\begin{gather*} \intertext {\texttt {... use J to implement ... \newline ... \newline ... \newline ... generate a tensor $S_{\beta\gamma\,.}^{\,\,\,\,\,\,\,\,\alpha}$ ... \newline ... where ... \newline ... $S_{\beta\gamma\,.}^{\,\,\,\,\,\,\,\,\alpha}$ has the characteristics of a torsion tensor ... \newline ... (i.e. skew-symmetric with respect to $\beta$ and $\gamma$) ... \newline ... \newline ... to enable covariant differentiation of tensors ... \newline ... use the transformation rule for the connection ... \newline ... $ \Gamma_{\beta\gamma}^{\,\alpha} = \Bigl\{ \begin{matrix} \alpha \\ \beta\gamma \end{matrix} \Bigr\} + \frac{1}{2} S_{\beta\gamma\,.}^{\,\,\,\,\,\,\,\,\alpha} $ ... } } \begin{align*} \begin{split} \Bigl\{ \begin{matrix} \nu \\ \lambda\mu \end{matrix} \Bigr\} + \frac{1}{2} S_{\lambda\mu\,.}^{\,\,\,\,\,\,\,\nu} = & + \frac {\partial x^\nu} {\partial y^\alpha} \frac {\partial y^\beta} {\partial x^\lambda} \frac {\partial y^\gamma} {\partial x^\mu} \Biggl( \Bigl\{ \begin{matrix} \alpha \\ \beta\gamma \end{matrix} \Bigr\} + \frac{1}{2} S_{\beta\gamma\,.}^{\,\,\,\,\,\,\,\,\alpha} \Biggr) + \frac {\partial x^\nu} {\partial y^\alpha} \frac {\partial^2 y^\alpha} {\partial x^\lambda\partial x^\mu} \end{split} \\ \intertext {\texttt {... so ... } } \begin{split} \Gamma_{\lambda\mu}^{\,\nu} - \Gamma_{\mu\lambda}^{\,\nu} = & + \frac {\partial x^\nu} {\partial y^\alpha} \frac {\partial y^\beta} {\partial x^\lambda} \frac {\partial y^\gamma} {\partial x^\mu} \Biggl( \Bigl\{ \begin{matrix} \alpha \\ \beta\gamma \end{matrix} \Bigr\} + \frac{1}{2} S_{\beta\gamma\,.}^{\,\,\,\,\,\,\,\,\alpha} \Biggr) + \frac {\partial x^\nu} {\partial y^\alpha} \frac {\partial^2 y^\alpha} {\partial x^\lambda\partial x^\mu} \\ & - \frac {\partial x^\nu} {\partial y^\alpha} \frac {\partial y^\gamma} {\partial x^\mu} \frac {\partial y^\beta} {\partial x^\lambda} \Biggl( \Bigl\{ \begin{matrix} \alpha \\ \gamma\beta \end{matrix} \Bigr\} + \frac{1}{2} S_{\gamma\beta\,.}^{\,\,\,\,\,\,\,\,\alpha} \Biggr) - \frac {\partial x^\nu} {\partial y^\alpha} \frac {\partial^2 y^\alpha} {\partial x^\mu\partial x^\lambda} \\ S_{\lambda\mu\,.}^{\,\,\,\,\,\,\,\,\nu} = & + \frac {\partial x^\nu} {\partial y^\alpha} \frac {\partial y^\beta} {\partial x^\lambda} \frac {\partial y^\gamma} {\partial x^\mu} S_{\beta\gamma\,.