86 |
87 | The Lorenz Equations
88 |
89 |
90 | \[\begin{aligned}
91 | \dot{x} & = \sigma(y-x) \\
92 | \dot{y} & = \rho x - y - xz \\
93 | \dot{z} & = -\beta z + xy
94 | \end{aligned} \]
95 |
96 |
97 |
98 |
99 | The Cauchy-Schwarz Inequality
100 |
101 |
102 | \[ \left( \sum_{k=1}^n a_k b_k \right)^2 \leq \left( \sum_{k=1}^n a_k^2 \right) \left( \sum_{k=1}^n b_k^2 \right) \]
103 |
104 |
105 |
106 |
107 | A Cross Product Formula
108 |
109 |
110 | \[\mathbf{V}_1 \times \mathbf{V}_2 = \begin{vmatrix}
111 | \mathbf{i} & \mathbf{j} & \mathbf{k} \\
112 | \frac{\partial X}{\partial u} & \frac{\partial Y}{\partial u} & 0 \\
113 | \frac{\partial X}{\partial v} & \frac{\partial Y}{\partial v} & 0
114 | \end{vmatrix} \]
115 |
116 |
117 |
118 |
119 | The probability of getting \(k\) heads when flipping \(n\) coins is
120 |
121 |
122 | \[P(E) = {n \choose k} p^k (1-p)^{ n-k} \]
123 |
124 |
125 |
126 |
127 | An Identity of Ramanujan
128 |
129 |
130 | \[ \frac{1}{\Bigl(\sqrt{\phi \sqrt{5}}-\phi\Bigr) e^{\frac25 \pi}} =
131 | 1+\frac{e^{-2\pi}} {1+\frac{e^{-4\pi}} {1+\frac{e^{-6\pi}}
132 | {1+\frac{e^{-8\pi}} {1+\ldots} } } } \]
133 |
134 |
135 |
136 |
137 | A Rogers-Ramanujan Identity
138 |
139 |
140 | \[ 1 + \frac{q^2}{(1-q)}+\frac{q^6}{(1-q)(1-q^2)}+\cdots =
141 | \prod_{j=0}^{\infty}\frac{1}{(1-q^{5j+2})(1-q^{5j+3})}\]
142 |
143 |
144 |
145 |
146 | Maxwell’s Equations
147 |
148 |
149 | \[ \begin{aligned}
150 | \nabla \times \vec{\mathbf{B}} -\, \frac1c\, \frac{\partial\vec{\mathbf{E}}}{\partial t} & = \frac{4\pi}{c}\vec{\mathbf{j}} \\ \nabla \cdot \vec{\mathbf{E}} & = 4 \pi \rho \\
151 | \nabla \times \vec{\mathbf{E}}\, +\, \frac1c\, \frac{\partial\vec{\mathbf{B}}}{\partial t} & = \vec{\mathbf{0}} \\
152 | \nabla \cdot \vec{\mathbf{B}} & = 0 \end{aligned}
153 | \]
154 |
155 |
156 |