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{}_{{}_{α}^{β}𝔄_{δ}^{γ}}^{{}_{ε}^{ζ}𝔅_{θ}^{η}}\prod_{{}_{ρ}^{σ}𝔈_{υ}^{τ}}^{{}_{ν}^{ξ}𝔇_{π}^{ο}}_{{}_{ϕ}^{χ}𝔉_{ω}^{ψ}}^{{}_{ι}^{κ}ℭ_{μ}^{λ}}

\forall A \exists P \forall B [B \in P ⟺ \forall C (C \in B \Rightarrow C \in A)]

Logic: \neg (p \land q) ⟺ (\neg p) \lor (\neg q)

Boolean algebra: \overline{⋃_{i = 1}^{n} A_{i}} = ⋂_{i = 1}^{n} \overline{A_{i}}

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

C (n, k) = C_{k}^{n} = {}_{n}C_{k} = (\binom{n}{k}) = \frac{n!}{k! (n - k)!}

\int_{0}^{1} x^{x} ⅆ x = \sum_{n = 1}^{∞} {(- 1)}^{n + 1} n^{- n}

c = \overset{complex number}{\overset{⏞}{\underset{real}{\underset{⏟}{a}} + \underset{imaginary}{\underset{⏟}{b ⅈ}}}}

M = [\begin{matrix} α_{1} & α_{1}^{q} & \dots & α_{1}^{q^{n - 1}} \\ α_{2} & α_{2}^{q} & \dots & α_{2}^{q^{n - 1}} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ α_{m} & α_{m}^{q} & \dots & α_{m}^{q^{n - 1}} \end{matrix}]

Spherical coordinates derivation of the volume of a sphere

(\frac{4}{3} π R^{3})

.
The formula

S

for a sphere of radius

R

in spherical coordinates is:

S = \{0 \leq ϕ \leq 2 π, 0 \leq θ \leq π, 0 \leq ρ \leq R\}

\begin{aligned} Volume & = \underset{S}{∭} ρ^{2} sin θ ⅆ ρ ⅆ θ ⅆ ϕ \\ = \int_{0}^{2 π} ⅆ ϕ \int_{0}^{π} sin θ ⅆ θ \int_{0}^{R} ρ^{2} ⅆ ρ \\ = ϕ |_{0}^{2 π} (- \cos θ) |_{0}^{π} \frac{1}{3} ρ^{3} |_{0}^{R} \\ = 2 π \times 2 \times \frac{1}{3} R^{3} \\ = \frac{4}{3} π R^{3} \end{aligned}

\frac{\sqrt{1 + \sqrt[3]{2 + \sqrt[5]{3 + \sqrt[7]{4 + \sqrt[11]{5 + \sqrt[13]{6 + \sqrt[17]{7 + \sqrt[19]{A}}}}}}}}}{ⅇ^{π}} = x^{‴}

(\begin{matrix} (\begin{matrix} a_{1} & a_{2} & a_{3} & a_{4} \\ a_{5} & a_{6} & a_{7} & a_{8} \end{matrix}) & (\begin{matrix} b_{1} \\ b_{2} \\ b_{3} \\ b_{4} \end{matrix}) \\ \begin{matrix} 0 & (\begin{matrix} c_{1} & c_{2} \\ c_{3} & c_{4} \end{matrix}) \end{matrix} \end{matrix})