}^{\,\,\,\,\,\,\,\,\alpha} \end{split} \\ \intertext {\texttt {... and ... } } \begin{split} & + \Bigl( B_{\,.\,\alpha\beta\gamma,\lambda}^{\epsilon} + S_{\beta\gamma\,.}^{\,\,\,\,\,\,\,\rho}\, B_{\,.\,\alpha\lambda\rho}^{\epsilon} \Bigr) \\ & + \Bigl( B_{\,.\,\alpha\gamma\lambda,\beta}^{\epsilon} + S_{\gamma\lambda\,.}^{\,\,\,\,\,\,\,\rho}\, B_{\,.\,\alpha\beta\rho}^{\epsilon} \Bigr) \\ & + \Bigl( B_{\,.\,\alpha\lambda\beta,\gamma}^{\epsilon} + S_{\lambda\beta\,.}^{\,\,\,\,\,\,\,\rho}\, B_{\,.\,\alpha\gamma\rho}^{\epsilon} \Bigr) = 0 \end{split} \end{align*} \end{gather*}

\begin{gather*} \intertext {\texttt {... \newline ... \newline ... derive the covariant derivative of a tensor $C_{\lambda}$ $(C_{\lambda,\mu})$ ... \newline ... \newline ... the connection ... } } \begin{align*} \begin{split} \frac {\partial^2 y^\epsilon} {\partial x^\lambda\partial x^\mu} = & + \frac {\partial y^\epsilon} {\partial x^\nu} \Biggl( \Bigl\{ \begin{matrix} \nu \\ \lambda\mu \end{matrix} \Bigr\} + \frac{1}{2} S_{\lambda\mu\,.}^{\,\,\,\,\,\,\,\nu} \Biggr) - \frac {\partial y^\beta} {\partial x^\lambda} \frac {\partial y^\gamma} {\partial x^\mu} \Biggl( \Bigl\{ \begin{matrix} \epsilon \\ \beta\gamma \end{matrix} \Bigr\} + \frac{1}{2} S_{\beta\gamma\,.}^{\,\,\,\,\,\,\,\,\epsilon} \Biggr) \end{split} \\ \intertext {\texttt {... } } \begin{split} C_\lambda = & + \frac{\partial y^\epsilon}{\partial x^\lambda} A_\epsilon \end{split} \\ \begin{split} \frac{\partial C_\lambda}{\partial x^\mu} = & + \frac{\partial y^\epsilon}{\partial x^\lambda} \frac{\partial y^\gamma}{\partial x^\mu} \frac{\partial A_\epsilon}{\partial y^\gamma} \\ & + \frac{\partial^2 y^\epsilon}{\partial x^\lambda\partial x^\mu} A_\epsilon \end{split} \\ \begin{split} = & + \frac{\partial y^\epsilon}{\partial x^\lambda} \frac{\partial y^\gamma}{\partial x^\mu} \frac{\partial A_\epsilon}{\partial y^\gamma} \\ & + \frac {\partial y^\epsilon} {\partial x^\nu} \Biggl( \Bigl\{ \begin{matrix} \nu \\ \lambda\mu \end{matrix} \Bigr\} + \frac{1}{2} S_{\lambda\mu\,.}^{\,\,\,\,\,\,\,\nu} \Biggr) A_\epsilon \\ & - \frac {\partial y^\beta} {\partial x^\lambda} \frac {\partial y^\gamma} {\partial x^\mu} \Biggl( \Bigl\{ \begin{matrix} \epsilon \\ \beta\gamma \end{matrix} \Bigr\} + \frac{1}{2} S_{\beta\gamma\,.}^{\,\,\,\,\,\,\,\,\epsilon} \Biggr) A_\epsilon \end{split} \\ \intertext {\texttt {... so ... } } \begin{split} \frac{\partial C_\lambda}{\partial x^\mu} - \Biggl( \Bigl\{ \begin{matrix} \nu \\ \lambda\mu \end{matrix} \Bigr\} & + \frac{1}{2} S_{\lambda\mu\,.