LaTeX vs MathPix image — 0707.4470

\begin{aligned} & E=E_{x} \mathrm{~d} x+E_{y} \mathrm{~d} y+E_{z} \mathrm{~d} z \\ & B=B_{x} \mathrm{~d} y \wedge \mathrm{~d} z+B_{y} \mathrm{~d} z \wedge \mathrm{~d} x+B_{z} \mathrm{~d} x \wedge \mathrm{~d} y \end{aligned}

\begin{aligned} E &= E _x \;\mathrm{d}x + E _y \;\mathrm{d}y + E _z \;\mathrm{d}z \\ B &= B _x \;\mathrm{d}y \wedge \mathrm{d} z + B _y \;\mathrm{d}z \wedge \mathrm{d} x + B _z \;\mathrm{d}x \wedge \mathrm{d} y, \end{aligned}

\oint_{C} \mathbf{E} \cdot \mathrm{~d} \mathbf{l}=-\frac{\mathrm{d}}{\mathrm{~d} t} \int_{S} \mathbf{B} \cdot \mathrm{~d} \mathbf{A},

\oint _C \mathbf{E} \cdot \mathrm{d} \mathbf{l} = -\frac{\mathrm{d}}{\mathrm{d}t} \int _S \mathbf{B} \cdot \mathrm{d} \mathbf{A},

\oint_{C} \mathbf{H} \cdot \mathrm{~d} \mathbf{l}=\frac{\mathrm{d}}{\mathrm{~d} t} \int_{S} \mathbf{D} \cdot \mathrm{~d} \mathbf{A},

\oint _C \mathbf{H} \cdot \mathrm{d} \mathbf{l} = \frac{\mathrm{d}}{\mathrm{d}t} \int _S \mathbf{D} \cdot \mathrm{d} \mathbf{A},

\mathbf{D}=\epsilon \mathbf{E}, \quad \mathbf{B}=\mu \mathbf{H} .

\mathbf{D} = \epsilon \mathbf{E}, \qquad \mathbf{B} = \mu \mathbf{H} .

F=E \wedge \mathrm{~d} t+B .

F = E \wedge \mathrm{d} t + B.

G=* F=H \wedge \mathrm{~d} t-D,

G = *F = H \wedge \mathrm{d} t - D ,

\mathscr{J}=J \wedge \mathrm{~d} t-\rho .

\mathcal{J} = J \wedge \mathrm{d} t - \rho .

\mathscr{L}=-\frac{1}{2} \mathrm{~d} A \wedge * \mathrm{~d} A+A \wedge \mathscr{J},

\mathcal{L} = - \frac{1}{2} \mathrm{d} A \wedge *\mathrm{d} A + A \wedge \mathcal{J} ,

S[A]=\int_{X} \mathscr{L} .

S[A] = \int _X \mathcal{L}.

\begin{aligned} \mathbf{d} S[A] \cdot \alpha & =\left.\frac{\mathrm{d}}{\mathrm{~d} \epsilon}\right|_{\epsilon=0} S[A+\epsilon \alpha] \\ & =\int_{X}(-\mathrm{d} \alpha \wedge * \mathrm{~d} A+\alpha \wedge \mathscr{J}) \\ & =\int_{X} \alpha \wedge(-\mathrm{d} * \mathrm{~d} A+\mathscr{J}), \end{aligned}

\begin{aligned} \mathbf{d} S [A] \cdot \alpha & = \left. \frac{\mathrm{d} }{\mathrm{d} \epsilon } \right\rvert _{ \epsilon = 0 } S[A + \epsilon \alpha ] \\ &= \int _X \left( - \mathrm{d} \alpha \wedge *\mathrm{d} A + \alpha \wedge \mathcal{J} \right) \\ &= \int _X \alpha \wedge \left( - \mathrm{d} {*\mathrm{d}A} + \mathcal{J} \right) , \end{aligned}

\mathrm{d} * \mathrm{~d} A=\mathscr{J} .

