MathML

$$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$

$$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$$

$$d\left(P_1,P_2\right)=\sqrt{{\left(x_2-<!--\; -\; -->x_1\right)}^2+{\left(y_2-<!--\; -\; -->y_1\right)}^2+{\left(z_2-<!--\; -\; -->z_1\right)}^2}$$

$$\cos (\theta +\varphi )=\cos (\theta )\cos (\varphi )-\sin (\theta )\sin (\varphi )$$

$$A\subseteq B\leftrightarrow a∊A\to a∊B$$

$$x∊\left\{\frac{-b+\sqrt{\left(b^2-4ac\right)}}{2a},\frac{-b-\sqrt{\left(b^2-4ac\right)}}{2a}\right\}$$

$\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 7& 8& 9\end{array}\right]\times \left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]=\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 7& 8& 9\end{array}\right]$

$$A_{m,n}=\left(\begin{array}{cccc}{a}_{1,1}& a_{1,2}& \cdots & a_{1,n}\\ a_{2,1}& a_{2,2}& \cdots & a_{2,n}\\ ⋮& ⋮& \ddots & ⋮\\ a_{m,1}& a_{m,2}& \cdots & a_{m,n}\end{array}\right)$$

$${𝐕}_1\times {𝐕}_2=\left|\begin{array}{ccc}𝐢& 𝐣& 𝐤\\ \frac{\partial X}{\partial u}& \frac{\partial Y}{\partial u}& 0\\ \frac{\partial X}{\partial v}& \frac{\partial Y}{\partial v}& 0\end{array}\right|$$

$$f(x)=\{\begin{array}{ll}1/3& \text{if}0\le x\le 1;\\ 2/3& \text{if}3\le x\le 4;\\ 0& \text{elsewhere}.\end{array}$$

$$\frac{n!}{k!(n-k)!}=\left(\genfrac{}{}{0px}{}{n}{k}\right)$$

$$\sum _{n=1}^{+\mathrm{\infty }}\frac{1}{n^2}=\frac{{\pi }^2}{6}$$

$$\sum _{n=0}^{\mathrm{\infty }}\frac{1}{n}=\mathrm{\infty }$$

$${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-x}\mathrm{d}x$$

${\int }_a^{b}f\left(x\right)\mathrm{dx}\frac{\partial }{\partial x}F(x,y)+\frac{\partial }{\partial y}F(x,y)$

$$\underset{E}{\iiint <!--\; \iiint \; -->}f\left(x,y,z\right)dV=\underset{D}{\iint <!--\; \iint \; -->}\left[{\int <!--\; \int \; -->}_{u_1\left(x,z\right)}^{u_2\left(x,z\right)}f\left(x,y,z\right)dy\right]dA$$

$$f(a)=\frac{1}{2\pi i}\oint \frac{f(z)}{z-a}dz$$

$${\int }_D(\mathrm{\nabla }\cdot F)dV={\int }_{\partial D}F\cdot ndS$$

$$\frac{\mathrm{d}}{\mathrm{d}x}\left(arcsin\left(x\right)\right)=\frac{1}{\sqrt{\left(1-x^2\right)}}$$

$$\frac{\partial V}{\partial t}+\frac{1}{2}{\sigma }^2{S}^2\frac{{\partial }^{2}V}{\partial S^2}+rS\frac{\partial V}{\partial S}-rV=0$$

$$\frac{{\mathrm{\partial <!--\; \partial \; -->}}^{2}u}{\mathrm{\partial <!--\; \partial \; -->}{t}^2}=c^2{\mathrm{\nabla <!--\; \nabla \; -->}}^{2}u$$

$$\overrightarrow{\mathrm{\nabla }}\times \overrightarrow{F}=(\frac{\partial F_z}{\partial y}-\frac{\partial F_y}{\partial z})\mathbf{i}+(\frac{\partial F_x}{\partial z}-\frac{\partial F_z}{\partial x})\mathbf{j}+(\frac{\partial F_y}{\partial x}-\frac{\partial F_x}{\partial y})\mathbf{k}$$

$$({\mathrm{\nabla }}_{X}Y{)}^k=X^i({\mathrm{\nabla }}_{i}Y{)}^k=X^i(\frac{\partial Y^k}{\partial x^i}+{\mathrm{\Gamma }}_{im}^k{Y}^m)$$