}^{\,\,\,\,\,\,\,\nu} \Biggr) C_\nu = \\ & + \frac{\partial y^\epsilon}{\partial x^\lambda} \frac{\partial y^\gamma}{\partial x^\mu} \Biggl( \frac{\partial A_\epsilon}{\partial y^\gamma} - \biggl( \Bigl\{ \begin{matrix} \alpha \\ \epsilon\gamma \end{matrix} \Bigr\} + \frac{1}{2} S_{\epsilon\gamma\,.}^{\,\,\,\,\,\,\,\alpha} \biggr) A_\alpha \Biggr) \end{split} \\ \end{align*} \end{gather*}

\begin{gather*} \intertext {\texttt {... \newline ... \newline ... derive $B_{\,.\,mnp}^k$ ... \newline ... \newline ... covariant derivative of $A_m$ with respect to $\Gamma_{j\,k}^{\,i}$ ... } } \begin{align*} \begin{split} A_{m,n} = & + \frac{\partial A_m}{\partial x^n} - \Gamma_{mn}^{\,\,k} A_k \end{split} \\ \intertext {\texttt {... $2^{nd}$ covariant derivative of $A_m$ with respect to $\Gamma_{j\,k}^{\,i}$ ... } } \begin{split} A_{m,np} = & + \frac{\partial A_{m,n}}{\partial x^p} \\ & - \Gamma_{mp}^{\,\,s} A_{s,n} \\ & - \Gamma_{np}^{\,\,s} A_{m,s} \\ = & + \frac{\partial^2 A_m}{\partial x^n\partial x^p} - \frac { \partial \Gamma_{mn}^{\,\,k} } { \partial x^p } A_k - \Gamma_{mn}^{\,\,k} \frac{\partial A_k}{\partial x^p} \\ & - \Gamma_{mp}^{\,\,s} \frac{\partial A_s}{\partial x^n} + \Gamma_{mp}^{\,\,s} \Gamma_{sn}^{\,k} A_k \\ & - \Gamma_{np}^{\,\,s} A_{m,s} \end{split} \\ \intertext {\texttt {... so (a tensor) ... } } \begin{split} A_{m,np} - A_{m,pn} = & + \frac{\partial^2 A_m}{\partial x^n\partial x^p} - \frac { \partial \Gamma_{mn}^{\,\,k} } { \partial x^p } A_k - \Gamma_{mn}^{\,\,k} \frac{\partial A_k}{\partial x^p} \\ & - \Gamma_{mp}^{\,\,s} \frac{\partial A_s}{\partial x^n} + \Gamma_{mp}^{\,\,s} \Gamma_{sn}^{\,k} A_k \\ & - \Gamma_{np}^{\,\,s} A_{m,s} \\ & - \frac{\partial^2 A_m}{\partial x^p\partial x^n} + \frac { \partial \Gamma_{mp}^{\,\,k} } { \partial x^n } A_k + \Gamma_{mp}^{\,\,k} \frac{\partial A_k}{\partial x^n} \\ & + \Gamma_{mn}^{\,\,s} \frac{\partial A_s}{\partial x^p} - \Gamma_{mn}^{\,\,s} \Gamma_{sp}^{\,k} A_k \\ & + \Gamma_{pn}^{\,\,s} A_{m,s} \\ = & + B_{\,.\,mnp}^k A_k + S_{pn\,.}^{\,\,\,\,\,\,\,\,s}\, A_{m,s} \end{split} \\ \intertext {\texttt {... where ... } } \begin{split} B_{\,.\,mnp}^k = & + \frac { \partial \Gamma_{mp}^{\,\,k} } { \partial x^n } - \frac { \partial \Gamma_{mn}^{\,\,k} } { \partial x^p } + \Gamma_{mp}^{\,\,s} \Gamma_{sn}^{\,k} - \Gamma_{mn}^{\,\,s} \Gamma_{sp}^{\,k} \end{split} \end{align*} \end{gather*}

