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- The Lorenz Equations
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- \[\begin{aligned}
- \dot{x} & = \sigma(y-x) \\
- \dot{y} & = \rho x - y - xz \\
- \dot{z} & = -\beta z + xy
- \end{aligned} \]
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- The Cauchy-Schwarz Inequality
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- \[ \left( \sum_{k=1}^n a_k b_k \right)^2 \leq \left( \sum_{k=1}^n a_k^2 \right) \left( \sum_{k=1}^n b_k^2 \right) \]
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- A Cross Product Formula
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- \[\mathbf{V}_1 \times \mathbf{V}_2 = \begin{vmatrix}
- \mathbf{i} & \mathbf{j} & \mathbf{k} \\
- \frac{\partial X}{\partial u} & \frac{\partial Y}{\partial u} & 0 \\
- \frac{\partial X}{\partial v} & \frac{\partial Y}{\partial v} & 0
- \end{vmatrix} \]
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- The probability of getting \(k\) heads when flipping \(n\) coins is
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- \[P(E) = {n \choose k} p^k (1-p)^{ n-k} \]
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- An Identity of Ramanujan
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- \[ \frac{1}{\Bigl(\sqrt{\phi \sqrt{5}}-\phi\Bigr) e^{\frac25 \pi}} =
- 1+\frac{e^{-2\pi}} {1+\frac{e^{-4\pi}} {1+\frac{e^{-6\pi}}
- {1+\frac{e^{-8\pi}} {1+\ldots} } } } \]
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- A Rogers-Ramanujan Identity
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- \[ 1 + \frac{q^2}{(1-q)}+\frac{q^6}{(1-q)(1-q^2)}+\cdots =
- \prod_{j=0}^{\infty}\frac{1}{(1-q^{5j+2})(1-q^{5j+3})}\]
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- Maxwell’s Equations
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- \[ \begin{aligned}
- \nabla \times \vec{\mathbf{B}} -\, \frac1c\, \frac{\partial\vec{\mathbf{E}}}{\partial t} & = \frac{4\pi}{c}\vec{\mathbf{j}} \\ \nabla \cdot \vec{\mathbf{E}} & = 4 \pi \rho \\
- \nabla \times \vec{\mathbf{E}}\, +\, \frac1c\, \frac{\partial\vec{\mathbf{B}}}{\partial t} & = \vec{\mathbf{0}} \\
- \nabla \cdot \vec{\mathbf{B}} & = 0 \end{aligned}
- \]
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