# MathML

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$\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1$
$x=\frac{-b±\sqrt{{b}^{2}-4ac}}{2a}$
$d\left({P}_{1},{P}_{2}\right)=\sqrt{{\left({x}_{2}-{x}_{1}\right)}^{2}+{\left({y}_{2}-{y}_{1}\right)}^{2}+{\left({z}_{2}-{z}_{1}\right)}^{2}}$
$\mathrm{cos}\left(\theta +\varphi \right)=\mathrm{cos}\left(\theta \right)\mathrm{cos}\left(\varphi \right)-\mathrm{sin}\left(\theta \right)\mathrm{sin}\left(\varphi \right)$
$A\subseteq B↔a∊A\to a∊B$
$x∊\left\{\frac{-b+\sqrt{\left({b}^{2}-4\phantom{\rule{0.16666666666666666em}{0ex}}a\phantom{\rule{0.16666666666666666em}{0ex}}c\right)}}{2\phantom{\rule{0.16666666666666666em}{0ex}}a},\phantom{\rule{0.16666666666666666em}{0ex}}\frac{-b-\sqrt{\left({b}^{2}-4\phantom{\rule{0.16666666666666666em}{0ex}}a\phantom{\rule{0.16666666666666666em}{0ex}}c\right)}}{2\phantom{\rule{0.16666666666666666em}{0ex}}a}\right\}$
$\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 7& 8& 9\end{array}\right]×\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]=\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 7& 8& 9\end{array}\right]$
${A}_{m,n}=\left(\begin{array}{cccc}{a}_{1,1}& {a}_{1,2}& \cdots & {a}_{1,n}\\ {a}_{2,1}& {a}_{2,2}& \cdots & {a}_{2,n}\\ ⋮& ⋮& \ddots & ⋮\\ {a}_{m,1}& {a}_{m,2}& \cdots & {a}_{m,n}\end{array}\right)$
${𝐕}_{1}×{𝐕}_{2}=|\begin{array}{ccc}𝐢& 𝐣& 𝐤\\ \frac{\partial X}{\partial u}& \frac{\partial Y}{\partial u}& 0\\ \frac{\partial X}{\partial v}& \frac{\partial Y}{\partial v}& 0\end{array}|$
$f\left(x\right)=\left\{\begin{array}{ll}1/3& \text{if}\phantom{\rule{1em}{0ex}}0\le x\le 1;\\ 2/3& \text{if}\phantom{\rule{1em}{0ex}}3\le x\le 4;\\ 0& \text{elsewhere}.\end{array}$
$\frac{n!}{k!\left(n-k\right)!}=\left(\genfrac{}{}{0px}{}{n}{k}\right)$
$\sum _{n=1}^{+\infty }\frac{1}{{n}^{2}}=\frac{{\pi }^{2}}{6}$
$\sum _{n=0}^{\infty }\frac{1}{n}=\infty$
${\int }_{0}^{\infty }{\mathrm{e}}^{-x}\phantom{\rule{0.16666666666666666em}{0ex}}\mathrm{d}x$
${\int }_{a}^{b}f\left(x\right)\mathrm{dx}\frac{\partial }{\partial x}F\left(x,y\right)+\frac{\partial }{\partial y}F\left(x,y\right)$
$\underset{E}{\iiint }f\left(x,y,z\right)\phantom{\rule{thinmathspace}{0ex}}dV=\underset{D}{\iint }\left[{\int }_{\phantom{\rule{thinmathspace}{0ex}}{u}_{1}\left(x,z\right)}^{\phantom{\rule{thinmathspace}{0ex}}{u}_{2}\left(x,z\right)}f\left(x,y,z\right)\phantom{\rule{thinmathspace}{0ex}}dy\right]\phantom{\rule{thinmathspace}{0ex}}dA$
$f\left(a\right)=\frac{1}{2\pi i}\oint \frac{f\left(z\right)}{z-a}dz$
${\int }_{D}\left(\mathrm{\nabla }\cdot F\right)dV={\int }_{\partial D}F\cdot ndS$
$\frac{\mathrm{d}}{\mathrm{d}x}\left(arcsin\left(x\right)\right)=\frac{1}{\sqrt{\left(1-{x}^{2}\right)}}$
$\frac{\partial V}{\partial t}+\frac{1}{2}{\sigma }^{2}{S}^{2}\frac{{\partial }^{2}V}{\partial {S}^{2}}+rS\frac{\partial V}{\partial S}-rV=0$
$\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{t}^{2}}={c}^{2}\phantom{\rule{thinmathspace}{0ex}}{\mathrm{\nabla }}^{2}u$
$\stackrel{\to }{\mathrm{\nabla }}×\stackrel{\to }{F}=\left(\frac{\partial {F}_{z}}{\partial y}-\frac{\partial {F}_{y}}{\partial z}\right)\mathbf{i}+\left(\frac{\partial {F}_{x}}{\partial z}-\frac{\partial {F}_{z}}{\partial x}\right)\mathbf{j}+\left(\frac{\partial {F}_{y}}{\partial x}-\frac{\partial {F}_{x}}{\partial y}\right)\mathbf{k}$
