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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.