10.6 Riemann-Christoffel Tensor (ISS Section 36)

\noindent \texttt {... in Cartesian coordinates $ \Bigl\{ \begin{matrix} s \\ ij \end{matrix} \Bigr\}_y $ vanish identically so ... } \begin{gather*} \begin{split} B_{i,jk} = & \frac{\partial^2 B^i}{\partial y^j\partial y^k} \end{split} \end{gather*} \noindent \texttt {... check using the 'D.' primitive ... }

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\noindent \texttt {... but (continuous functions) ... } \begin{gather*} \begin{align*} \begin{split} \frac{\partial^2 B^i}{\partial y^j\partial y^k} = & \frac{\partial^2 B^i}{\partial y^k\partial y^j} \end{split} \\ \intertext {\texttt {... so ... } } \begin{split} B_{i,jk} = & B_{i,kj} \end{split} \end{align*} \end{gather*}

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\noindent \texttt {... a tensor equation ... } \begin{gather*} \begin{split} B_{i,jk} - B_{i,kj} = & \frac{\partial x^m}{\partial y^i} \frac{\partial x^n}{\partial y^j} \frac{\partial x^p}{\partial y^k} \Bigl( A_{m,np} - A_{m,pn} \Bigr) \end{split} \end{gather*} \noindent \texttt {... all components are zero ... \newline ... the tensor $ \Bigl( B_{i,jk} - B_{i,kj} \Bigr) $ ... \newline ... and therefore the tensor $ \Bigl( A_{m,np} - A_{m,pn} \Bigr) $ ... }

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\noindent \texttt {... covariant derivative of $A_m$ ... } \begin{gather*} \begin{align*} \begin{split} A_{m,n} = & \frac{\partial A_m}{\partial x^n} - \Bigl\{ \begin{matrix} k \\ mn \end{matrix} \Bigr\}_x A_k \end{split} \\ \intertext {\texttt {... covariant derivative of $A_{m,n}$ ... } } \begin{split} A_{m,np} = & \frac{\partial A_{m,n}}{\partial x^p} \\ & - \Bigl\{ \begin{matrix} s \\ mp \end{matrix} \Bigr\}_x A_{s,n} \\ & - \Bigl\{ \begin{matrix} s \\ np \end{matrix} \Bigr\}_x A_{m,s} \\ = & \frac{\partial^2 A_m}{\partial x^n\partial x^p} - \frac { \partial \Bigl\{ \begin{matrix} k \\ mn \end{matrix} \Bigr\}_x } { \partial x^p } A_k - \Bigl\{ \begin{matrix} k \\ mn \end{matrix} \Bigr\}_x \frac{\partial A_k}{\partial x^p} \\ & - \Bigl\{ \begin{matrix} s \\ mp \end{matrix} \Bigr\}_x \frac{\partial A_s}{\partial x^n} + \Bigl\{ \begin{matrix} s \\ mp \end{matrix} \Bigr\}_x \Bigl\{ \begin{matrix} k \\ sn \end{matrix} \Bigr\}_x A_k \\ & - \Bigl\{ \begin{matrix} s \\ np \end{matrix} \Bigr\}_x \frac{\partial A_m}{\partial x^s} + \Bigl\{ \begin{matrix} s \\ np \end{matrix} \Bigr\}_x \Bigl\{ \begin{matrix} k \\ ms \end{matrix} \Bigr\}_x A_k \end{split} \\ \intertext {\texttt {... so ... } } \begin{split} A_{m,np} - A_{m,pn} = & \frac{\partial^2 A_m}{\partial x^n\partial x^p} - \frac { \partial \Bigl\{ \begin{matrix} k \\ mn \end{matrix} \Bigr\}_x } { \partial x^p } A_k - \Bigl\{ \begin{matrix} k \\ mn \end{matrix} \Bigr\}_x \frac{\partial A_k}{\partial x^p} \\ & - \Bigl\{ \begin{matrix} s \\ mp \end{matrix} \Bigr\}_x \frac{\partial A_s}{\partial x^n} + \Bigl\{ \begin{matrix} s \\ mp \end{matrix} \Bigr\}_x \Bigl\{ \begin{matrix} k \\ sn \end{matrix} \Bigr\}_x A_k \\ & - \Bigl\{ \begin{matrix} s \\ np \end{matrix} \Bigr\}_x \frac{\partial A_m}{\partial x^s} + \Bigl\{ \begin{matrix} s \\ np \end{matrix} \Bigr\}_x \Bigl\{ \begin{matrix} k \\ ms \end{matrix} \Bigr\}_x A_k \\ & - \frac{\partial^2 A_m}{\partial x^p\partial x^n} + \frac { \partial \Bigl\{ \begin{matrix} k \\ mp \end{matrix} \Bigr\}_x } { \partial x^n } A_k + \Bigl\{ \begin{matrix} k \\ mp \end{matrix} \Bigr\}_x \frac{\partial A_k}{\partial x^n} \\ & + \Bigl\{ \begin{matrix} s \\ mn \end{matrix} \Bigr\}_x \frac{\partial A_s}{\partial x^p} - \Bigl\{ \begin{matrix} s \\ mn \end{matrix} \Bigr\}_x \Bigl\{ \begin{matrix} k \\ sp \end{matrix} \Bigr\}_x A_k \\ & + \Bigl\{ \begin{matrix} s \\ pn \end{matrix} \Bigr\}_x \frac{\partial A_m}{\partial x^s} - \Bigl\{ \begin{matrix} s \\ pn \end{matrix} \Bigr\}_x \Bigl\{ \begin{matrix} k \\ ms \end{matrix} \Bigr\}_x A_k \end{split} \end{align*} \end{gather*} \noindent \texttt {... \newline ... \newline ... but (continuous functions) ... } \begin{gather*} \begin{split} \frac{\partial^2 A_m}{\partial x^n\partial x^p} = & \frac{\partial^2 A_m}{\partial x^p\partial x^n} \end{split} \end{gather*}