$\left({\mathrm{\nabla }}_{X}Y{\right)}^{k}={X}^{i}\left({\mathrm{\nabla }}_{i}Y{\right)}^{k}={X}^{i}\left(\frac{\partial {Y}^{k}}{\partial {x}^{i}}+{\mathrm{\Gamma }}_{im}^{k}{Y}^{m}\right)$
${f}^{\prime }\left(x\right)=\underset{h\to 0}{lim}\frac{f\left(x+h\right)-f\left(x\right)}{h}$
$\underset{n\to \infty }{lim}{\left(1+\frac{r}{n}\right)}^{\frac{t}{n}}={\mathrm{e}}^{r\phantom{\rule{0.16666666666666666em}{0ex}}t}$
$\mathrm{\Gamma }\left(t\right)=\underset{n\to \infty }{lim}\frac{n!\phantom{\rule{0.2777777777777778em}{0ex}}{n}^{t}}{t\phantom{\rule{0.2777777777777778em}{0ex}}\left(t+1\right)\cdots \left(t+n\right)}=\frac{1}{t}\prod _{n=1}^{\infty }\frac{{\left(1+\frac{1}{n}\right)}^{t}}{1+\frac{t}{n}}=\frac{{e}^{-\gamma t}}{t}\prod _{n=1}^{\infty }{\left(1+\frac{t}{n}\right)}^{-1}{e}^{\frac{t}{n}}$
$\mathrm{\Gamma }\left(t\right)={\int }_{0}^{+\infty }{x}^{t-1}{e}^{-x}dx=\frac{1}{t}\prod _{n=1}^{\infty }\frac{{\left(1+\frac{1}{n}\right)}^{t}}{1+\frac{t}{n}}\sim \sqrt{\frac{2\pi }{t}}{\left(\frac{t}{e}\right)}^{t}$
$\begin{array}{}\text{(1)}& \mathcal{L}\left\{f\left(t\right)\right\}={\int }_{\phantom{\rule{thinmathspace}{0ex}}0}^{\phantom{\rule{thinmathspace}{0ex}}\mathrm{\infty }}{\mathbf{e}}^{-s\phantom{\rule{thinmathspace}{0ex}}t}f\left(t\right)\phantom{\rule{thinmathspace}{0ex}}dt\end{array}$
$f\left(x\right)=\sum _{n=-\infty }^{\infty }{c}_{n}{e}^{2\pi i\left(n/T\right)x}=\sum _{n=-\infty }^{\infty }\stackrel{^}{f}\left({\xi }_{n}\right){e}^{2\pi i{\xi }_{n}x}\mathrm{\Delta }\xi$
$\begin{array}{rl}\stackrel{˙}{x}& =\sigma \left(y-x\right)\\ \stackrel{˙}{y}& =\rho x-y-xz\\ \stackrel{˙}{z}& =-\beta z+xy\end{array}$
${\left(\sum _{k=1}^{n}{a}_{k}{b}_{k}\right)}^{2}\le \left(\sum _{k=1}^{n}{a}_{k}^{2}\right)\left(\sum _{k=1}^{n}{b}_{k}^{2}\right)$
$\begin{array}{rl}\nabla ×\stackrel{⇀}{𝐁}-\phantom{\rule{0.16666666666666666em}{0ex}}\frac{1}{c}\phantom{\rule{0.16666666666666666em}{0ex}}\frac{\partial \stackrel{⇀}{𝐄}}{\partial t}& =\frac{4\pi }{c}\stackrel{⇀}{𝐣}\\ \nabla \cdot \stackrel{⇀}{𝐄}& =4\pi \rho \\ \nabla ×\stackrel{⇀}{𝐄}\phantom{\rule{0.16666666666666666em}{0ex}}+\phantom{\rule{0.16666666666666666em}{0ex}}\frac{1}{c}\phantom{\rule{0.16666666666666666em}{0ex}}\frac{\partial \stackrel{⇀}{𝐁}}{\partial t}& =\stackrel{⇀}{𝟎}\\ \nabla \cdot \stackrel{⇀}{𝐁}& =0\end{array}$
${a}_{0}+\frac{1}{{a}_{1}+\frac{1}{{a}_{2}+\frac{1}{{a}_{3}+\frac{1}{{a}_{4}}}}}$
$\frac{1}{\left(\sqrt{\varphi \sqrt{5}}-\varphi \right){e}^{\frac{25}{\pi }}}=1+\frac{{e}^{-2\pi }}{1+\frac{{e}^{-4\pi }}{1+\frac{{e}^{-6\pi }}{1+\frac{{e}^{-8\pi }}{1+\dots }}}}$
$f\left(x\right)∊O\left(g\left(x\right)\right)↔\exists x{}_{0}∊ℤ\phantom{\rule{0.2777777777777778em}{0ex}}\exists C∊ℝ\phantom{\rule{0.2777777777777778em}{0ex}}\forall x>x{}_{0}\phantom{\rule{0.2777777777777778em}{0ex}}|f\left(x\right)|\le C\phantom{\rule{0.16666666666666666em}{0ex}}|g\left(x\right)|$
$\sigma =\sqrt{\frac{1}{N}\sum _{i=1}^{N}\left({x}_{i}-\mu {\right)}^{2}}$
${\mathrm{CH}}_{4}+2{O}_{2}\to {\mathrm{CO}}_{2}+2{H}_{2}O$
${\mathrm{Ca}}^{2+}+2{\mathrm{Cl}}^{-}+2{\mathrm{Ag}}^{+}+2{{\mathrm{NO}}_{3}}^{-}\to {\mathrm{Ca}}^{2+}+2{{\mathrm{NO}}_{3}}^{-}+2\mathrm{Ag}\mathrm{Cl}\mathrm{\left(s\right)}$