\begin{gather*} \intertext {\texttt {... \newline ... \newline ... derive $B_{\,.\,mnp}^k$ ... \newline ... \newline ... covariant derivative of $A_m$ with respect to $\Gamma_{j\,k}^{\,i}$ ... } } \begin{align*} \begin{split} A_{m,n} = & + \frac{\partial A_m}{\partial x^n} - \Gamma_{mn}^{\,\,k} A_k \end{split} \\ \intertext {\texttt {... $2^{nd}$ covariant derivative of $A_m$ with respect to $\Gamma_{j\,k}^{\,i}$ ... } } \begin{split} A_{m,np} = & + \frac{\partial A_{m,n}}{\partial x^p} \\ & - \Gamma_{mp}^{\,\,s} A_{s,n} \\ & - \Gamma_{np}^{\,\,s} A_{m,s} \\ = & + \frac{\partial^2 A_m}{\partial x^n\partial x^p} - \frac { \partial \Gamma_{mn}^{\,\,k} } { \partial x^p } A_k - \Gamma_{mn}^{\,\,k} \frac{\partial A_k}{\partial x^p} \\ & - \Gamma_{mp}^{\,\,s} \frac{\partial A_s}{\partial x^n} + \Gamma_{mp}^{\,\,s} \Gamma_{sn}^{\,k} A_k \\ & - \Gamma_{np}^{\,\,s} A_{m,s} \end{split} \\ \intertext {\texttt {... so (a tensor) ... } } \begin{split} A_{m,np} - A_{m,pn} = & + \frac{\partial^2 A_m}{\partial x^n\partial x^p} - \frac { \partial \Gamma_{mn}^{\,\,k} } { \partial x^p } A_k - \Gamma_{mn}^{\,\,k} \frac{\partial A_k}{\partial x^p} \\ & - \Gamma_{mp}^{\,\,s} \frac{\partial A_s}{\partial x^n} + \Gamma_{mp}^{\,\,s} \Gamma_{sn}^{\,k} A_k \\ & - \Gamma_{np}^{\,\,s} A_{m,s} \\ & - \frac{\partial^2 A_m}{\partial x^p\partial x^n} + \frac { \partial \Gamma_{mp}^{\,\,k} } { \partial x^n } A_k + \Gamma_{mp}^{\,\,k} \frac{\partial A_k}{\partial x^n} \\ & + \Gamma_{mn}^{\,\,s} \frac{\partial A_s}{\partial x^p} - \Gamma_{mn}^{\,\,s} \Gamma_{sp}^{\,k} A_k \\ & + \Gamma_{pn}^{\,\,s} A_{m,s} \\ = & + B_{\,.\,mnp}^k A_k + S_{pn\,.}^{\,\,\,\,\,\,\,\,s}\, A_{m,s} \end{split} \\ \intertext {\texttt {... where ... } } \begin{split} B_{\,.\,mnp}^k = & + \frac { \partial \Gamma_{mp}^{\,\,k} } { \partial x^n } - \frac { \partial \Gamma_{mn}^{\,\,k} } { \partial x^p } + \Gamma_{mp}^{\,\,s} \Gamma_{sn}^{\,k} - \Gamma_{mn}^{\,\,s} \Gamma_{sp}^{\,k} \end{split} \end{align*} \end{gather*}

