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On the invariant level $L^2=2E{I}_2$, the torque-free rigid-body orbit is a separatrix joining the two intermediate-axis rotations. Withone orientation has ${\omega }_2(t)=\mu \tanh (\lambda (t-t_0))$; the other components are constant multiples of $\mathrm{sech}(\lambda (t-t_0))$.

Let $R\subseteq A$ be an integral extension of domains with $R$ integrally closed. Given primes ${\mathfrak{p}}_0\subseteq {\mathfrak{p}}_1$ of $R$ and a prime ${\mathfrak{q}}_1$ of $A$ over ${\mathfrak{p}}_1$, there is a prime ${\mathfrak{q}}_0\subseteq {\mathfrak{q}}_1$ over ${\mathfrak{p}}_0$.

The integral closure of $R$ in an $R$-algebra $A$ is the subring of elements of $A$ integral over $R$.

An integral domain is integrally closed when every element of its fraction field that is integral over the domain already belongs to the domain.

If $a_1,\dots ,a_n$ are integral over $R$, then $R[a_1,\dots ,a_n]$ is a finitely generated $R$-module. In particular, every element of this subalgebra is integral over $R$.

For an integral extension $A\subseteq B$ and $\mathfrak{p}\in \mathrm{Spec}A$, localization gives a bijectionIt combines the prime ideal correspondence for localization with the contraction of a maximal ideal under an integral extension.

If $A\subseteq B$ is integral and $\mathfrak{n}$ is a maximal ideal of $B$, then $\mathfrak{n}\cap A$ is maximal in $A$. Indeed, $B/\mathfrak{n}$ is an integral domain integral over $A/(\mathfrak{n}\cap A)$, and a subring over which a field is integral is itself a field.

The Gaussian integral gives the exact area under a Gaussian function. For a positive real number $a$,This is a finite improper integral over the whole real line. For example, $I(1)=\sqrt{\pi }\approx 1.772454$ and $I(4)=\sqrt{\pi }/2\approx 0.886227$. Increasing $a$ from one to four narrows the curve and halves its area, while its height at zero stays one. The change of variables $u=\sqrt{a}x$ explains the factor $a^{-1/2}$.

The useful consequence is an exact normalization. For $a>0$ and any real center $\mu$,This is a probability density function for a normal distribution with expected value $\mu$ and variance $1/(2a)$. A narrower probability density function must be taller to keep total probability one. In the usual notation $a=1/(2{\sigma }^2)$, the prefactor is $1/(\sigma \sqrt{2\pi })$, where $\sigma$ is the standard deviation.

The same area calculation makes the one-dimensional heat kernelintegrate to one in $x$, with $D$ the diffusion coefficient. A spreading Gaussian function can therefore preserve total mass while its height falls. This links the Gaussian integral with both the normal distribution and the diffusion equation.

Finite intervals lead to the error function, rather than this single constant:The complementary tail is described by the complementary error function. These definitions and their real-axis limits are given in the NIST reference. ^[ref]

The polar-coordinate proof of the Gaussian integral explains why $\pi$ appears in an integral of a function defined on a line: squaring its area creates a rotationally symmetric integral on a plane. Substitution $t=x^2$ also gives $\Gamma (1/2)=2{\int }_0^{\infty }{e}^{-x^2}dx=\sqrt{\pi }$, connecting the result with the Gamma function.

An antiderivative recovers a function from its local rate of change. For a real-valued function $g$ on an open subset $U$ of the real line, a differentiable function $F$ is an antiderivative whenFor example, $g(x)=x^2$ has the antiderivatives $F_C(x)=x^3/3+C$ for every real additive constant $C$. Their derivatives agree, while their values can differ. On an interval, the constant of integration describes the entire freedom in the answer. ^[ref]

The indefinite-integral notation $\int g(x)dx=F(x)+C$ records this family on an interval. For the illustrated family, the extra condition $F(1)=2$ selects $C=5/3$. Thus reversing differentiation requires both a rate and a reference value if a unique function is wanted.

The fundamental theorem of calculus supplies a construction when $g$ is continuous on an interval $I$. Choose $x_0\in I$ and set $F(x)={\int }_{x_0}^{x}g(t)dt$; then $F^{\prime }=g$. Integrability alone does not guarantee a classical antiderivative. The Heaviside step function is Riemann integrable on bounded intervals, but cannot be a derivative throughout an interval containing its jump: Darboux theorem for derivatives requires every intermediate value. ^[ref]

If measurable functions satisfy $f\le g$ almost everywhere, thenwhenever the two sides are defined.

A double integral integrates a function over a two-dimensional region. A change of variables formula can replace the region and area element using a Jacobian determinant.

It is a multiple integral with two integration variables.

A complex $n$-manifold has charts to ${\mathbb{C}}^n$ with holomorphic transition maps. It is a real $2n$-manifold with an integrable almost complex structure.

The Newlander-Nirenberg theorem says that a smooth almost complex structure is integrable if and only if its Nijenhuis tensor vanishes.

The alephs enumerate the infinite initial ordinals: ${\aleph }_0=\omega$, successor indices select the next initial ordinal, and limit indices select the least initial ordinal above all earlier values.

The $i$th $k$-ary projection returns its $i$th input:

The successor function is

The zero function has the constant valueDummy variables give a zero function of any finite arity.

Radial null curves comprise the ingoing family $v=\text{constant}$ and the outgoing familyOutside the horizon the outgoing family moves to larger $r$; inside it both future-directed families move to smaller $r$.

At a distribution with positive density $f$ at its median $m$, the median influence function is $\mathrm{sgn}(x-m)/(2f(m))$ away from $m$.

Gross-error sensitivity is the supremum of the absolute influence function over contamination points.