\mathrm{d} {*\mathrm{d} A} = \mathcal{J} .

\begin{aligned} & \mathrm{d} F=0 \\ & \mathrm{~d} G=\mathscr{J} \end{aligned}

\begin{aligned} \mathrm{d} F & = 0 \\ \mathrm{d} G & = \mathcal{J} \end{aligned}

\begin{aligned} \nabla \times \mathbf{E}+\partial_{t} \mathbf{B} & =0 \\ \nabla \cdot \mathbf{B} & =0 . \end{aligned}

\begin{aligned} \nabla \times \mathbf{E} + \partial _t \mathbf{B} & = 0 \\ \nabla \cdot \mathbf{B} &= 0 . \end{aligned}

\begin{aligned} \nabla \times \mathbf{H}-\partial_{t} \mathbf{D} & =\mathbf{J} \\ \nabla \cdot \mathbf{D} & =\rho . \end{aligned}

\begin{aligned} \nabla \times \mathbf{H} - \partial _t \mathbf{D} & = \mathbf{J} \\ \nabla \cdot \mathbf{D} & = \rho . \end{aligned}

S[A, F, G]=\int_{X}\left[-\frac{1}{2} F \wedge * F+A \wedge \mathscr{J}+(F-\mathrm{d} A) \wedge G\right] .

S[A,F,G] = \int _X \left[ - \frac{1}{2} F \wedge *F + A \wedge \mathcal{J} + \left( F - \mathrm{d} A \right) \wedge G \right] .

\begin{aligned} \mathbf{d} S[A, F, G] \cdot(\alpha, \phi, \gamma) & =\int_{X}[-\phi \wedge * F+\alpha \wedge \mathscr{J}+(\phi-\mathrm{d} \alpha) \wedge G+(F-\mathrm{d} A) \wedge \gamma] \\ & =\int_{X}[\alpha \wedge(\mathscr{J}-\mathrm{d} G)+\phi \wedge(G-* F)+(F-\mathrm{d} A) \wedge \gamma] \end{aligned}

\begin{aligned} \mathbf{d} S [A,F,G] \cdot \left( \alpha, \phi , \gamma \right) &= \int _X \left[ - \phi \wedge * F + \alpha \wedge \mathcal{J} + \left( \phi - \mathrm{d} \alpha \right) \wedge G + \left( F - \mathrm{d} A \right) \wedge \gamma \right] \\ &= \int _X \left[ \alpha \wedge \left( \mathcal{J} - \mathrm{d} G \right) + \phi \wedge \left( G - *F \right) + \left( F - \mathrm{d} A \right) \wedge \gamma \right] . \end{aligned}

\mathrm{d} G=\mathscr{J}, \quad G=* F, \quad F=\mathrm{d} A .

\mathrm{d} G = \mathcal{J} , \qquad G = *F, \qquad F = \mathrm{d} A .

\partial_{t}(\nabla \cdot \mathbf{B})=0, \quad \partial_{t}(\nabla \cdot \mathbf{D})+\nabla \cdot \mathbf{J}=\partial_{t}(\nabla \cdot \mathbf{D}-\rho)=0 .

\partial _t \left( \nabla \cdot \mathbf{B}\right) = 0, \qquad \partial _t \left( \nabla \cdot \mathbf{D} \right) + \nabla \cdot \mathbf{J} = \partial _t \left( \nabla \cdot \mathbf{D} - \rho \right) = 0 .

\langle\alpha, \sigma\rangle \equiv \int_{\sigma} \alpha .

\left\langle \alpha , \sigma \right\rangle \equiv \int _{ \sigma } \alpha .

\int_{\sigma} \mathrm{d} \alpha=\int_{\partial \sigma} \alpha .

\int _\sigma \mathrm{d} \alpha = \int _{ \partial \sigma } \alpha .

\langle\mathrm{d} \alpha, \sigma\rangle=\langle\alpha, \partial \sigma\rangle,

\left\langle \mathrm{d} \alpha , \sigma \right\rangle = \left\langle \alpha , \partial \sigma \right\rangle ,

\frac{1}{|* \sigma|}\langle * \alpha, * \sigma\rangle=\kappa(\sigma) \frac{1}{|\sigma|}\langle\alpha, \sigma\rangle,

