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103 lines
2.9 KiB
HTML
103 lines
2.9 KiB
HTML
<!DOCTYPE html>
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<html>
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<!-- Copyright (c) 2012 Design Science, Inc. -->
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<head>
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<title>Math Test Page</title>
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<meta http-equiv="content-type" content="text/html; charset=UTF-8" />
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<!-- This script tag is needed to make calibre's ebook-viewer recpgnize that this file needs math typesetting -->
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<script type="text/x-mathjax-config">
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// This line adds numbers to all equations automatically, unless explicitly suppressed.
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MathJax.Hub.Config({ TeX: { equationNumbers: {autoNumber: "all"} } });
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</script>
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<style>
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h1 {text-align:center}
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h2 {
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font-weight: bold;
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background-color: #DDDDDD;
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padding: .2em .5em;
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margin-top: 1.5em;
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border-top: 3px solid #666666;
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border-bottom: 2px solid #999999;
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}
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</style>
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</head>
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<body>
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<h1>Sample Equations</h1>
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<h2>The Lorenz Equations</h2>
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<p>
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\begin{align}
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\dot{x} & = \sigma(y-x) \label{lorenz}\\
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\dot{y} & = \rho x - y - xz \\
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\dot{z} & = -\beta z + xy
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\end{align}
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</p>
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<h2>The Cauchy-Schwarz Inequality</h2>
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<p>\[
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\left( \sum_{k=1}^n a_k b_k \right)^{\!\!2} \leq
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\left( \sum_{k=1}^n a_k^2 \right) \left( \sum_{k=1}^n b_k^2 \right)
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\]</p>
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<h2>A Cross Product Formula</h2>
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<p>\[
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\mathbf{V}_1 \times \mathbf{V}_2 =
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\begin{vmatrix}
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\mathbf{i} & \mathbf{j} & \mathbf{k} \\
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\frac{\partial X}{\partial u} & \frac{\partial Y}{\partial u} & 0 \\
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\frac{\partial X}{\partial v} & \frac{\partial Y}{\partial v} & 0 \\
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\end{vmatrix}
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\]</p>
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<h2>The probability of getting \(k\) heads when flipping \(n\) coins is:</h2>
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<p>\[P(E) = {n \choose k} p^k (1-p)^{ n-k} \]</p>
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<h2>An Identity of Ramanujan</h2>
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<p>\[
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\frac{1}{(\sqrt{\phi \sqrt{5}}-\phi) e^{\frac25 \pi}} =
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1+\frac{e^{-2\pi}} {1+\frac{e^{-4\pi}} {1+\frac{e^{-6\pi}}
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{1+\frac{e^{-8\pi}} {1+\ldots} } } }
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\]</p>
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<h2>A Rogers-Ramanujan Identity</h2>
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<p>\[
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1 + \frac{q^2}{(1-q)}+\frac{q^6}{(1-q)(1-q^2)}+\cdots =
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\prod_{j=0}^{\infty}\frac{1}{(1-q^{5j+2})(1-q^{5j+3})},
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\quad\quad \text{for $|q|<1$}.
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\]</p>
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<h2>Maxwell's Equations</h2>
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<p>
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\begin{align}
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\nabla \times \vec{\mathbf{B}} -\, \frac1c\, \frac{\partial\vec{\mathbf{E}}}{\partial t} & = \frac{4\pi}{c}\vec{\mathbf{j}} \\
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\nabla \cdot \vec{\mathbf{E}} & = 4 \pi \rho \\
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\nabla \times \vec{\mathbf{E}}\, +\, \frac1c\, \frac{\partial\vec{\mathbf{B}}}{\partial t} & = \vec{\mathbf{0}} \\
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\nabla \cdot \vec{\mathbf{B}} & = 0
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\end{align}
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</p>
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<h2>In-line Mathematics</h2>
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<p>While display equations look good for a page of samples, the
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ability to mix math and text in a paragraph is also important. This
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expression \(\sqrt{3x-1}+(1+x)^2\) is an example of an inline equation. As
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you see, equations can be used this way as well, without unduly
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disturbing the spacing between lines.</p>
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<h2>References to equations</h2>
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<p>Here is a reference to the Lorenz Equations (\ref{lorenz}). Clicking on the equation number will take you back to the equation.</p>
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</body>
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</html>
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