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## Mathematical & Chemical API Integration

December 23, 2018

## MathML Integral Formula

$\begin{array}{r}f\left(a\right)=\frac{1}{2\pi i}{\oint }_{\gamma }\frac{f\left(z\right)}{z-a}dz\end{array}$

## or Tex base ... Integral Formula

\begin{align} f(a) = \frac{1}{2\pi i} \oint_{\gamma}\frac{f(z)}{z-a}dz \end{align}

## Dynamic Equations - step by step

Expand the following:

\begin{align} (x+1)^2 &\cssId{Step1}{= (x+1)(x+1)}\\ &\cssId{Step2}{= x(x+1) + 1(x+1)}\\ &\cssId{Step3}{= (x^2+x) + (x+1)}\\ &\cssId{Step4}{= x^2 + (x + x) + 1}\\ &\cssId{Step5}{= x^2+2x+1}\\ \end{align}

\begin{align} When $a \ne 0$, there are two solutions to $ax^2 bx c = 0$ and they are \end{align}

\begin{align} x = {-b \pm \sqrt{b^2-4ac} \over 2a}. \end{align}

## Double angle formula for Cosines

\begin{align} \cos(θ+φ)=\cos(θ)\cos(φ)−\sin(θ)\sin(φ) \end{align}

## Gauss' Divergence Theorem

\begin{align} \int_D ({\nabla\cdot} F)dV=\int_{\partial D} F\cdot ndS \end{align}

## Curl of a Vector Field

\begin{align} \vec{\nabla} \times \vec{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} \end{align}

## Standard Deviation

\begin{align} \sigma = \sqrt{ \frac{1}{N} \sum_{i=1}^N (x_i -\mu)^2} \end{align}

## Definition of Christoffel Symbols

\begin{align} (\nabla_X Y)^k = X^i (\nabla_i Y)^k = X^i \left( \frac{\partial Y^k}{\partial x^i} + \Gamma_{im}^k Y^m \right) \end{align}

## The Lorenz Equations

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

## Cauchy's Integral Formula

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

## The Cauchy-Schwarz Inequality

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

## A Cross Product Formula

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

## The probability of getting $$k$$ heads when flipping $$n$$ coins is:

$P(E) = {n \choose k} p^k (1-p)^{ n-k}$

## An Identity of Ramanujan

$\frac{1}{(\sqrt{\phi \sqrt{5}}-\phi) e^{\frac25 \pi}} = 1 \frac{e^{-2\pi}} {1 \frac{e^{-4\pi}} {1 \frac{e^{-6\pi}} {1 \frac{e^{-8\pi}} {1 \ldots} } } }$

## A Rogers-Ramanujan Identity

$1 \frac{q^2}{(1-q)} \frac{q^6}{(1-q)(1-q^2)} \cdots = \prod_{j=0}^{\infty}\frac{1}{(1-q^{5j 2})(1-q^{5j 3})}, \quad\quad \text{for |q|<1}.$

## Maxwell's Equations

\begin{align} \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{align}

## In-line Mathematics

Finally, while display equations look good for a page of samples, the ability to mix math and text in a paragraph is also important. This expression $$\sqrt{3x-1} (1 x)^2$$ is an example of an inline equation. As you see, equations can be used this way as well, without unduly disturbing the spacing between lines.

## Trig Identity Formulas

Use these fundemental formulas of trigonometry to help solve problems by re-writing expressions in another equivalent form.

### Basic Identities

$\sin(x)=\frac{1}{\csc(x)}$

$\cos(x)=\frac{1}{\sec(x)}$

$\tan(x)=\frac{1}{\cot(x)}$

$\sec(x)=\frac{1}{\cos(x)}$

$\csc(x)=\frac{1}{\sin(x)}$

$\cot(x)=\frac{1}{\tan(x)}$

$\tan(x)=\frac{\sin(x)}{\cos(x)}$

$\sin(-x)=-\sin(x)$

$\cos(-x)=\cos(x)$

$\tan(-x)=-\tan(x)$

### Pythagorean Identities

$\sin^2(x) \cos^2(x)=1$

$1 \tan^2(x)=\sec^2(x)$

$1 \cot^2(x)=\csc^2(x)$

### Sum and Difference Formulas

$\sin(a b)=\sin(a)\cos(b) \cos(a)\sin(b)$

$\sin(a-b)=\sin(a)\cos(b)-\cos(a)\sin(b)$

$\cos(a b)=\cos(a)\cos(b)-\sin(a)\sin(b)$

$\cos(a-b)=\cos(a)\cos(b) \sin(a)\sin(b)$

$\tan(a b)=\frac{\tan(a) \tan(b)}{1-\tan(a)\tan(b)}$

$\tan(a-b)=\frac{\tan(a)-\tan(b)}{1 \tan(a)\tan(b)}$

$\sin(x) \sin(y)=2\sin(\frac{x y}{2})\cos(\frac{x-y}{2})$

$\sin(x)-\sin(y)=2\cos(\frac{x y}{2})\sin(\frac{x-y}{2})$

$\cos(x) \cos(y)=2\cos(\frac{x y}{2})\cos(\frac{x-y}{2})$

$\cos(x)-\cos(y)=-2\sin(\frac{x y}{2})\sin(\frac{x-y}{2})$

### Double Angle Formulas

$\sin(2x)=2\sin(x)\cos(x)$

$\cos(2x)=\cos^2(x)-\sin^2(x)=1-2\sin^2(x) = 2\cos^2(x)-1$

### Half Angle Formulas

$\sin(\frac{x}{2})=\pm\sqrt{\frac{1-\cos(x)}{2}}$

$\cos(\frac{x}{2})=\pm\sqrt{\frac{1 \cos(x)}{2}}$

$\tan(\frac{x}{2})=\pm\sqrt{\frac{1-\cos(x)}{1 \cos(x)}}=\frac{1-\cos(x)}{\sin(x)}=\frac{\sin(x)}{1 \cos(x)}$

### Trigonometric Products

$\sin(x)\cos(y)=\frac{\sin(x y) \sin(x-y)}{2}$

$\cos(x)\cos(y)=\frac{\cos(x y) \cos(x-y)}{2}$

$\sin(x)\sin(y)=\frac{\cos(x-y)-\cos(x y)}{2}$

## In-line Chemical Equations

Finally, while display equations look good for a page of samples, the ability to mix Chemical Equations and text in a paragraph is also important. This expression $$\ce{Zn^2 <=>[ 2OH-][ 2H ] \underset{\text{amphoteres Hydroxid}}{\ce{Zn(OH)2 v}} <=>[ 2OH-][ 2H ] \underset{\text{Hydroxozikat}}{\ce{[Zn(OH)4]^2-}}}$$ is an example of an inline equation. As you see, equations can be used this way as well, without unduly disturbing the spacing between lines.