\frac{1}{\left\lvert *\sigma \right\rvert } \left\langle *\alpha, * \sigma \right\rangle = \kappa (\sigma ) \frac{1}{\left\lvert \sigma \right\rvert } \left\langle \alpha , \sigma \right\rangle,

\begin{aligned} (\alpha, \beta) & =\sum_{\sigma^{k}} \kappa(\sigma)\binom{n}{k} \frac{|\mathrm{CH}(\sigma, * \sigma)|}{|\sigma|^{2}}\langle\alpha, \sigma\rangle\langle\beta, \sigma\rangle \\ & =\sum_{\sigma^{k}} \kappa(\sigma) \frac{|* \sigma|}{|\sigma|}\langle\alpha, \sigma\rangle\langle\beta, \sigma\rangle \end{aligned}

\begin{aligned} \left( \alpha, \beta \right) &= \sum_{\sigma ^k} \kappa (\sigma) {n\choose k} \frac{ \left\lvert \operatorname{CH}(\sigma, *\sigma) \right\rvert }{ \left\lvert \sigma \right\rvert ^2 } \left\langle \alpha, \sigma \right\rangle \left\langle \beta, \sigma \right\rangle \\ &= \sum_{\sigma ^k} \kappa (\sigma ) \frac{\left\lvert *\sigma \right\rvert }{\left\lvert \sigma \right\rvert } \left\langle \alpha, \sigma \right\rangle \left\langle \beta, \sigma \right\rangle \end{aligned}

(\mathrm{d} \alpha, \beta)=(\alpha, \delta \beta)+\langle\alpha \wedge * \beta, \partial K\rangle .

\left( \mathrm{d} \alpha , \beta \right) = \left( \alpha , \delta \beta \right) + \left\langle \alpha \wedge *\beta , \partial K \right\rangle .

\begin{aligned} & F=E_{x} \mathrm{~d} x \wedge \mathrm{~d} t+E_{y} \mathrm{~d} y \wedge \mathrm{~d} t+E_{z} \mathrm{~d} z \wedge d t \\ & \quad+B_{x} \mathrm{~d} y \wedge \mathrm{~d} z+B_{y} \mathrm{~d} z \wedge \mathrm{~d} x+B_{z} \mathrm{~d} x \wedge \mathrm{~d} y \end{aligned}

\begin{aligned} F &= E _x \;\mathrm{d}x \wedge \mathrm{d} t + E _y \;\mathrm{d}y \wedge \mathrm{d} t + E _z \;\mathrm{d}z \wedge d t \\ &\qquad + B _x \;\mathrm{d}y \wedge \mathrm{d} z + B _y \;\mathrm{d}z \wedge \mathrm{d} x + B _z \;\mathrm{d}x \wedge \mathrm{d} y, \end{aligned}

\begin{gathered} x t \text {-face : }\left.E_{x}\right|_{k+\frac{1}{2}, l, m} ^{n+\frac{1}{2}} \Delta x \Delta t \\ y t \text {-face : }\left.E_{y}\right|_{k, l+\frac{1}{2}, m} ^{n+\frac{1}{2}} \Delta y \Delta t \\ z t \text {-face : }\left.E_{z}\right|_{k, l, m+\frac{1}{2}} ^{n+\frac{1}{2}} \Delta z \Delta t \\ y z \text {-face : }\left.B_{x}\right|_{k, l+\frac{1}{2}, m+\frac{1}{2}} ^{n} \Delta y \Delta z \\ x z \text {-face : }\left.B_{y}\right|_{k+\frac{1}{2}, l, m+\frac{1}{2}} ^{n} \Delta z \Delta x \\ x y \text {-face : }\left.B_{z}\right|_{k+\frac{1}{2}, l+\frac{1}{2}, m} ^{n} \Delta x \Delta y \end{gathered}

\mathrm{d} F=0, \quad \mathrm{~d} G=\mathscr{J},

\mathrm{d} F = 0, \qquad \mathrm{d} G = \mathcal{J} ,

\begin{aligned} x y t \text {-face : } & -\left(\left.E_{x}\right|_{k+\frac{1}{2}, l+1, m} ^{n+\frac{1}{2}}-\left.E_{x}\right|_{k+\frac{1}{2}, l, m} ^{n+\frac{1}{2}}\right) \Delta x \Delta t \\ & +\left(\left.E_{y}\right|_{k+1, l+\frac{1}{2}, m} ^{n+\frac{1}{2}}-\left.E_{y}\right|_{k, l+\frac{1}{2}, m} ^{n+\frac{1}{2}}\right) \Delta y \Delta t \\ & +\left(\left.B_{z}\right|_{k+\frac{1}{2}, l+\frac{1}{2}, m} ^{n+1}-\left.B_{z}\right|_{k+\frac{1}{2}, l+\frac{1}{2}, m} ^{n}\right) \Delta x \Delta y \end{aligned}

