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Axiom of power set ∀ A ∃ P ∀ B [ B ∈ P ⟺ ∀ C ( C ∈ B ⇒ C ∈ A ) ]
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De Morgan's law:
Logic: ¬ ( p ∧ q ) ⟺ ( ¬ p ) ∨ ( ¬ q ) Boolean algebra: ⋃ i = 1 n A i ‾ = ⋂ i = 1 n A i ‾ -
Quadratic formula:
x = − b ± b 2 − 4 a c 2 a -
Binomial coefficient:
C ( n , k ) = C k n = C k n = ( n k ) = n ! k ! ( n − k ) ! -
Sophomore's law
∫ 0 1 x x ⅆ x = ∑ n = 1 ∞ ( − 1 ) n + 1 n − n -
Divergence
∇ · v → = ∂ v x ∂ x + ∂ v y ∂ y + ∂ v z ∂ z -
Complex number
c = a ⏟ real + b ⅈ ⏟ imaginary ⏞ complex number -
Moore determinant
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 ] -
Sphere volume
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 = { 0 ≤ ϕ ≤ 2 π , 0 ≤ θ ≤ π , 0 ≤ ρ ≤ R }
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Schwinger-Dyson equation
⟨ ψ | 𝒯 { δ δ ϕ F [ ϕ ] } | ψ ⟩ = − ⅈ ⟨ ψ | 𝒯 { F [ ϕ ] δ δ ϕ S [ ϕ ] } | ψ ⟩ -
Differentiable manifold (tangent vector)
γ 1 ≡ γ 2 ⟺ { γ 1 ( 0 ) = γ 2 ( 0 ) = p , and ⅆ ⅆ t ϕ ∘ γ 1 ( t ) | t = 0 = ⅆ ⅆ t ϕ ∘ γ 2 ( t ) | t = 0 -
Cichoń's diagram
cov ( ℒ ) ⟶ non ( 𝒦 ) ⟶ cof ( 𝒦 ) ⟶ cof ( ℒ ) ⟶ 2 ℵ 0 ↑ ↑ ↑ ↑ 𝔟 ⟶ 𝔡 ↑ ↑ ℵ 1 ⟶ add ( ℒ ) ⟶ add ( 𝒦 ) ⟶ cov ( 𝒦 ) ⟶ non ( ℒ ) -
Multiscripts & Greek alphabet
∏ 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 -
Nested roots
1 + 2 + 3 + 4 + 5 + 6 + 7 + A 19 17 13 11 7 5 3 ⅇ π = x ‴ -
Nested matrices
( ( 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 ) ) -
font sizes
scriptlevel : − 3 , − 2 , − 1 , 0 , 1
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