\noindent \texttt {... covariant derivative of $A_m$ ... } \begin{gather*} \begin{align*} \begin{split} A_{m,n} = & \frac{\partial A_m}{\partial x^n} - \Bigl\{ \begin{matrix} k \\ mn \end{matrix} \Bigr\}_x A_k \end{split} \\ \intertext {\texttt {... covariant derivative of $A_{m,n}$ ... } } \begin{split} A_{m,np} = & \frac{\partial A_{m,n}}{\partial x^p} \\ & - \Bigl\{ \begin{matrix} s \\ mp \end{matrix} \Bigr\}_x A_{s,n} \\ & - \Bigl\{ \begin{matrix} s \\ np \end{matrix} \Bigr\}_x A_{m,s} \\ = & \frac{\partial^2 A_m}{\partial x^n\partial x^p} - \frac { \partial \Bigl\{ \begin{matrix} k \\ mn \end{matrix} \Bigr\}_x } { \partial x^p } A_k - \Bigl\{ \begin{matrix} k \\ mn \end{matrix} \Bigr\}_x \frac{\partial A_k}{\partial x^p} \\ & - \Bigl\{ \begin{matrix} s \\ mp \end{matrix} \Bigr\}_x \frac{\partial A_s}{\partial x^n} + \Bigl\{ \begin{matrix} s \\ mp \end{matrix} \Bigr\}_x \Bigl\{ \begin{matrix} k \\ sn \end{matrix} \Bigr\}_x A_k \\ & - \Bigl\{ \begin{matrix} s \\ np \end{matrix} \Bigr\}_x \frac{\partial A_m}{\partial x^s} + \Bigl\{ \begin{matrix} s \\ np \end{matrix} \Bigr\}_x \Bigl\{ \begin{matrix} k \\ ms \end{matrix} \Bigr\}_x A_k \end{split} \\ \intertext {\texttt {... so ... } } \begin{split} A_{m,np} - A_{m,pn} = & \frac{\partial^2 A_m}{\partial x^n\partial x^p} - \frac { \partial \Bigl\{ \begin{matrix} k \\ mn \end{matrix} \Bigr\}_x } { \partial x^p } A_k - \Bigl\{ \begin{matrix} k \\ mn \end{matrix} \Bigr\}_x \frac{\partial A_k}{\partial x^p} \\ & - \Bigl\{ \begin{matrix} s \\ mp \end{matrix} \Bigr\}_x \frac{\partial A_s}{\partial x^n} + \Bigl\{ \begin{matrix} s \\ mp \end{matrix} \Bigr\}_x \Bigl\{ \begin{matrix} k \\ sn \end{matrix} \Bigr\}_x A_k \\ & - \Bigl\{ \begin{matrix} s \\ np \end{matrix} \Bigr\}_x \frac{\partial A_m}{\partial x^s} + \Bigl\{ \begin{matrix} s \\ np \end{matrix} \Bigr\}_x \Bigl\{ \begin{matrix} k \\ ms \end{matrix} \Bigr\}_x A_k \\ & - \frac{\partial^2 A_m}{\partial x^p\partial x^n} + \frac { \partial \Bigl\{ \begin{matrix} k \\ mp \end{matrix} \Bigr\}_x } { \partial x^n } A_k + \Bigl\{ \begin{matrix} k \\ mp \end{matrix} \Bigr\}_x \frac{\partial A_k}{\partial x^n} \\ & + \Bigl\{ \begin{matrix} s \\ mn \end{matrix} \Bigr\}_x \frac{\partial A_s}{\partial x^p} - \Bigl\{ \begin{matrix} s \\ mn \end{matrix} \Bigr\}_x \Bigl\{ \begin{matrix} k \\ sp \end{matrix} \Bigr\}_x A_k \\ & + \Bigl\{ \begin{matrix} s \\ pn \end{matrix} \Bigr\}_x \frac{\partial A_m}{\partial x^s} - \Bigl\{ \begin{matrix} s \\ pn \end{matrix} \Bigr\}_x \Bigl\{ \begin{matrix} k \\ ms \end{matrix} \Bigr\}_x A_k \end{split} \end{align*} \end{gather*} \noindent \texttt {... \newline ... \newline ... but (continuous functions) ... } \begin{gather*} \begin{split} \frac{\partial^2 A_m}{\partial x^n\partial x^p} = & \frac{\partial^2 A_m}{\partial x^p\partial x^n} \end{split} \end{gather*}

