For a smooth, locally invertible material map $\mathbf{x}=\mathbf{X}(\mathbf{a},t)$ with ${\mathbf{X}}_t=\mathbf{u}(\mathbf{X},t)$ and positive Jacobian determinant $J=\det (\partial \mathbf{X}/\partial \mathbf{a})$, Euler's volume identity states that $J_t=J\nabla \cdot \mathbf{u}$. Indeed, $F=\partial \mathbf{X}/\partial \mathbf{a}$ obeys $F_t=(\nabla \mathbf{u})F$, so the determinant derivative is $J_t=J\mathrm{tr}(F^{-1}{F}_t)=J\mathrm{tr}(\nabla \mathbf{u})$. This form of the Jacobian evolution of a smooth flow applies also to explicitly time-dependent velocity fields. It gives the material-volume Reynolds transport theorem by change of variables to the fixed particle labels. It is distinct from the complex-exponential Euler's formula.

For a partial function $f:D\to X$ with $D\subseteq X$, its $n$th iterated function is defined only when each intermediate application is defined. Put $D_0=X$, $f^0={\mathrm{id}}_X$, and use the displayed recursion for the domain of a function. Simplifying an algebraic expression for $f^n$ does not restore points excluded by an intermediate application. For example, $f(x)=1/x$ has $f^2(x)=x$ on $\mathbb{R}\setminus \{0\}$; the simplified expression $x$ has a larger natural domain of a function.

A constrained extremal is normal when the constraint functional has a nonzero first variation in some allowed direction. The implicit function theorem then permits a compensating variation, yielding a multiplier equation with objective coefficient normalized to one. The Euler-Lagrange equation applies to the objective integrand minus the multiplier times the constraint integrand.

If the first variation of the constraint vanishes in every allowed direction, the objective coefficient in a general multiplier equation can vanish. One must then allow ${\lambda }_{0}F-{\lambda }_{1}G$ with coefficients not both zero. Normalizing ${\lambda }_0$ to one requires the regularity condition used for a normal isoperimetric extremal.

For a nonnegative graph joining $(0,0)$ and $(2a,0)$ with length between $2a$ and $\pi a$, the area-maximizing smooth graph is a circular arch. The Euler-Lagrange equation integrates to the displayed circle formula. Its radius satisfies $l=2R\arcsin (a/R)$ with $R>a$. Concavity of $y-R\sqrt{1+(y^{\prime }{)}^2}$ gives global maximality on the fixed-length class.

Let $f\in \mathbb{Z}[X]$ be primitive polynomial and let $p$ be a prime number not dividing its leading coefficient. If its coefficientwise reduction is irreducible in ${\mathbb{F}}_p[X]$, then $f$ is irreducible in $\mathbb{Q}[X]$. Indeed, a rational factorization would give an integer factorization by Gauss's lemma. The leading coefficients of both factors remain nonzero modulo $p$, so reduction preserves their positive degrees, contradicting irreducibility of the reduction. For monic polynomials the leading-coefficient condition is automatic.

The ferric chloride, historically called perchloride of iron; it is not a perchlorate.

The ferric nitrate, historically called pernitrate of iron.

The ferric sulfate, historically called persulphate of iron in pharmaceutical monographs.

The ferrous sulfate; the ordinary 1867 preparation is the heptahydrate.

The ferrous carbonate, used in protected or freshly prepared historical formulations.

The ferrous iodide, supplied in historical solid, syrup and pill preparations.

An iron phosphate; a precise description must state the oxidation state and hydration or mixed composition.

Multicast routing establishes and maintains the forwarding decisions that deliver group-addressed packets to interested receivers.

Triiodide is the anion ${\mathrm{I}}_3^{-}$, formed in the equilibrium ${\mathrm{I}}_2+{\mathrm{I}}^{-}\rightleftharpoons {\mathrm{I}}_3^{-}$. Formation of triiodide can increase the amount of iodine accommodated in an aqueous iodide-containing solution.

The total dissolved iodine inventory therefore need not equal the concentration of free molecular iodine. Interpreting an optical or titration measurement requires the relevant species balance.

Iodate is the oxyanion $\mathrm{I}{\mathrm{O}}_3^{-}$, with iodine formally in oxidation state +5. It differs from iodide, in which iodine has oxidation state −1.

Reduction can convert iodate into lower-oxidation-state iodine species. Iodate occurs in some natural deposits and industrial feedstocks, connecting iodine extraction with oxidation-reduction chemistry.