k$ commute in 
$p^{i'}_{ik}\,p^k_{k'}$ (see exercise~23).
\bigskip

%\noindent 
{\bf Definition:}
\begin{eqnarray} \label{covdifdef}
  F^i_{~j;k} = F^{i'}_{~j',k'}\,p^i_{i'}\,p^{j'}_j\,p^{k'}_k\,.
\end{eqnarray}
Here $_{;k}$ indicates covariant differentiation, $_{,k'}$ is
ordinary differentiation. The LHS is in an arbitrary coordinate
system $\{x^i\}$ and the RHS in a geodesic (freely falling)
coordinate system $\{x^{i'}\}$. Let us re-write the derivative
$F^{i'}_{~j',k'}$ of the RHS in terms of 2-tensors in the $\{x^i\}$
frame:
\begin{eqnarray} \nonumber 
  F^{i'}_{~j',k'} &=& \frac{\partial~}{\partial x^{k'}}\,F^{i'}_{~j'}
                 = \frac{\partial~}{\partial x^{k'}}\,\left(
                   F^l_{~m}\,p^{i'}_l\,p^m_{j'}\,\right) 
  \\ \label{Fprime}&=& F^l_{~m,n}\,p^{i'}_l\,p^m_{j'}\,p^n_{k'} 
                 + F^l_{~m}\,p^{i'}_{l\,n}\,p^n_{k'}\,p^m_{j'}
                 + F^l_{~m}\,p^{i'}_l\,p^m_{j'\,k'}\,.
\end{eqnarray}
Now we work out the contractions of the $p$ factors in (\ref{Fprime}) 
with the $p^i_{i'}\,p^{j'}_j\,p^{k'}_k$ factor in (\ref{covdifdef}):
\begin{eqnarray} \label{p1}
  p^{i'}_l\,p^m_{j'}\,p^n_{k'}\ p^i_{i'}\,p^{j'}_j\,p^{k'}_k 
  &=& \delta^i_l\,\delta^m_j\,\delta^n_k \\ \label{p2}
  p^{i'}_{l\,n}\,p^n_{k'}\,p^m_{j'}\ p^i_{i'}\,p^{j'}_j\,p^{k'}_k
  &=& p^i_{i'}\,p^{i'}_{l\,n}\,\delta^n_k\,\delta^m_j 
  = \Gamma^i_{~l\,k}\,\delta^m_j \\ \label{p3}
  p^{i'}_l\,p^m_{j'\,k'}\ p^i_{i'}\,p^{j'}_j\,p^{k'}_k
  &=& p^m_{j'\,k'}\,p^{j'}_j\,p^{k'}_k\,\delta^i_l 
  = - \Gamma^m_{~j\,k}\,\delta^i_l\,.
\end{eqnarray}
Carrying out the $\delta$ contractions we find:
\begin{eqnarray} \nonumber
  F^i_{~j;k} = F^{i'}_{~j',k'}\,p^i_{i'}\,p^{j'}_j\,p^{k'}_k 
  &=& F^l_{~m,n}\,\delta^i_l\,\delta^m_j\,\delta^n_k + F^l_{~m}\,
  \Gamma^i_{~l\,k}\,\delta^m_j-F^l_{~m}\,\Gamma^m_{~j\,k}\,\delta^i_l
  \\ \label{covdif} &=& F^i_{~j,k} + F^l_{~j}\,\Gamma^i_{~l\,k} 
                     -              F^i_{~m}\,\Gamma^m_{~j\,k}
\end{eqnarray}
in agreement with Rindler (10.28), p.211. Notably, the auxiliary geodesic
system has been eliminated. Application of Eq.~(\ref{covdif}) are given in
the following. \medskip

\noindent Scalar:
\begin{eqnarray} 
  \Phi_{;i} = \Phi_{,i}\,.
\end{eqnarray}
Contravariant Vector:
\begin{eqnarray} 
  V^i_{~;j} = V^i_{~,j} + V^a\,\Gamma^i_{~aj}\,.
\end{eqnarray}
Covariant Vector:
\begin{eqnarray} 
  V_{i;j} = V_{i,j} - V_a\,\Gamma^a_{~ij}\,.
\end{eqnarray}
Contravariant rank two tensor:
\begin{eqnarray} 
  F^{ij}_{~~;k} = F^{ij}_{~~,k} + F^{aj}\,\Gamma^i_{~ak}
                                + F^{ia}\,\Gamma^j_{~ak}\,.
\end{eqnarray}
Covariant rank two tensor:
\begin{eqnarray} 
  F_{ij;k} = F_{ij,k} - F_{aj}\,\Gamma^a_{~ik}
                      - F_{ia}\,\Gamma^a_{~jk}\,.
\end{eqnarray}
Contravariant rank three tensor:
\begin{eqnarray} 
  T^{i_1i_2i_3}_{~~~~~;j} = T^{i_1i_2i_3}_{~~~~~,j} 
                          + T^{ai_2i_3}\,\Gamma^{i_1}_{~aj}
                          + T^{i_1ai_3}\,\Gamma^{i_2}_{~aj}
                          + T^{i_1i_2a}\,\Gamma^{i_3}_{~aj}
\end{eqnarray}
Covariant rank three tensor:
\begin{eqnarray} 
  T_{i_1i_2i_3;j} = T_{i_1i_2i_3,j} - T_{ai_2i_3}\,\Gamma^a_{~i_1j}
                                    - T_{i_1ai_3}\,\Gamma^a_{~i_2j}
                                    - T_{i_1i_2a}\,\Gamma^a_{~i_3j}\,.
\end{eqnarray}
For mixed tensors the positive and negative contributions are simply
added up, e.g.,
\begin{eqnarray} 
  T^{i_1i_2i_3}_{j_1j_2j_3;k}\ =\ T^{i_1i_2i_3}_{j_1j_2j_3,k} 
             &+&T^{ai_2i_3}_{j_1j_2j_3}\,\Gamma^{i_1}_{~ak}\
              + T^{i_1ai_3}_{j_1j_2j_3}\,\Gamma^{i_2}_{~ak}\
              + T^{i_1i_2a}_{j_1j_2j_3}\,\Gamma^{i_3}_{~ak} \\ 
             &-&T^{i_1i_2i_3}_{aj_2j_3}\,\Gamma^a_{~j_1k}
              - T^{i_1i_2i_3}_{j_1aj_3}\,\Gamma^a_{~j_2k}
              - T^{i_1i_2i_3}_{j_1j_2a}\,\Gamma^a_{~j_3k}\,.
\end{eqnarray}
The order of contravariant indices with respect to one another matters
as does the order of covariant indices with respect to one another,
while the relative order of contravariant to covariant indices is
not relevant for the contraction rules (but matters of course for
the tensor itself). Generalization to rank $n$ tensors should now
be obvious.
\vfill
\end{document}