\begin{gather*} \intertext {\texttt {... \newline ... \newline ... \newline ... the metric tensor ... \newline ... \newline ... the covariant derivative $h_{ij,k} $ w.r.t.\,$\Gamma_{mn}^{\,\,p}$ is not zero ... \newline ... $ ( \frac {\partial h_{ij}} {\partial x^k} - \Gamma_{ik}^n h_{nj} - \Gamma_{jk}^n h_{in} ) \neq 0 $ ... } } \begin{align*} \begin{split} C_{i,k} = & + \bigl( h_{ij} C^j \bigr)_{,k} \end{split} \\ \begin{split} \frac { \partial C_i } { \partial x^k } - \Gamma_{ik}^n C_n = & + h_{ij} \Bigl( \frac { \partial C^j } { \partial x^k } + \Gamma_{nk}^j C^n \Bigr) \\ & + \Bigl( \frac {\partial h_{ij}} {\partial x^k} - \Gamma_{ik}^n h_{nj} - \Gamma_{jk}^n h_{in} \Bigr) C^j \end{split} \\ \begin{split} \frac { \partial C_i } { \partial x^k } - \Bigl( \Bigl\{ \begin{matrix} n \\ ik \end{matrix} \Bigr\} + \frac{1}{2} S_{ik\,.}^{\,\,\,\,\,\,n} \Bigr) C_n = & + h_{ij} \Biggl( \frac { \partial C^j } { \partial x^k } + \Bigl( \Bigl\{ \begin{matrix} j \\ nk \end{matrix} \Bigr\} + \frac{1}{2} S_{nk\,.}^{\,\,\,\,\,\,\,\,j} \Bigr) C^n \Biggr) \\ & + \Biggl( \, + \frac {\partial h_{ij}} {\partial x^k} \\ & \qquad - \Bigl( \Bigl\{ \begin{matrix} n \\ ik \end{matrix} \Bigr\} + \frac{1}{2} S_{ik\,.}^{\,\,\,\,\,\,n} \Bigr) h_{nj} \\ & \qquad - \Bigl( \Bigl\{ \begin{matrix} n \\ jk \end{matrix} \Bigr\} + \frac{1}{2} S_{jk\,.}^{\,\,\,\,\,\,n} \Bigr) h_{in} \\ & \qquad \Biggr) C^j \end{split} \\ \intertext {\texttt {... so ... } } \begin{split} - \frac{1}{2} S_{ik\,.}^{\,\,\,\,\,\,n} C_n = & + \frac{1}{2} h_{ij} S_{nk\,.}^{\,\,\,\,\,\,\,\,j} C^n \\ & - \frac{1}{2} h_{nj} S_{ik\,.}^{\,\,\,\,\,\,n} C^j \\ & - \frac{1}{2} h_{in} S_{jk\,.}^{\,\,\,\,\,\,n} C^j \end{split} \\ \begin{split} - \frac{1}{2} S_{ik\,.}^{\,\,\,\,\,\,n} C_n = & - \frac{1}{2} S_{ik\,.}^{\,\,\,\,\,\,n} C_n \end{split} \\ \intertext {\texttt {... \newline ... the covariant derivative $h^{ij}_{\,\,\,\,,k} $ w.r.t.\,$\Gamma_{mn}^{\,\,p}$ is not zero ... \newline ... $ ( \frac {\partial h^{ij}} {\partial x^k} + \Gamma_{nk}^{\,\,i} h^{nj} + \Gamma_{nk}^{\,\,j} h^{in} ) \neq 0 $ ... } } \begin{split} C_{\,,k}^i = & + \bigl( h^{ij} C_j \bigr)_{,k} \end{split} \\ \begin{split} \frac { \partial C^i } { \partial x^k } + \Gamma_{nk}^{\,\,i} C^n = & + h^{ij} \Bigl( \frac { \partial C_j } { \partial x^k } - \Gamma_{jk}^n C_n \Bigr) \\ & + \Bigl( \frac {\partial h^{ij}} {\partial x^k} + \Gamma_{nk}^{\,\,i} h^{nj} + \Gamma_{nk}^{\,\,j} h^{in} \Bigr) C_j \end{split} \\ \begin{split} \frac { \partial C^i } { \partial x^k } + \Bigl( \Bigl\{ \begin{matrix} i \\ nk \end{matrix} \Bigr\} + \frac{1}{2} S_{nk\,.