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On a homogeneous background with $X=\bar{X}$, a $P(X,\varphi )$ perturbation has sound speedThe denominator is its time-kinetic coefficient and $P_X$ is its spatial-gradient coefficient.

Connection one-forms are defined by $\nabla e_b={\omega }^a{}_b\otimes e_a$. For the Levi-Civita connection they obey Cartan's torsion-free equation $d{\theta }^a+{\omega }^a{}_b\wedge {\theta }^b=0$ and metric compatibility ${\omega }_{ab}=-{\omega }_{ba}$.

A Jacobi matrix is a symmetric tridiagonal matrix whose characteristic polynomials obey the orthogonal-polynomial three-term recurrence.

A degree-n orthogonal polynomial for a positive interval weight has n simple zeros in the interval interior.

A monic orthogonal polynomial has leading coefficient one. For a fixed positive weight, there is a unique monic orthogonal polynomial of each degree.

A matrix is skew-symmetric when $A^T=-A$.

The special orthogonal group isEvery element of $SO(2)$ is a rotation in the plane. Every element of $SO(3)$ fixes an axis and restricts to a planar rotation on its perpendicular plane.

A real square matrix is orthogonal when $R^{T}R=I$, equivalently when its columns form an orthonormal basis.

An orientation of a surface induces the boundary direction used in Stokes theorem. Walking in the positive boundary direction keeps an upward-oriented surface on the left.

A multiplicative subset $S$ is left Ore when, for every $s\in S$ and $r\in R$, there are $s^{\prime }\in S$ and $r^{\prime }\in R$ such that $s^{\prime }r=r^{\prime }s$.

The sequence $\lfloor n/2\rfloor$ has generating function

A first integral is a function of time, state, and derivatives that remains constant along every solution.

A linear ordinary differential equation is linear in the unknown function and its derivatives.

A differentiable real function satisfyingon an interval has $f(0)=0$ and $f^{\prime }=C(1+f^2)$, hence $f(x)=\tan (Cx)$ wherever the tangent remains finite.

A Bernoulli differential equation is a first-order ordinary differential equation of the displayed form, with $P$ and $Q$ continuous functions on an interval and a fixed real exponent $n$. A nonlinear power can hide a first-order linear differential equation: on a suitable branch, the Bernoulli power substitution $z=y^{1-n}$ removes it exactly. The transformed solution of a differential equation can be completely regular even when recovering $y$ produces finite-time blowup.

For example, the reciprocal Bernoulli equation with finite-time blowup $y^{\prime }+y=e^x{y}^2$ becomes $z^{\prime }-z=-e^x$ under $z=1/y$. The initial conditions $y(0)=1/2$ and $y(0)=-1$ select a positive trajectory blowing up at $x=2$ and a negative trajectory tending to zero as $x\to \infty$, respectively. The integrating factor exposes both behaviors with the same short calculation.

The cases $n=0$ and $n=1$ are already first-order linear differential equations: respectively $y^{\prime }+Py=Q$, with the zeroth-power term interpreted as the constant $1$, and $y^{\prime }+(P-Q)y=0$. There is no need for a nonlinear change of variable; for $n=1$, $y^{1-n}=1$ would discard the unknown entirely. For other exponents, the substitution must respect the domain of exponentiation and may lose the zero solution of a differential equation.^[ref]

A differential inequality bounds derivatives rather than specifying them exactly; comparison and integration turn it into bounds on the function.

Differentiating a parameter-dependent initial-value problem with respect to its parameter gives a linear inhomogeneous equation for the solution sensitivity, with initial data obtained by differentiating the original initial condition.

For $y_0>0$, the initial-value problemhas the continuous piecewise differentiable solutionThe vector field is not Lipschitz at zero, but the direction on each side prevents any solution from leaving zero, so this solution is unique in the piecewise differentiable class.

A boundary condition prescribes values of an unknown function or its derivatives at the boundary of the domain of a differential equation.