Foundations

If columns of invertible P are an ordered basis B in standard coordinates, v=P[v]B.

From independent real vectors v₁,…,vₖ form wⱼ=vⱼ−Σ_(i<j)[(vⱼ·wᵢ)/(wᵢ·wᵢ)]wᵢ.

For uᵢ₊₁=uᵢ+hφ(tᵢ,uᵢ,h), if the exact-value one-step defect is bounded by Chᵖ⁺¹ and |∂ᵤφ|≤L, L>0, then exact initialization gives |u(tᵢ)−uᵢ|≤(Chᵖ/L)(eᴸ⁽ᵗⁱ⁻ᵃ⁾−1).

An invertible affine transformation of ℝⁿ has form T(x)=Ax+b with A invertible.

Every distance-preserving bijection of ℝⁿ has form T(x)=Qx+b with QᵀQ=I; conversely each such map is an isometry.

A bijection of ℝⁿ uniformly scaling all distances by s>0 is a similarity and has form T(x)=sQx+b with Q orthogonal.

The symmetries of a figure S⊂ℝⁿ are the Euclidean isometries g with g(S)=S.

For a one-to-one C¹ transformation T mapping a planar or spatial integration region G onto R, with nonzero interior Jacobian, ∫R f(x)dx=∫G f(T(u))|det DT(u)|du.

A regular C¹ plane or space curve r(t), a≤t≤b, has length ∫ₐᵇ‖r′(t)‖dt.

A C¹ regular parameterization r(u,v) has rᵤ×rᵥ≠0 and area element dS=‖rᵤ×rᵥ‖du dv.

The subgroup generated by a is ⟨a⟩={aᵏ:k∈Z}; its order is the smallest positive n with aⁿ=e, or infinite if none exists.

Over a field F, f=gq+r has unique q,r for nonzero g, with r=0 or deg r<deg g.

A nonconstant polynomial over F is irreducible if it cannot factor into two positive-degree polynomials over F.

The sum of squares of k independent standard normal variables has chi-square law χ²ₖ=Gamma(k/2,1/2), with mean k and variance 2k.

If Z∼N(0,1) and U∼χ²ν are independent, Z/√(U/ν) has tν law.

Recompute statistic T on many size-n simulated data sets.

For eventually nonnegative terms aₙ≤bₙ, convergence of Σbₙ implies convergence of Σaₙ; divergence of Σaₙ implies divergence of Σbₙ.

If |aₙ₊₁/aₙ|→ρ, or |aₙ|¹⁄ⁿ→ρ, then Σaₙ converges absolutely for ρ<1 and diverges for ρ>1.

If differentiable real fₙ on (a,b) converge pointwise to f and fₙ′ converge uniformly to g, then f is differentiable and f′=g.

If Riemann-integrable fₙ on a finite interval [a,b] converge uniformly to f, then f is Riemann integrable and ∫ₐᵇfₙ dx→∫ₐᵇf dx.

For Σaₙ(x−c)ⁿ the radius R is 1/lim sup|aₙ|¹⁄ⁿ, with 1/0=∞ and 1/∞=0.

Disjoint finite alternatives have total count equal to the sum of their counts.

For finite A,B, |A∪B|=|A|+|B|−|A∩B|.

From n distinct objects choose k without repetition.

A unit has a multiplicative inverse.

For integral domain R, fractions a/b with b≠0 are equivalence classes under a/b=c/d iff ad=bc.

For a commutative ring with identity and proper ideal I, I is prime iff ab∈I implies a∈I or b∈I, equivalently R/I is an integral domain.

A field has characteristic zero if no positive n satisfies n·1=0; otherwise its characteristic is a prime p.

In an integral domain irreducibles are nonzero nonunits that cannot split into two nonunits; primes divide a product only by dividing a factor.

A Euclidean domain has a nonnegative integer size d on nonzero elements permitting b=aq+r with r=0 or d(r)<d(a), and d(a)≤d(ab).