Bernoulli Trials P ( E ) = n k p k ( 1 - p ) n - k Cauchy-Schwarz Inequality k = 1 n a k b k 2 ≤ k = 1 n a k 2 k = 1 n b k 2 Cauchy Formula f ( z )   · Ind γ ( z ) = 1 2 π i γ f ( ξ ) ξ - z   d ξ Cross Product V 1 × V 2 = i j k ∂ X ∂ u ∂ Y ∂ u 0 ∂ X ∂ v ∂ Y ∂ v 0 Vandermonde Determinant 1 1 ⋯ 1 v 1 v 2 ⋯ v n v 1 2 v 2 2 ⋯ v n 2 ⋮ ⋮ ⋱ ⋮ v 1 n - 1 v 2 n - 1 ⋯ v n n - 1 = 1 ≤ i < j ≤ n ( v j - v i ) Lorenz Equations x = σ ( y - x ) y = ρ x - y - x z z = - β z + x y Maxwell's Equations ∇ × B -   1 c   ∂ E ∂ t = 4 π c   j ∇ · E = 4 π ρ ∇ × E   +   1 c   ∂ B ∂ t = 0 ∇ · B = 0 Einstein Field Equations R μ ν - 1 2   g μ ν   R = 8 π G c 4   T μ ν Ramanujan Identity 1 ( φ 5 - φ ) e 25 π = 1 + e - 2 π 1 + e - 4 π 1 + e - 6 π 1 + e - 8 π 1 + … Another Ramanujan identity k = 1 ∞ 1 2 ⌊ k · φ ⌋ = 1 2 0 + 1 2 1 + ⋯ Rogers-Ramanujan Identity 1 + k = 1 ∞ q k 2 + k ( 1 - q ) ( 1 - q 2 ) ⋯ ( 1 - q k ) = j = 0 ∞ 1 ( 1 - q 5 j + 2 ) ( 1 - q 5 j + 3 ) ,       f o r   | q | < 1 . Commutative Diagram H ← K ↓ ↑ H → K