mirror of
https://github.com/kovidgoyal/calibre.git
synced 2025-10-31 02:27:01 -04:00
103 lines
2.8 KiB
HTML
103 lines
2.8 KiB
HTML
<!DOCTYPE html>
|
|
<html>
|
|
<!-- Copyright (c) 2012 Design Science, Inc. -->
|
|
<head>
|
|
<title>Math Test Page</title>
|
|
<meta http-equiv="content-type" content="text/html; charset=UTF-8" />
|
|
|
|
<!-- This script tag is needed to make calibre's ebook-viewer recpgnize that this file needs math typesetting -->
|
|
<script type="text/x-mathjax-config">
|
|
// This line adds numbers to all equations automatically, unless explicitly suppressed.
|
|
MathJax.tex = {tags: 'all'};
|
|
</script>
|
|
|
|
<style>
|
|
h1 {text-align:center}
|
|
h2 {
|
|
font-weight: bold;
|
|
background-color: #DDDDDD;
|
|
padding: .2em .5em;
|
|
margin-top: 1.5em;
|
|
border-top: 3px solid #666666;
|
|
border-bottom: 2px solid #999999;
|
|
}
|
|
</style>
|
|
</head>
|
|
<body>
|
|
|
|
<h1>Sample Equations</h1>
|
|
|
|
<h2>The Lorenz Equations</h2>
|
|
|
|
<p>
|
|
\begin{align}
|
|
\dot{x} & = \sigma(y-x) \label{lorenz}\\
|
|
\dot{y} & = \rho x - y - xz \\
|
|
\dot{z} & = -\beta z + xy
|
|
\end{align}
|
|
</p>
|
|
|
|
<h2>The Cauchy-Schwarz Inequality</h2>
|
|
|
|
<p>\[
|
|
\left( \sum_{k=1}^n a_k b_k \right)^{\!\!2} \leq
|
|
\left( \sum_{k=1}^n a_k^2 \right) \left( \sum_{k=1}^n b_k^2 \right)
|
|
\]</p>
|
|
|
|
<h2>A Cross Product Formula</h2>
|
|
|
|
<p>\[
|
|
\mathbf{V}_1 \times \mathbf{V}_2 =
|
|
\begin{vmatrix}
|
|
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
|
|
\frac{\partial X}{\partial u} & \frac{\partial Y}{\partial u} & 0 \\
|
|
\frac{\partial X}{\partial v} & \frac{\partial Y}{\partial v} & 0 \\
|
|
\end{vmatrix}
|
|
\]</p>
|
|
|
|
<h2>The probability of getting \(k\) heads when flipping \(n\) coins is:</h2>
|
|
|
|
<p>\[P(E) = {n \choose k} p^k (1-p)^{ n-k} \]</p>
|
|
|
|
<h2>An Identity of Ramanujan</h2>
|
|
|
|
<p>\[
|
|
\frac{1}{(\sqrt{\phi \sqrt{5}}-\phi) e^{\frac25 \pi}} =
|
|
1+\frac{e^{-2\pi}} {1+\frac{e^{-4\pi}} {1+\frac{e^{-6\pi}}
|
|
{1+\frac{e^{-8\pi}} {1+\ldots} } } }
|
|
\]</p>
|
|
|
|
<h2>A Rogers-Ramanujan Identity</h2>
|
|
|
|
<p>\[
|
|
1 + \frac{q^2}{(1-q)}+\frac{q^6}{(1-q)(1-q^2)}+\cdots =
|
|
\prod_{j=0}^{\infty}\frac{1}{(1-q^{5j+2})(1-q^{5j+3})},
|
|
\quad\quad \text{for $|q|<1$}.
|
|
\]</p>
|
|
|
|
<h2>Maxwell's Equations</h2>
|
|
|
|
<p>
|
|
\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}
|
|
</p>
|
|
|
|
<h2>Inline Mathematics</h2>
|
|
|
|
<p>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.</p>
|
|
|
|
<h2>References to equations</h2>
|
|
|
|
<p>Here is a reference to the Lorenz Equations (\ref{lorenz}). Clicking on the equation number will take you back to the equation.</p>
|
|
|
|
</body>
|
|
</html>
|