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As a quadrature rule, the trapezoidal rule replaces an integrand by the linear interpolation joining its two endpoint values:For nonnegative endpoint values, this is the area of a trapezoid; in general it is a signed area. The quadrature nodes are $a,b$, each with quadrature weight $(b-a)/2$. Unlike the midpoint quadrature rule, it uses endpoint samples, which can be reused between adjacent panels.

Every affine function $f(x)=\alpha x+\beta$ agrees with its linear interpolation, so the trapezoidal rule integrates it exactly. On $[0,2]$, $f(x)=2x+1$ gives $T(f)=(2/2)(1+5)=6=I(f)$. On $[0,1]$, the polynomial $f(x)=x^2$ instead gives $T(f)=1/2$, exceeding $I(f)=1/3$ by $1/6$. Its degree of precision of a quadrature rule is exactly one.

If $f\in C^2[a,b]$, its signed approximation error satisfiesA convex function lies below its endpoint chord, so the trapezoidal rule overestimates its integral; a concave function gives the reverse inequality. The error formula assumes that the second derivative exists and is a continuous function throughout the closed interval.^[ref] A short derivation is the identityobtained by two integrations by parts. The nonnegative Peano kernel has integral $(b-a{)}^3/12$, and the weighted mean value theorem for integrals gives the formula.

The trapezoidal rule for an ordinary differential equation averages vector fields at the old and new states and is A-stable.

This time-stepping use applies the same trapezoidal rule to the integral of the derivative in an ordinary differential equation. That connection explains its place under implicit Runge-Kutta methods; its quadrature behavior and the nonlinear issue in trapezoidal rule fails B-stability concern different uses of the same endpoint averaging.

The Karoubi envelope freely splits idempotents. Its objects are pairs $(A,e)$ and a morphism $(A,e)\to (B,f)$ is a map $u:A\to B$ satisfying $u=fue$.

A steady axisymmetric magnetohydrodynamic wind follows nested magnetic-flux surfaces and has conserved mass loading, field-line angular velocity, angular momentum, entropy, and Bernoulli quantities along each surface.

A magnetohydrodynamic shock is a discontinuity satisfying conservation of mass, momentum, energy, normal magnetic field, and tangential electric field.

The Alfvén speed associated with magnetic-field magnitude $B$ and density $\rho$ is $v_A=B/\sqrt{{\mu }_0\rho }$.

The radical of an ideal $I$ isAn ideal equal to its radical is a radical ideal.

An ideal in a graded ring is homogeneous when it is generated by homogeneous elements, equivalently when every homogeneous component of each of its elements also belongs to the ideal.

Reduction modulo an ideal $I$ is the quotient homomorphism $R\to R/I$.

For finitely many distinct prime ideals, one can prescribe finite valuation data simultaneously by the Chinese remainder theorem.

A prime-power ideal has the form ${\mathfrak{p}}^m$ for a prime ideal $\mathfrak{p}$ and positive integer $m$.

Ideals $I$ and $J$ are comaximal when $I+J=R$.

The set-theoretic intersection of any family of ideals is again an ideal.

The product $IJ$ consists of finite sums of products $xy$ with $x\in I$ and $y\in J$.

A principal ideal is generated by one element: $(a)=\{ra:r\in R\}$.

A set $S\subseteq I$ generates an ideal $I$ when every element of $I$ is a finite sum of ring multiples of elements of $S$.

An ideal $I$ is primary when $ab\in I$ and $a\notin I$ imply $b^n\in I$ for some positive integer $n$.

For a geodesic hyperbolic $n$-gon of curvature $-1$ with interior angles ${\alpha }_1,\dots ,{\alpha }_n$, triangulation or the Gauss--Bonnet theorem gives

The identity ${\cosh }^{2}x-{\sinh }^{2}x=1$ implies

The identity follows directly from the exponential definitions of the hyperbolic functions.

The Taylor series at zero beginsso $\tanh x\sim x$ as $x\to 0$.

The gravitational scattering angle is the change between incoming and outgoing asymptotic velocity directions.