\begin{aligned} x z t \text {-face : } & -\left(\left.E_{x}\right|_{k+\frac{1}{2}, l, m+1} ^{n+\frac{1}{2}}-\left.E_{x}\right|_{k+\frac{1}{2}, l, m} ^{n+\frac{1}{2}}\right) \Delta x \Delta t \\ & +\left(\left.E_{z}\right|_{k+1, l, m+\frac{1}{2}} ^{n+\frac{1}{2}}-\left.E_{z}\right|_{k, l, m+\frac{1}{2}} ^{n+\frac{1}{2}}\right) \Delta z \Delta t \\ & -\left(\left.B_{y}\right|_{k+\frac{1}{2}, l, m+\frac{1}{2}} ^{n+1}-\left.B_{y}\right|_{k+\frac{1}{2}, l, m+\frac{1}{2}} ^{n}\right) \Delta x \Delta z \end{aligned}

\begin{aligned} y z t \text {-face: } & -\left(\left.E_{y}\right|_{k, l+\frac{1}{2}, m+1} ^{n+\frac{1}{2}}-\left.E_{y}\right|_{k, l+\frac{1}{2}, m} ^{n+\frac{1}{2}}\right) \Delta y \Delta t \\ & +\left(\left.E_{z}\right|_{k, l+1, m+\frac{1}{2}} ^{n+\frac{1}{2}}-\left.E_{z}\right|_{k, l, m+\frac{1}{2}} ^{n+\frac{1}{2}}\right) \Delta z \Delta t \\ & +\left(\left.B_{x}\right|_{k, l+\frac{1}{2}, m+\frac{1}{2}} ^{n+1}-\left.B_{x}\right|_{k, l+\frac{1}{2}, m+\frac{1}{2}} ^{n}\right) \Delta y \Delta z \end{aligned}

\begin{aligned} x y z \text {-face : } & \left(\left.B_{x}\right|_{k+1, l+\frac{1}{2}, m+\frac{1}{2}} ^{n}-\left.B_{x}\right|_{k, l+\frac{1}{2}, m+\frac{1}{2}} ^{n}\right) \Delta y \Delta z \\ & +\left(\left.B_{y}\right|_{k+\frac{1}{2}, l+1, m+\frac{1}{2}} ^{n}-\left.B_{y}\right|_{k+\frac{1}{2}, l, m+\frac{1}{2}} ^{n}\right) \Delta x \Delta z \\ & +\left(\left.B_{z}\right|_{k+\frac{1}{2}, l+\frac{1}{2}, m+1} ^{n}-\left.B_{z}\right|_{k+\frac{1}{2}, l+\frac{1}{2}, m} ^{n}\right) \Delta x \Delta y \end{aligned}

\begin{aligned} & \frac{\left.B_{x}\right|_{k, l+\frac{1}{2}, m+\frac{1}{2}} ^{n+1}-\left.B_{x}\right|_{k, l+\frac{1}{2}, m+\frac{1}{2}} ^{n}}{\Delta t}= \\ & \frac{\left.E_{y}\right|_{k, l+\frac{1}{2}, m+1} ^{n+\frac{1}{2}}-\left.E_{y}\right|_{k, l+\frac{1}{2}, m} ^{n+\frac{1}{2}}}{\Delta z}-\frac{\left.E_{z}\right|_{k, l+1, m+\frac{1}{2}} ^{n+\frac{1}{2}}-\left.E_{z}\right|_{k, l, m+\frac{1}{2}} ^{n+\frac{1}{2}}}{\Delta y} \\ & \frac{\left.B_{y}\right|_{k+\frac{1}{2}, l, m+\frac{1}{2}} ^{n+1}-\left.B_{y}\right|_{k+\frac{1}{2}, l, m+\frac{1}{2}} ^{n}}{\Delta t}= \\ & \frac{\left.E_{z}\right|_{k+1, l, m+\frac{1}{2}} ^{n+\frac{1}{2}}-\left.E_{z}\right|_{k, l, m+\frac{1}{2}} ^{n+\frac{1}{2}}}{\Delta x}-\frac{\left.E_{x}\right|_{k+\frac{1}{2}, l, m+1} ^{n+\frac{1}{2}, m}-\left.E_{x}\right|_{k+\frac{1}{2}, l, m} ^{n+\frac{1}{2}}}{\Delta z} \\ & \frac{\left.E_{x}\right|_{k+\frac{1}{2}, l+1, m} ^{n+\frac{1}{2}}-\left.E_{x}\right|_{k+\frac{1}{2}, l, m} ^{n+\frac{1}{2}}}{\Delta y}-\frac{\left.E_{y}\right|_{k+1, l+\frac{1}{2}, m} ^{n+\frac{1}{2}}-\left.E_{y}\right|_{k, l+\frac{1}{2}, m} ^{n+\frac{1}{2}}}{\Delta x} \end{aligned}

