Topology, Geometry, and Gauge Fields
$$| F|^2=\int_M (F_A,F_A)$$
\[\boxed{ S = \int\nolimits_{t_1}^{t_2} \left\{ \int\nolimits_{\varSigma_t} N \left(R+K_{ij}K^{ij} -K^2\right) \sqrt{\gamma} \; \hbox{d}^3 x \right\} \hbox{d} t }
\]
Let us consider the standard $ \mathbf{\textit{Hilbert;,action}} $ for general relativity (see e.g. [32, 12]):
\[S = \int\nolimits_{{\fancyscript{V}}} {}^4\!R \sqrt{-g} \; \hbox{d}^4 x,
\]
\[\begin{aligned}[b] R_{ij}& = \frac{\partial {{\Gamma^k}_{ij}}}{\partial x^k} - \frac{\partial{{\Gamma^k}_{ik}}}{\partial x^j} + \cdots \\ & = {\frac{1}{2}} \frac{\partial}{\partial x^k} \left[ \gamma^{kl} \left( \frac{\partial{\gamma_{lj}}}{\partial x^i} + \frac{\partial{\gamma_{il}}}{\partial x^j} - \frac{\partial{\gamma_{ij}}}{\partial x^l} \right) \right] - {\frac{1}{2}} \frac{\partial}{\partial x^j} \left[ \gamma^{kl} \left( \frac{\partial{\gamma_{lk}}}{\partial x^i} + \frac{\partial{\gamma_{il}}}{\partial x^k} - \frac{\partial{\gamma_{ik}}}{\partial x^l} \right) \right] + \cdots \\ & = {\frac{1}{2}} \gamma^{kl} \left( \frac{\partial^2 {\gamma_{lj}}}{\partial {x^k}\partial x^i} + \frac{\partial^2 {\gamma_{il}}}{\partial {x^k}\partial x^j} - \frac{\partial^2 {\gamma_{ij}}}{\partial {x^k}\partial x^l} - \frac{\partial^2 {\gamma_{lk}}}{\partial {x^j}\partial x^i}- \frac{\partial^2 {\gamma_{il}}}{\partial {x^j}\partial x^k}+ \frac{\partial^2 {\gamma_{ik}}}{\partial {x^j}\partial x^l} \right) + \cdots \\ R_{ij} & = - {\frac{1}{2}} \gamma^{kl} \left( \frac{\partial^2 {\gamma_{ij}}}{\partial {x^k}\partial x^l}+ \frac{\partial^2 {\gamma_{kl}}}{\partial {x^i}\partial x^j}- \frac{\partial^2 {\gamma_{lj}}}{\partial {x^i}\partial x^k}- \frac{\partial^2 {\gamma_{il}}}{\partial {x^j}\partial x^k}\right) + {\fancyscript{Q}}_{ij}\left(\gamma_{kl}, \frac{\partial {\gamma_{kl}}}{\partial x^m} \right), \end{aligned}
\]
\[\left\{
\begin{array}{c}
a_1x+b_1y+c_1z=d_1 \\
a_2x+b_2y+c_2z=d_2 \\
a_3x+b_3y+c_3z=d_3
\end{array}
\right.
\]
\[\begin{align}
v + w & = 0 &\text{Given} \tag 1\\
-w & = -w + 0 & \text{additive identity} \tag 2\\
-w + 0 & = -w + (v + w) & \text{equations $(1)$ and $(2)$}
\end{align}
\]
\[\begin{matrix} 1 & 2 \\ 3 & 4 \\ \end{matrix}
\]
\[\begin{pmatrix} 1 & 2 \\ 3 & 4 \\ \end{pmatrix}
\]
\[\begin{bmatrix} 1 & 2 \\ 3 & 4 \\ \end{bmatrix}
\]
\[\begin{Bmatrix} 1 & 2 \\ 3 & 4 \\ \end{Bmatrix}
\]
\[\begin{vmatrix} 1 & 2 \\ 3 & 4 \\ \end{vmatrix}
\]
\[\begin{Vmatrix} 1 & 2 \\ 3 & 4 \\ \end{Vmatrix}
\]
\[\require{cancel}\begin{array}{rl}
\verb|y+\cancel{x}| & y+\cancel{x}\\
\verb|\cancel{y+x}| & \cancel{y+x}\\
\verb|y+\bcancel{x}| & y+\bcancel{x}\\
\verb|y+\xcancel{x}| & y+\xcancel{x}\\
\verb|y+\cancelto{0}{x}| & y+\cancelto{0}{x}\\
\verb+\frac{1\cancel9}{\cancel95} = \frac15+& \frac{1\cancel9}{\cancel95} = \frac15 \\
\end{array}
\]
