Math Notes

LaTex is pretty messy when read textually and I don’t use it very often so I copied a nice page from the MathJax website to use as reference whenever I need to write math equations on the blog.

The Lorenz Equations

\[ \begin{aligned} \dot{x} & = \sigma(y-x) \cr \dot{y} & = \rho x - y - xz \cr \dot{z} & = -\beta z + xy \end{aligned} \]
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\[
	\begin{aligned}
		\dot{x} & = \sigma(y-x) \cr
		\dot{y} & = \rho x - y - xz \cr
		\dot{z} & = -\beta z + xy
	\end{aligned} 
\]

The Cauchy-Schwarz Inequality

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

A Cross Product Formula

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

The probability of getting \(k\) heads when flipping \(n\) coins is

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

An Identity of Ramanujan

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

A Rogers-Ramanujan Identity

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

Maxwell’s Equations

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

	\nabla \times \vec{\mathbf{B}} -\, \frac1c\,
	\frac{\partial\vec{\mathbf{E}}}{\partial t} &
	=
	\frac{4\pi}{c}\vec{\mathbf{j}}

	\cr

	\nabla \cdot \vec{\mathbf{E}} &
	=
	4 \pi \rho

	\cr

	\nabla \times \vec{\mathbf{E}}\, +\, \frac1c\,
	\frac{\partial\vec{\mathbf{B}}}{\partial t} & = \vec{\mathbf{0}}

	\cr

	\nabla \cdot \vec{\mathbf{B}} & 
	= 
	0 

	\end{aligned}
\]

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, MathJax equations can be used this way as well, without unduly disturbing the spacing between lines.

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This expression \(\sqrt{3x-1}+(1+x)^2\) is an example of an inline equation.