\begin{aligned} & \frac{\left.B_{x}\right|_{k+1, l+\frac{1}{2}, m+\frac{1}{2}} ^{n}-\left.B_{x}\right|_{k, l+\frac{1}{2}, m+\frac{1}{2}} ^{n}}{\Delta x}+\frac{\left.B_{y}\right|_{k+\frac{1}{2}, l+1, m+\frac{1}{2}} ^{n}-\left.B_{y}\right|_{k+\frac{1}{2}, l, m+\frac{1}{2}} ^{n}}{\Delta y} \\ & \quad+\frac{\left.B_{z}\right|_{k+\frac{1}{2}, l+\frac{1}{2}, m+1} ^{n}-\left.B_{z}\right|_{k+\frac{1}{2}, l+\frac{1}{2}, m} ^{n}}{\Delta z}=0 . \end{aligned}

\partial_{t} \mathbf{B}=-\nabla \times \mathbf{E}, \quad \nabla \cdot \mathbf{B}=0 .

\partial _t \mathbf{B} = - \nabla \times \mathbf{E} , \qquad \nabla \cdot \mathbf{B} = 0.

\begin{aligned} & \frac{\left.D_{x}\right|_{k+\frac{1}{2}, l, m} ^{n+\frac{1}{2}}-\left.D_{x}\right|_{k+\frac{1}{2}, l, m} ^{n-\frac{1}{2}}}{\Delta t}= \\ & \quad \frac{\left.H_{z}\right|_{k+\frac{1}{2}, l+\frac{1}{2}, m} ^{n}-\left.H_{z}\right|_{k+\frac{1}{2}, l-\frac{1}{2}, m} ^{n}}{\Delta y}-\frac{\left.H_{y}\right|_{k+\frac{1}{2}, l, m+\frac{1}{2}} ^{n}-\left.H_{y}\right|_{k+\frac{1}{2}, l, m-\frac{1}{2}} ^{n}}{\Delta z}-\left.J_{x}\right|_{k+\frac{1}{2}, l, m} ^{n} \\ & \quad \frac{\left.D_{y}\right|_{k, l+\frac{1}{2}, m} ^{n+\frac{1}{2}}-\left.D_{y}\right|_{k, l+\frac{1}{2}, m} ^{n-\frac{1}{2}}}{\Delta t}= \\ & \quad \frac{\left.H_{x}\right|_{k, l+\frac{1}{2}, m+\frac{1}{2}} ^{n}-\left.H_{x}\right|_{k, l+\frac{1}{2}, m-\frac{1}{2}} ^{n}}{\Delta z}-\frac{\left.H_{z}\right|_{k+\frac{1}{2}, l+\frac{1}{2}, m} ^{n}-\left.H_{z}\right|_{k-\frac{1}{2}, l+\frac{1}{2}, m} ^{n}-\left.D_{z}\right|_{k, l, m+\frac{1}{2}} ^{n-\frac{1}{2}}}{\Delta x}=\left.J_{y}\right|_{k, l+\frac{1}{2}, m} ^{n} \\ & \quad \frac{\left.H_{y}\right|_{k+\frac{1}{2}, l, m+\frac{1}{2}} ^{n}-\left.H_{y}\right|_{k-\frac{1}{2}, l, m+\frac{1}{2}} ^{n}}{\Delta x}-\frac{\left.H_{x}\right|_{k, l+\frac{1}{2}, m+\frac{1}{2}} ^{n}-\left.H_{x}\right|_{k, l-\frac{1}{2}, m+\frac{1}{2}} ^{n}}{\Delta y}-\left.J_{z}\right|_{k, l, m+\frac{1}{2}} ^{n} \end{aligned}