$$f^{\prime }\left(x\right)=\underset{h\to 0}{lim}\frac{f\left(x+h\right)-f\left(x\right)}{h}$$

$$\underset{n\to \mathrm{\infty }}{lim}{\left(1+\frac{r}{n}\right)}^{\frac{t}{n}}={\mathrm{e}}^{rt}$$

$$\mathrm{\Gamma }(t)=\underset{n\to \mathrm{\infty }}{lim}\frac{n!n^t}{t(t+1)\cdots (t+n)}=\frac{1}{t}\prod _{n=1}^{\mathrm{\infty }}\frac{{\left(1+\frac{1}{n}\right)}^t}{1+\frac{t}{n}}=\frac{e^{-\gamma t}}{t}\prod _{n=1}^{\mathrm{\infty }}{\left(1+\frac{t}{n}\right)}^{-1}{e}^{\frac{t}{n}}$$

$$\mathrm{\Gamma }(t)={\int }_0^{+\mathrm{\infty }}{x}^{t-1}{e}^{-x}dx=\frac{1}{t}\prod _{n=1}^{\mathrm{\infty }}\frac{{\left(1+\frac{1}{n}\right)}^t}{1+\frac{t}{n}}\sim \sqrt{\frac{2\pi }{t}}{\left(\frac{t}{e}\right)}^t$$

$$\begin{array}{}\text{(1)}& \mathcal{L}\left\{f\left(t\right)\right\}={\int <!--\; \int \; -->}_0^{\mathrm{\infty <!--\; \infty \; -->}}{\mathbf{e}}^{-<!--\; -\; -->st}f\left(t\right)dt\end{array}$$

$$f(x)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{c}_n{e}^{2\pi i(n/T)x}=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\hat{f}({\xi }_n)e^{2\pi i{\xi }_{n}x}\mathrm{\Delta }\xi$$

$$\begin{array}{rl}\dot{x}& =\sigma (y-x)\\ \dot{y}& =\rho x-y-xz\\ \dot{z}& =-\beta z+xy\end{array}$$

$${\left(\sum _{k=1}^n{a}_k{b}_k\right)}^2\le \left(\sum _{k=1}^n{a}_k^2\right)\left(\sum _{k=1}^n{b}_k^2\right)$$

$$\begin{array}{rl}\nabla \times \stackrel{\rightharpoonup }{𝐁}-\frac{1}{c}\frac{\partial \stackrel{\rightharpoonup }{𝐄}}{\partial t}& =\frac{4\pi }{c}\stackrel{\rightharpoonup }{𝐣}\\ \nabla \cdot \stackrel{\rightharpoonup }{𝐄}& =4\pi \rho \\ \nabla \times \stackrel{\rightharpoonup }{𝐄}+\frac{1}{c}\frac{\partial \stackrel{\rightharpoonup }{𝐁}}{\partial t}& =\stackrel{\rightharpoonup }{\mathrm{𝟎}}\\ \nabla \cdot \stackrel{\rightharpoonup }{𝐁}& =0\end{array}$$

$${\displaystyle a_0+\frac{1}{{\displaystyle a_1+\frac{1}{{\displaystyle a_2+\frac{1}{{\displaystyle a_3+\frac{1}{{\displaystyle a_4}}}}}}}}}$$

$$\frac{1}{(\sqrt{\varphi \sqrt{5}}-\varphi )e^{\frac{25}{\pi }}}=1+\frac{e^{-2\pi }}{1+\frac{e^{-4\pi }}{1+\frac{e^{-6\pi }}{1+\frac{e^{-8\pi }}{1+\dots }}}}$$

$$f\left(x\right)∊O\left(g\left(x\right)\right)\leftrightarrow \exists x{}_0∊\mathbb{Z}\exists C∊ℝ\forall x>x{}_0\left|f\left(x\right)\right|\le C\left|g\left(x\right)\right|$$

$$\sigma =\sqrt{\frac{1}{N}\sum _{i=1}^N(x_i-\mu {)}^2}$$

${\mathrm{CH}}_4+2O_2\to {\mathrm{CO}}_2+2H_{2}O$

${\mathrm{Ca}}^{2+}+2{\mathrm{Cl}}^{-}+2{\mathrm{Ag}}^{+}+2{{\mathrm{NO}}_3}^{-}\to {\mathrm{Ca}}^{2+}+2{{\mathrm{NO}}_3}^{-}+2\mathrm{Ag}\mathrm{Cl}\mathrm{(s)}$