\noindent \texttt {... covariant derivative of $A_m$ ... } \begin{gather*} \begin{align*} \begin{split} A_{m,n} = & \frac{\partial A_m}{\partial x^n} - \Bigl\{ \begin{matrix} k \\ mn \end{matrix} \Bigr\}_x A_k \end{split} \\ \intertext {\texttt {... covariant derivative of $A_{m,n}$ ... } } \begin{split} A_{m,np} = & \frac{\partial A_{m,n}}{\partial x^p} \\ & - \Bigl\{ \begin{matrix} s \\ mp \end{matrix} \Bigr\}_x A_{s,n} \\ & - \Bigl\{ \begin{matrix} s \\ np \end{matrix} \Bigr\}_x A_{m,s} \\ = & \frac{\partial^2 A_m}{\partial x^n\partial x^p} - \frac { \partial \Bigl\{ \begin{matrix} k \\ mn \end{matrix} \Bigr\}_x } { \partial x^p } A_k - \Bigl\{ \begin{matrix} k \\ mn \end{matrix} \Bigr\}_x \frac{\partial A_k}{\partial x^p} \\ & - \Bigl\{ \begin{matrix} s \\ mp \end{matrix} \Bigr\}_x \frac{\partial A_s}{\partial x^n} + \Bigl\{ \begin{matrix} s \\ mp \end{matrix} \Bigr\}_x \Bigl\{ \begin{matrix} k \\ sn \end{matrix} \Bigr\}_x A_k \\ & - \Bigl\{ \begin{matrix} s \\ np \end{matrix} \Bigr\}_x \frac{\partial A_m}{\partial x^s} + \Bigl\{ \begin{matrix} s \\ np \end{matrix} \Bigr\}_x \Bigl\{ \begin{matrix} k \\ ms \end{matrix} \Bigr\}_x A_k \end{split} \\ \intertext {\texttt {... so ... } } \begin{split} A_{m,np} - A_{m,pn} = & \frac{\partial^2 A_m}{\partial x^n\partial x^p} - \frac { \partial \Bigl\{ \begin{matrix} k \\ mn \end{matrix} \Bigr\}_x } { \partial x^p } A_k - \Bigl\{ \begin{matrix} k \\ mn \end{matrix} \Bigr\}_x \frac{\partial A_k}{\partial x^p} \\ & - \Bigl\{ \begin{matrix} s \\ mp \end{matrix} \Bigr\}_x \frac{\partial A_s}{\partial x^n} + \Bigl\{ \begin{matrix} s \\ mp \end{matrix} \Bigr\}_x \Bigl\{ \begin{matrix} k \\ sn \end{matrix} \Bigr\}_x A_k \\ & - \Bigl\{ \begin{matrix} s \\ np \end{matrix} \Bigr\}_x \frac{\partial A_m}{\partial x^s} + \Bigl\{ \begin{matrix} s \\ np \end{matrix} \Bigr\}_x \Bigl\{ \begin{matrix} k \\ ms \end{matrix} \Bigr\}_x A_k \\ & - \frac{\partial^2 A_m}{\partial x^p\partial x^n} + \frac { \partial \Bigl\{ \begin{matrix} k \\ mp \end{matrix} \Bigr\}_x } { \partial x^n } A_k + \Bigl\{ \begin{matrix} k \\ mp \end{matrix} \Bigr\}_x \frac{\partial A_k}{\partial x^n} \\ & + \Bigl\{ \begin{matrix} s \\ mn \end{matrix} \Bigr\}_x \frac{\partial A_s}{\partial x^p} - \Bigl\{ \begin{matrix} s \\ mn \end{matrix} \Bigr\}_x \Bigl\{ \begin{matrix} k \\ sp \end{matrix} \Bigr\}_x A_k \\ & + \Bigl\{ \begin{matrix} s \\ pn \end{matrix} \Bigr\}_x \frac{\partial A_m}{\partial x^s} - \Bigl\{ \begin{matrix} s \\ pn \end{matrix} \Bigr\}_x \Bigl\{ \begin{matrix} k \\ ms \end{matrix} \Bigr\}_x A_k \end{split} \end{align*} \end{gather*} \noindent \texttt {... \newline ... \newline ... but (continuous functions) ... } \begin{gather*} \begin{split} \frac{\partial^2 A_m}{\partial x^n\partial x^p} = & \frac{\partial^2 A_m}{\partial x^p\partial x^n} \end{split} \end{gather*}

