sample-mathematics.html

The Lorenz Equations

x ˙ y ˙ z ˙ = σ (y - x) = ρ x - y - x z = - β z + x y

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

The Cauchy-Schwarz Inequality

(\sum k = 1 n a k b k) 2 \leq (\sum k = 1 n a 2 k) (\sum k = 1 n b 2 k)

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

A Cross Product Formula

V 1 \times V 2 = ∣ ∣ ∣ ∣ ∣ ∣ ∣ i \partial X \partial u \partial X \partial v j \partial Y \partial u \partial Y \partial v k 00 ∣ ∣ ∣ ∣ ∣ ∣ ∣

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

The probability of getting
$k$ $k$
heads when flipping
$n$ $n$
coins is

P (E) = (n k) p k (1 - p) n - k

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

An Identity of Ramanujan

1 ( ϕ 5 \sqrt - - - - \sqrt - ϕ ) e 2 5 π = 1 + e - 2 π 1 + e - 4 π 1 + e - 6 π 1 + e - 8 π 1 + \dots

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

A Rogers-Ramanujan Identity

1+q2(1−q)+q6(1−q)(1−q2)+⋯=∏j=0∞1(1−q5j+2)(1−q5j+3),for |q|\<1.

$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

\nabla \times B ⃗ - 1 c \partial E ⃗ \partial t \nabla \cdot E ⃗ \nabla \times E ⃗ + 1 c \partial B ⃗ \partial t \nabla \cdot B ⃗ = 4 π c j ⃗ = 4 π ρ = 0 ⃗ = 0

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

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

3 x - 1 - - - - - \sqrt + (1 + x) 2

$\sqrt{3x-1}+(1+x)^2$