\begin{aligned} & \frac{\left.D_{x}\right|_{k+\frac{1}{2}, l, m} ^{n+\frac{1}{2}}-\left.D_{x}\right|_{k-\frac{1}{2}, l, m} ^{n+\frac{1}{2}}}{\Delta x}+\frac{\left.D_{y}\right|_{k, l+\frac{1}{2}, m} ^{n+\frac{1}{2}}-\left.D_{y}\right|_{k, l-\frac{1}{2}, m} ^{n+\frac{1}{2}}}{\Delta y} \\ & \quad+\frac{\left.D_{z}\right|_{k, l, m+\frac{1}{2}} ^{n+\frac{1}{2}}-\left.D_{z}\right|_{k, l, m-\frac{1}{2}} ^{n+\frac{1}{2}}}{\Delta z}=\left.\rho\right|_{k, l, m} ^{n+\frac{1}{2}} . \end{aligned}

\partial_{t} \mathbf{D}=\nabla \times \mathbf{H}-\mathbf{J}, \quad \nabla \cdot \mathbf{D}=\rho .

\partial _t \mathbf{D} = \nabla \times \mathbf{H} - \mathbf{J} , \qquad \nabla \cdot \mathbf{D} = \rho .

F=E \wedge \mathrm{~d} t+B .

F = E \wedge \mathrm{d} t + B.

\frac{B^{n+1}-B^{n}}{\Delta t}=-\mathrm{d}_{1} E^{n+1 / 2} .

\frac{ B^{n+1} - B ^n }{ \Delta t } = - \mathrm{d} _1 E^{n + 1/2 }.

\frac{D^{n+1 / 2}-D^{n-1 / 2}}{\Delta t}=\mathrm{d}_{1}^{T} H^{n}-J^{n} .

\frac{ D ^{ n + 1/2 } - D ^{ n - 1/2 } }{ \Delta t } = \mathrm{d} _1 ^T H ^n - J ^n .

\Theta_{\sigma}=\left\{t_{\sigma}^{0}<\cdots<t_{\sigma}^{N_{\sigma}}\right\} .

\Theta _\sigma = \left\{ t ^0 _\sigma < \cdots < t ^{ N _\sigma } _\sigma \right\}.

\Theta_{e}=\bigcup_{\sigma \ni e} \Theta_{\sigma}=\left\{t_{e}^{0} \leq \cdots \leq t_{e}^{N_{e}}\right\} .

\Theta _e = \bigcup _{ \sigma \ni e } \Theta _\sigma = \left\{ t ^0 _e \leq \cdots \leq t ^{ N _e } _e \right\}.

\Theta_{e}^{\prime}=\left\{t_{e}^{1 / 2} \leq \cdots \leq t_{e}^{N_{e}-1 / 2}\right\},

\Theta _{e} ^\prime = \left\{ t ^{1/2} _e \leq \cdots \leq t ^{ N _e - 1/2 } _e \right\},

\Theta_{* \sigma}=\Theta_{\sigma}, \quad \Theta_{* e}=\Theta_{e} .

\Theta _{*\sigma} = \Theta _\sigma , \qquad \Theta _{*e} = \Theta _{e}.

\frac{B_{\sigma}^{n+1}-B_{\sigma}^{n}}{t_{\sigma}^{n+1}-t_{\sigma}^{n}}=-\mathrm{d}_{1} \sum\left\{E_{e}^{m+1 / 2}: t_{\sigma}^{n}<t_{e}^{m+1 / 2}<t_{\sigma}^{n+1}\right\} .

\frac{ B^{n+1}_\sigma - B ^n_\sigma }{ t^{n+1}_\sigma - t^n_\sigma } = - \mathrm{d} _1 \sum \left\{ E^{m+1/2}_e : t^n_\sigma < t^{m+1/2}_e < t^{n+1}_\sigma \right\} .