}^{\,\,\,\,\,\,\,\,i} \Bigr) C^n = & + h^{ij} \Biggl( \frac { \partial C_j } { \partial x^k } - \Bigl( \Bigl\{ \begin{matrix} n \\ jk \end{matrix} \Bigr\} + \frac{1}{2} S_{jk\,.}^{\,\,\,\,\,\,n} \Bigr) C_n \Biggr) \\ & + \Biggl( \, + \frac {\partial h^{ij}} {\partial x^k} \\ & \qquad + \Bigl( \Bigl\{ \begin{matrix} i \\ nk \end{matrix} \Bigr\} + \frac{1}{2} S_{nk\,.}^{\,\,\,\,\,\,\,\,i} \Bigr) h^{nj} \\ & \qquad + \Bigl( \Bigl\{ \begin{matrix} j \\ nk \end{matrix} \Bigr\} + \frac{1}{2} S_{nk\,.}^{\,\,\,\,\,\,\,\,j} \Bigr) h^{in} \\ & \qquad \Biggr) C_j \end{split} \\ \intertext {\texttt {... so ... } } \begin{split} + \frac{1}{2} S_{nk\,.}^{\,\,\,\,\,\,\,\,i} C^n = & - \frac{1}{2} h^{ij} S_{jk\,.}^{\,\,\,\,\,\,n} C_n \\ & + \frac{1}{2} h^{nj} S_{nk\,.}^{\,\,\,\,\,\,\,\,i} C_j \\ & + \frac{1}{2} h^{in} S_{nk\,.}^{\,\,\,\,\,\,\,\,j} C_j \end{split} \\ \begin{split} + \frac{1}{2} S_{nk\,.}^{\,\,\,\,\,\,\,\,i} C^n = & + \frac{1}{2} S_{nk\,.}^{\,\,\,\,\,\,\,\,i} C^n \end{split} \\ \end{align*} \end{gather*}

\begin{gather*} \intertext {\texttt {... \newline ... \newline ... \newline ... the metric tensor ... \newline ... \newline ... the covariant derivative $h_{ij,k} $ w.r.t.\,$\Gamma_{mn}^{\,\,p}$ is not zero ... \newline ... $ ( \frac {\partial h_{ij}} {\partial x^k} - \Gamma_{ik}^n h_{nj} - \Gamma_{jk}^n h_{in} ) \neq 0 $ ... } } \begin{align*} \begin{split} C_{i,k} = & + \bigl( h_{ij} C^j \bigr)_{,k} \end{split} \\ \begin{split} \frac { \partial C_i } { \partial x^k } - \Gamma_{ik}^n C_n = & + h_{ij} \Bigl( \frac { \partial C^j } { \partial x^k } + \Gamma_{nk}^j C^n \Bigr) \\ & + \Bigl( \frac {\partial h_{ij}} {\partial x^k} - \Gamma_{ik}^n h_{nj} - \Gamma_{jk}^n h_{in} \Bigr) C^j \end{split} \\ \begin{split} \frac { \partial C_i } { \partial x^k } - \Bigl( \Bigl\{ \begin{matrix} n \\ ik \end{matrix} \Bigr\} + \frac{1}{2} S_{ik\,.}^{\,\,\,\,\,\,n} \Bigr) C_n = & + h_{ij} \Biggl( \frac { \partial C^j } { \partial x^k } + \Bigl( \Bigl\{ \begin{matrix} j \\ nk \end{matrix} \Bigr\} + \frac{1}{2} S_{nk\,.}^{\,\,\,\,\,\,\,\,j} \Bigr) C^n \Biggr) \\ & + \Biggl( \, + \frac {\partial h_{ij}} {\partial x^k} \\ & \qquad - \Bigl( \Bigl\{ \begin{matrix} n \\ ik \end{matrix} \Bigr\} + \frac{1}{2} S_{ik\,.}^{\,\,\,\,\,\,n} \Bigr) h_{nj} \\ & \qquad - \Bigl( \Bigl\{ \begin{matrix} n \\ jk \end{matrix} \Bigr\} + \frac{1}{2} S_{jk\,.