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\noindent \texttt {... and terms are equal ... } \begin{gather*} \begin{align*} \begin{split} \Bigl\{ \begin{matrix} k \\ mn \end{matrix} \Bigr\}_x \frac{\partial A_k}{\partial x^p} = & \Bigl\{ \begin{matrix} s \\ mn \end{matrix} \Bigr\}_x \frac{\partial A_s}{\partial x^p} \\ \Bigl\{ \begin{matrix} s \\ mp \end{matrix} \Bigr\}_x \frac{\partial A_s}{\partial x^n} = & \Bigl\{ \begin{matrix} k \\ mp \end{matrix} \Bigr\}_x \frac{\partial A_k}{\partial x^n} \end{split} \\ \intertext {\texttt {... also $ \Bigl\{ \begin{matrix} s \\ np \end{matrix} \Bigr\}_x $ is symmetric ... \newline ... with respect to the indices $n$ and $p$ ... } } \begin{split} \Bigl\{ \begin{matrix} s \\ np \end{matrix} \Bigr\}_x = & \Bigl\{ \begin{matrix} s \\ pn \end{matrix} \Bigr\}_x \end{split} \end{align*} \end{gather*}

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\noindent \texttt {... so ... } \begin{gather*} \begin{align*} \begin{split} \Bigl\{ \begin{matrix} s \\ np \end{matrix} \Bigr\}_x \frac{\partial A_m}{\partial x^s} = & \Bigl\{ \begin{matrix} s \\ pn \end{matrix} \Bigr\}_x \frac{\partial A_m}{\partial x^s} \end{split} \\ \intertext {\texttt {... and ... } } \begin{split} \Bigl\{ \begin{matrix} s \\ np \end{matrix} \Bigr\}_x \Bigl\{ \begin{matrix} k \\ ms \end{matrix} \Bigr\}_x A_k = & \Bigl\{ \begin{matrix} s \\ pn \end{matrix} \Bigr\}_x \Bigl\{ \begin{matrix} k \\ ms \end{matrix} \Bigr\}_x A_k \end{split} \end{align*} \end{gather*} \noindent \texttt {... \newline ... \newline ... therefore ... } \begin{gather*} \begin{split} A_{m,np} - A_{m,pn} = & - \frac { \partial \Bigl\{ \begin{matrix} k \\ mn \end{matrix} \Bigr\}_x } { \partial x^p } A_k + \Bigl\{ \begin{matrix} s \\ mp \end{matrix} \Bigr\}_x \Bigl\{ \begin{matrix} k \\ sn \end{matrix} \Bigr\}_x A_k \\ & + \frac { \partial \Bigl\{ \begin{matrix} k \\ mp \end{matrix} \Bigr\}_x } { \partial x^n } A_k - \Bigl\{ \begin{matrix} s \\ mn \end{matrix} \Bigr\}_x \Bigl\{ \begin{matrix} k \\ sp \end{matrix} \Bigr\}_x A_k \\ = & R_{mnp}^k A_k \end{split} \end{gather*} \noindent \texttt {... \newline ... the Riemann-Christoffel tensor of the second kind ... \newline ... $R_{mnp}^k$ ... } \begin{gather*} \begin{split} R_{mnp}^k = & \frac { \partial \Bigl\{ \begin{matrix} k \\ mp \end{matrix} \Bigr\}_x } { \partial x^n } - \frac { \partial \Bigl\{ \begin{matrix} k \\ mn \end{matrix} \Bigr\}_x } { \partial x^p } + \Bigl\{ \begin{matrix} s \\ mp \end{matrix} \Bigr\}_x \Bigl\{ \begin{matrix} k \\ sn \end{matrix} \Bigr\}_x - \Bigl\{ \begin{matrix} s \\ mn \end{matrix} \Bigr\}_x \Bigl\{ \begin{matrix} k \\ sp \end{matrix} \Bigr\}_x \end{split} \end{gather*}

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R=:(dhC2kdx((1&|:@(1&|:)@[(-tz)2&|:@[)(+tz)2&|:@(1&|:)@](-tz)])axs@((2;hC2k),0;hC2k))"1

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