\frac{D_{e}^{m+1 / 2}-D_{e}^{m-1 / 2}}{t_{e}^{m+1 / 2}-t_{e}^{m-1 / 2}}=\mathrm{d}_{1}^{T}\left(H_{\sigma}^{n} \nVdash\left\{t_{\sigma}^{n}=t_{e}^{m}\right\}\right)-J_{e}^{m},

\frac{ D ^{ m + 1/2 }_e - D ^{ m - 1/2 }_e }{ t^{m+1/2}_e - t^{m-1/2}_e } = \mathrm{d} _1 ^T \left( H ^n_\sigma \, \mathbb{1}_{ \left\{t^n_\sigma = t^m_e \right\}} \right) - J ^m _e ,

\mathscr{L}_{d}=-\frac{1}{2} \mathrm{~d} A \wedge * \mathrm{~d} A+A \wedge \mathscr{J},

\mathcal{L} _d = -\frac{1}{2} \mathrm{d} A \wedge *\mathrm{d} A + A \wedge \mathcal{J} ,

S_{d}[A]=\left\langle\mathscr{L}_{d}, K\right\rangle .

S _d [A] = \left\langle \mathcal{L} _d , K \right\rangle .

\mathbf{d} S_{d}[A] \cdot \alpha=\langle-\mathrm{d} \alpha \wedge * \mathrm{~d} A+\alpha \wedge \mathscr{J}, K\rangle=\langle\alpha \wedge(-\mathrm{d} * \mathrm{~d} A+\mathscr{J}), K\rangle .

\mathbf{d} S _d [A] \cdot \alpha = \left\langle - \mathrm{d} \alpha \wedge *\mathrm{d} A + \alpha \wedge \mathcal{J} , K \right\rangle = \left\langle \alpha \wedge \left( - \mathrm{d} {*\mathrm{d} A} + \mathcal{J} \right) , K \right\rangle .

\mathbf{d} S_{d}[A] \cdot \alpha=\langle\alpha \wedge(-\mathrm{d} * \mathrm{~d} A+\mathscr{J}), K\rangle+\langle\alpha \wedge * d A, \partial K\rangle .

\mathbf{d} S _d [A] \cdot \alpha = \left\langle \alpha \wedge \left( - \mathrm{d} {*\mathrm{d} A } + \mathcal{J} \right) , K \right\rangle + \left\langle \alpha \wedge *d A, \partial K \right\rangle .

\mathbf{d} S_{d}(A) \cdot \alpha=\langle\alpha \wedge * \mathrm{~d} A, \partial K\rangle

\mathbf{d} S _d (A) \cdot \alpha = \left\langle \alpha \wedge *\mathrm{d} A,\partial K \right\rangle

\theta_{\mathscr{L}_{d}} \cdot \alpha=\alpha \wedge * \mathrm{~d} A .

\theta _{ \mathcal{L} _d } \cdot \alpha = \alpha \wedge *\mathrm{d} A.

\mathbf{d}^{2} S_{d}[A] \cdot \alpha \cdot \beta=\langle\mathbf{d} \theta \cdot \alpha \cdot \beta, \partial K\rangle .

\mathbf{d} ^2 S _d [A] \cdot \alpha \cdot \beta = \left\langle \mathbf{d} \theta \cdot \alpha \cdot \beta , \partial K \right\rangle.

\left\langle\omega_{\mathscr{L}_{d}} \cdot \alpha \cdot \beta, \partial K\right\rangle=0

\left\langle \omega _{ \mathcal{L} _d } \cdot \alpha \cdot \beta , \partial K \right\rangle = 0

A=A_{x} \mathrm{~d} x+A_{y} \mathrm{~d} y+A_{z} \mathrm{~d} z .

A = A _x \;\mathrm{d}x + A _y \;\mathrm{d}y + A _z \;\mathrm{d}z .