}^{\,\,\,\,\,\,n} \Bigr) h_{in} \\ & \qquad \Biggr) C^j \end{split} \\ \intertext {\texttt {... so ... } } \begin{split} - \frac{1}{2} S_{ik\,.}^{\,\,\,\,\,\,n} C_n = & + \frac{1}{2} h_{ij} S_{nk\,.}^{\,\,\,\,\,\,\,\,j} C^n \\ & - \frac{1}{2} h_{nj} S_{ik\,.}^{\,\,\,\,\,\,n} C^j \\ & - \frac{1}{2} h_{in} S_{jk\,.}^{\,\,\,\,\,\,n} C^j \end{split} \\ \begin{split} - \frac{1}{2} S_{ik\,.}^{\,\,\,\,\,\,n} C_n = & - \frac{1}{2} S_{ik\,.}^{\,\,\,\,\,\,n} C_n \end{split} \\ \intertext {\texttt {... \newline ... the covariant derivative $h^{ij}_{\,\,\,\,,k} $ w.r.t.\,$\Gamma_{mn}^{\,\,p}$ is not zero ... \newline ... $ ( \frac {\partial h^{ij}} {\partial x^k} + \Gamma_{nk}^{\,\,i} h^{nj} + \Gamma_{nk}^{\,\,j} h^{in} ) \neq 0 $ ... } } \begin{split} C_{\,,k}^i = & + \bigl( h^{ij} C_j \bigr)_{,k} \end{split} \\ \begin{split} \frac { \partial C^i } { \partial x^k } + \Gamma_{nk}^{\,\,i} C^n = & + h^{ij} \Bigl( \frac { \partial C_j } { \partial x^k } - \Gamma_{jk}^n C_n \Bigr) \\ & + \Bigl( \frac {\partial h^{ij}} {\partial x^k} + \Gamma_{nk}^{\,\,i} h^{nj} + \Gamma_{nk}^{\,\,j} h^{in} \Bigr) C_j \end{split} \\ \begin{split} \frac { \partial C^i } { \partial x^k } + \Bigl( \Bigl\{ \begin{matrix} i \\ nk \end{matrix} \Bigr\} + \frac{1}{2} S_{nk\,.}^{\,\,\,\,\,\,\,\,i} \Bigr) C^n = & + h^{ij} \Biggl( \frac { \partial C_j } { \partial x^k } - \Bigl( \Bigl\{ \begin{matrix} n \\ jk \end{matrix} \Bigr\} + \frac{1}{2} S_{jk\,.}^{\,\,\,\,\,\,n} \Bigr) C_n \Biggr) \\ & + \Biggl( \, + \frac {\partial h^{ij}} {\partial x^k} \\ & \qquad + \Bigl( \Bigl\{ \begin{matrix} i \\ nk \end{matrix} \Bigr\} + \frac{1}{2} S_{nk\,.}^{\,\,\,\,\,\,\,\,i} \Bigr) h^{nj} \\ & \qquad + \Bigl( \Bigl\{ \begin{matrix} j \\ nk \end{matrix} \Bigr\} + \frac{1}{2} S_{nk\,.}^{\,\,\,\,\,\,\,\,j} \Bigr) h^{in} \\ & \qquad \Biggr) C_j \end{split} \\ \intertext {\texttt {... so ... } } \begin{split} + \frac{1}{2} S_{nk\,.}^{\,\,\,\,\,\,\,\,i} C^n = & - \frac{1}{2} h^{ij} S_{jk\,.}^{\,\,\,\,\,\,n} C_n \\ & + \frac{1}{2} h^{nj} S_{nk\,.}^{\,\,\,\,\,\,\,\,i} C_j \\ & + \frac{1}{2} h^{in} S_{nk\,.}^{\,\,\,\,\,\,\,\,j} C_j \end{split} \\ \begin{split} + \frac{1}{2} S_{nk\,.}^{\,\,\,\,\,\,\,\,i} C^n = & + \frac{1}{2} S_{nk\,.}^{\,\,\,\,\,\,\,\,i} C^n \end{split} \\ \end{align*} \end{gather*}


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Essays/Torsion Tensor/TorsionTensor02 (last edited 2010-09-01 04:40:37 by TomAllen)