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Examples

Einstein said $R_{\mu \nu} - {1 \over 2}g_{\mu \nu}\,R + g_{\mu \nu} \Lambda = {8 \pi G \over c^4} T_{\mu \nu}$, where $R$ is the Ricci scalar. This is known as the "Einstein field equation".

How about Maxwell equations...$x^2$

\begin{aligned} \nabla \times \vec{\mathbf{B}} -\, \frac1c\, \frac{\partial\vec{\mathbf{E}}}{\partial t} & = \frac{4\pi}{c}\vec{\mathbf{j}} \\ \nabla \cdot \vec{\mathbf{E}} & = 4 \pi \rho \\ \nabla \times \vec{\mathbf{E}}\, +\, \frac1c\, \frac{\partial\vec{\mathbf{B}}}{\partial t} & = \vec{\mathbf{0}} \\ \nabla \cdot \vec{\mathbf{B}} & = 0 \end{aligned}

Here is another frame

This Identity of Ramanujan is really cool!

\begin{aligned} \frac{1}{\Bigl(\sqrt{\phi \sqrt{5}}-\phi\Bigr) e^{\frac25 \pi}} = 1+\frac{e^{-2\pi}} {1+\frac{e^{-4\pi}} {1+\frac{e^{-6\pi}} {1+\frac{e^{-8\pi}} {1+\ldots} } } } \end{aligned}

\begin{aligned} \frac{1}{\Bigl(\sqrt{\phi \sqrt{5}}-\phi\Bigr) e^{\frac25 \pi}} = 1+\frac{e^{-2\pi}} {1+\frac{e^{-4\pi}} {1+\frac{e^{-6\pi}} {1+\frac{e^{-8\pi}} {1+\ldots} } } } \end{aligned}

or blue

and now we have $x^7+x+1$ and $\sin(\sqrt{x^2+1})$ and within the line or

$\displaystyle{\int_0^{x} \frac{dt}{\ln(t)}\qquad \partial_t u=\frac{\partial u}{\partial t}}$

\begin{align*} a^2 &= 1 \\ a &= \pm 1 \end{align*}

This is a nice one.

$\displaystyle{1 +\frac{q^2}{ (1-q)}+\frac{q^6}{ (1-q)(1-q^2)}+\cdots =\prod_{j=0}^{\infty}\frac{1}{ (1-q^{ 5j+2})(1-q^{ 5j+3})}.}$

Here a set of ode's

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

Cauchy-Schwarz

$\displaystyle{\left( \sum_{k=1}^n a_k b_k \right)^2 \leq \left( \sum_{k=1}^n a_k^2 \right) \left( \sum_{k=1}^n b_k^2 \right)}$

Cross product...

$$\mathbf{V}_1 \times \mathbf{V}_2 = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \frac{\partial X}{\partial u} & \frac{\partial Y}{\partial u} & 0 \\ \frac{\partial X}{\partial v} & \frac{\partial Y}{\partial v} & 0 \end{vmatrix}$$

Here is other matrix example...

$$\mathbf{X}= \left( \begin{array}{ccc} x_1 & x_2 & \ldots \\ x_3 & x_4 & \ldots \\ \vdots & \vdots & \ddots \end{array} \right)$$