\mathrm{d}_{t} A=E \wedge \mathrm{~d} t, \quad \mathrm{~d}_{s} A=B .

\begin{aligned} \mathrm{d} _t A = E \wedge \mathrm{d} t, && \mathrm{d} _s A = B. \end{aligned}

\begin{aligned} \mathscr{L} & =-\frac{1}{2}\left(\mathrm{~d}_{t} A+\mathrm{d}_{s} A\right) \wedge *\left(\mathrm{~d}_{t} A+\mathrm{d}_{s} A\right)+A \wedge \mathscr{J} \\ & =-\frac{1}{2}\left(\mathrm{~d}_{t} A \wedge * \mathrm{~d}_{t} A+\mathrm{d}_{s} A \wedge * \mathrm{~d}_{s} A\right)+A \wedge J \wedge \mathrm{~d} t \end{aligned}

\begin{aligned} \mathcal{L} &= -\frac{1}{2} \left( \mathrm{d} _t A + \mathrm{d} _s A \right) \wedge *\left( \mathrm{d} _t A + \mathrm{d} _s A \right) + A \wedge \mathcal{J} \\ &= -\frac{1}{2} \left( \mathrm{d} _t A \wedge * \mathrm{d} _t A + \mathrm{d} _s A \wedge * \mathrm{d} _s A \right) + A \wedge J \wedge \mathrm{d} t \end{aligned}

\begin{aligned} \mathbf{d} S[A] \cdot \alpha & =\int_{X}\left(\mathrm{~d}_{t} \alpha \wedge D-\mathrm{d}_{s} \alpha \wedge H \wedge \mathrm{~d} t+\alpha \wedge J \wedge \mathrm{~d} t\right) \\ & =\int_{X} \alpha \wedge\left(\mathrm{~d}_{t} D-\mathrm{d}_{s} H \wedge \mathrm{~d} t+J \wedge \mathrm{~d} t\right) \end{aligned}

\begin{aligned} \mathbf{d} S[A] \cdot \alpha &= \int _X \left( \mathrm{d} _t \alpha \wedge {D} - \mathrm{d} _s \alpha \wedge H \wedge \mathrm{d} t + \alpha \wedge J \wedge \mathrm{d} t \right) \\ &= \int _X \alpha \wedge \left( \mathrm{d} _t D - \mathrm{d} _s H \wedge \mathrm{d} t + J \wedge \mathrm{d} t \right) . \end{aligned}

\mathbf{d} S[A] \cdot \alpha=\left.\int_{\Sigma} \alpha \wedge D\right|_{t_{0}} ^{t_{f}},

\mathbf{d} S [A] \cdot \alpha = \left. \int _{\Sigma} \alpha \wedge {D} \,\right\rvert _{t _0 } ^{ t _f } ,

\mathbf{d} S[A] \cdot \mathrm{d}_{s} f=\left.\int_{\Sigma} \mathrm{d}_{s} f \wedge D\right|_{t_{0}} ^{t_{f}}=-\left.\int_{\Sigma} f \wedge \mathrm{~d}_{s} D\right|_{t_{0}} ^{t_{f}}

\mathbf{d} S [A] \cdot \mathrm{d} _s f = \left. \int _\Sigma \mathrm{d} _s f \wedge {D} \,\right\rvert _{t _0 } ^{ t _f } = - \left. \int _\Sigma f \wedge \mathrm{d} _s {D} \,\right\rvert _{t _0 } ^{ t _f }

\mathbf{d} S[A] \cdot \mathrm{d}_{s} f=\int_{X} \mathrm{~d}_{s} f \wedge J \wedge \mathrm{~d} t=-\int_{X} f \wedge \mathrm{~d}_{s} J \wedge \mathrm{~d} t=-\int_{X} f \wedge \mathrm{~d}_{t} \rho=-\left.\int_{\Sigma} f \wedge \rho\right|_{t_{0}} ^{t_{f}} .

\mathbf{d} S [A] \cdot \mathrm{d} _s f = \int _X \mathrm{d} _s f \wedge J \wedge \mathrm{d} t = - \int _X f \wedge \mathrm{d} _s J \wedge \mathrm{d} t = - \int _X f \wedge \mathrm{d} _t \rho = - \left. \int _\Sigma f \wedge \rho \,\right\rvert _{ t _0 } ^{ t _f } .

\left.\left(\mathrm{d}_{s} D-\rho\right)\right|_{t_{0}} ^{t_{f}}=0 .

\left. \left( \mathrm{d} _s D - \rho \right) \right\rvert _{t_0}^{t_f} = 0.