$∏ 𝔈 υ τ ρ σ 𝔇 π ο ν ξ 𝔄 δ γ α β 𝔅 θ η ε ζ 𝔉 ω ψ ϕ χ ℭ μ λ ι κ$ $∀ A ∃ P ∀ B B ∈ P ⟺ ∀ C C ∈ B ⇒ C ∈ A$ $x = − b ± b 2 − 4 ⁢ a ⁢ c 2 ⁢ a$ $C n k = C k n = C k n = n k = n ! k ! ⁢ n − k !$ $∫ 0 1 x x ⁢ ⅆ x = ∑ n = 1 ∞ − 1 n + 1 ⁢ n − n$ $c = a ⏟ real + b ⁢ ⅈ ⏟ imaginary ⏞ complex number$ $M = α 1 α 1 q … α 1 q n − 1 α 2 α 2 q … α 2 q n − 1 ⋮ ⋮ ⋱ ⋮ α m α m q … α m q n − 1$     Spherical coordinates derivation of the volume of a sphere $4 3 ⁢ π R 3$ .
The formula $S$ for a sphere of radius $R$ in spherical coordinates is:
$S=\left\{0\le \varphi \le 2\pi ,0\le \theta \le \pi ,0\le \rho \le R\right\}$
$Volume = ∭ S ρ 2 ⁢ sin ⁡ θ ⁢ ⅆ ρ ⁢ ⅆ θ ⁢ ⅆ ϕ = ∫ 0 2 ⁢ π ⅆ ϕ ⁢ ∫ 0 π sin ⁡ θ ⁢ ⅆ θ ⁢ ∫ 0 R ρ 2 ⁢ ⅆ ρ = ϕ | 0 2 ⁢ π ⁢ − cos ⁡ θ | 0 π ⁢ 1 3 ⁢ ρ 3 | 0 R = 2 ⁢ π × 2 × 1 3 ⁢ R 3 = 4 3 ⁢ π R 3$
$1 + 2 + 3 + 4 + 5 + 6 + 7 + A 19 17 13 11 7 5 3 ⅇ π = x ‴$ $a 1 a 2 a 3 a 4 a 5 a 6 a 7 a 8 b 1 b 2 b 3 b 4 0 c 1 c 2 c 3 c 4$