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\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1

x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}

d (P_{1}, P_{2}) = \sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2} + {(z_{2} - z_{1})}^{2}}

\cos (θ + ϕ) = \cos (θ) \cos (ϕ) - \sin (θ) \sin (ϕ)

A \subseteq B \leftrightarrow a ∊ A \to a ∊ B

x ∊ {\frac{- b + \sqrt{(b^{2} - 4 a c)}}{2 a}, \frac{- b - \sqrt{(b^{2} - 4 a c)}}{2 a}}

[\begin{matrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{matrix}] \times [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}] = [\begin{matrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{matrix}]

A_{m, n} = (\begin{matrix} a_{1, 1} & a_{1, 2} & \dots & a_{1, n} \\ a_{2, 1} & a_{2, 2} & \dots & a_{2, n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ a_{m, 1} & a_{m, 2} & \dots & a_{m, n} \end{matrix})

𝐕_{1} \times 𝐕_{2} = | \begin{matrix} 𝐢 & 𝐣 & 𝐤 \\ \frac{\partial X}{\partial u} & \frac{\partial Y}{\partial u} & 0 \\ \frac{\partial X}{\partial v} & \frac{\partial Y}{\partial v} & 0 \end{matrix} |

f (x) = {\begin{cases} 1 / 3 & if 0 \leq x \leq 1; \\ 2 / 3 & if 3 \leq x \leq 4; \\ 0 & elsewhere . \end{cases}

\frac{n!}{k! (n - k)!} = (\binom{n}{k})

\sum_{n = 1}^{+ ∞} \frac{1}{n^{2}} = \frac{π^{2}}{6}

\sum_{n = 0}^{∞} \frac{1}{n} = ∞

\int_{0}^{∞} e^{- x} d x

\int_{a}^{b} f (x) dx \frac{\partial}{\partial x} F (x, y) + \frac{\partial}{\partial y} F (x, y)

\underset{E}{∭} f (x, y, z) d V = \iint_{D} [\int_{u_{1} (x, z)}^{u_{2} (x, z)} f (x, y, z) d y] d A

f (a) = \frac{1}{2 π i} \oint \frac{f (z)}{z - a} d z

\int_{D} (\nabla \cdot F) d V = \int_{\partial D} F \cdot n d S

\frac{d}{d x} (arcsin (x)) = \frac{1}{\sqrt{(1 - x^{2})}}

\frac{\partial V}{\partial t} + \frac{1}{2} σ^{2} S^{2} \frac{\partial^{2} V}{\partial S^{2}} + r S \frac{\partial V}{\partial S} - r V = 0

\frac{\partial^{2} u}{\partial t^{2}} = c^{2} \nabla^{2} u

\vec{\nabla} \times \vec{F} = (\frac{\partial F_{z}}{\partial y} - \frac{\partial F_{y}}{\partial z}) i + (\frac{\partial F_{x}}{\partial z} - \frac{\partial F_{z}}{\partial x}) j + (\frac{\partial F_{y}}{\partial x} - \frac{\partial F_{x}}{\partial y}) k

(\nabla_{X} Y)^{k} = X^{i} (\nabla_{i} Y)^{k} = X^{i} (\frac{\partial Y^{k}}{\partial x^{i}} + Γ_{i m}^{k} Y^{m})

f^{'} (x) = lim_{h \to 0} \frac{f (x + h) - f (x)}{h}

lim_{n \to ∞} {(1 + \frac{r}{n})}^{\frac{t}{n}} = e^{r t}

Γ (t) = lim_{n \to ∞} \frac{n! n^{t}}{t (t + 1) \dots (t + n)} = \frac{1}{t} \prod_{n = 1}^{∞} \frac{{(1 + \frac{1}{n})}^{t}}{1 + \frac{t}{n}} = \frac{e^{- γ t}}{t} \prod_{n = 1}^{∞} {(1 + \frac{t}{n})}^{- 1} e^{\frac{t}{n}}

Γ (t) = \int_{0}^{+ ∞} x^{t - 1} e^{- x} d x = \frac{1}{t} \prod_{n = 1}^{∞} \frac{{(1 + \frac{1}{n})}^{t}}{1 + \frac{t}{n}} \sim \sqrt{\frac{2 π}{t}} {(\frac{t}{e})}^{t}

\begin{matrix} (1) & L {f (t)} = \int_{0}^{\infty} e^{- s t} f (t) d t \end{matrix}

f (x) = \sum_{n = - ∞}^{∞} c_{n} e^{2 π i (n / T) x} = \sum_{n = - ∞}^{∞} \hat{f} (ξ_{n}) e^{2 π i ξ_{n} x} Δ ξ

\begin{aligned} \dot{x} & = σ (y - x) \\ \dot{y} & = ρ x - y - x z \\ \dot{z} & = - β z + x y \end{aligned}

{(\sum_{k = 1}^{n} a_{k} b_{k})}^{2} \leq (\sum_{k = 1}^{n} a_{k}^{2}) (\sum_{k = 1}^{n} b_{k}^{2})

\begin{aligned} \nabla \times \overset{⇀}{𝐁} - \frac{1}{c} \frac{\partial \overset{⇀}{𝐄}}{\partial t} & = \frac{4 π}{c} \overset{⇀}{𝐣} \\ \nabla \cdot \overset{⇀}{𝐄} & = 4 π ρ \\ \nabla \times \overset{⇀}{𝐄} + \frac{1}{c} \frac{\partial \overset{⇀}{𝐁}}{\partial t} & = \overset{⇀}{𝟎} \\ \nabla \cdot \overset{⇀}{𝐁} & = 0 \end{aligned}

a_{0} + \frac{1}{a_{1} + \frac{1}{a_{2} + \frac{1}{a_{3} + \frac{1}{a_{4}}}}}

\frac{1}{(\sqrt{ϕ \sqrt{5}} - ϕ) e^{\frac{25}{π}}} = 1 + \frac{e^{- 2 π}}{1 + \frac{e^{- 4 π}}{1 + \frac{e^{- 6 π}}{1 + \frac{e^{- 8 π}}{1 + \dots}}}}

f (x) ∊ O (g (x)) \leftrightarrow \exists x_{0} ∊ ℤ \exists C ∊ ℝ \forall x > x_{0} | f (x) | \leq C | g (x) |

σ = \sqrt{\frac{1}{N} \sum_{i = 1}^{N} (x_{i} - μ)^{2}}

{CH}_{4} + 2 O_{2} \to {CO}_{2} + 2 H_{2} O

{Ca}^{2+} + 2 {Cl}^{-} + 2 {Ag}^{+} + 2 {NO}_{3}^{-} \to {Ca}^{2+} + 2 {NO}_{3}^{-} + 2 Ag Cl (s)