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For $a_t+(V(s)a{)}_s=0$ with $V>0$, the flux perturbation $b=Va$ solves $b_t+V{b}_s=0$. The travel coordinate transforms it into unit-speed transport. Thus the flux perturbation is constant on characteristics while area amplitude changes as $1/V$. In a supplied power-law river, $A_{\mathrm{eq}}^{\prime }=M/V$, giving travel time $[A_{\mathrm{eq}}(l)-A_{\mathrm{eq}}(0)]/M$. At a zero-speed head, this time can remain integrable even though the amplitude formula is not uniform.

For nodes $a=x_0<\cdots <x_n=b$ and prescribed real node values, let $s$ be the linear spline joining them. Among all absolutely continuous functions $u$ with the same node values and square-integrable derivative, $s$ uniquely minimizes the Dirichlet energy ${\int }_a^b|u^{\prime }{|}^2$.

On each node interval, $s^{\prime }$ is constant and $u-s$ vanishes at both endpoints. The fundamental theorem of calculus therefore gives $\int s^{\prime }(u^{\prime }-s^{\prime })=s^{\prime }[(u-s)(x_{i+1})-(u-s)(x_i)]=0$. Expanding the square and summing proves the displayed energy decomposition. Its final term is nonnegative; equality forces $u^{\prime }=s^{\prime }$ almost everywhere and the endpoint values fix the additive constant, proving uniqueness. In particular, the minimum is ${\sum }_i(u(x_{i+1})-u(x_i){)}^2/(x_{i+1}-x_i)$. If all interior node values are zero and there is at least one interior node, only the two endpoint intervals contribute to this minimum.

For $y^{\prime }+ay=g$ with real $a>0$ and $2\pi$-periodic forcing, comparing Fourier coefficients gives $(a+in){\hat{y}}_n={\hat{g}}_n$. There is no vanishing denominator, including the constant mode. With real cosine and sine coefficients $a_n,b_n$ for $g$ and $A_n,B_n$ for $y$, the equivalent equations giveWhenever termwise differentiation is justified, these formulas construct a periodic particular solution. For a forcing series with coefficients $O(n^{-2})$, the differentiated response coefficients are also summable, which supplies such a justification by uniform convergence. All other solutions add the homogeneous solution $C{e}^{-ax}$. Since this exponential cannot be periodic unless $C=0$, the periodic solution is unique. This is also a simple example of how a linear equation changes the amplitude and phase of each harmonic independently.

A fundamental system for a homogeneous linear ordinary differential equation of order $n$ on an interval is a basis of its vector space of homogeneous solutions. Every solution has a unique expression ${\sum }_{j=1}^n{c}_j{y}_j$ with constant coefficients. Under the usual continuity assumptions and a nonvanishing leading coefficient, a nonzero Wronskian at one point proves that $n$ solutions form such a system: their initial-value vectors are independent and span all possible initial data, and the Picard-Lindelöf theorem supplies uniqueness. For a first-order vector system, the corresponding columns form a fundamental matrix.

A characteristic exponent is a value $\lambda$ for which an exponential mode $e^{\lambda t}$ solves the homogeneous constant-coefficient linear equation or system. Repeated exponents can give polynomial factors multiplying that exponential.

An invertible linear transformation replaces one system of coordinates by another. If $x=Pu$ in a linear differential equation $\dot{x}=Ax$, then $\dot{u}=P^{-1}APu$. The similar matrices $A$ and $P^{-1}AP$ have the same characteristic polynomial and eigenvalues, although their coordinate descriptions of the eigenvectors differ. Such a change can turn a planar linear system with eigenvalues $\alpha \pm i\beta$ into $\dot{u}=\alpha u-\beta v$, $\dot{v}=\beta u+\alpha v$.

A rational-linear map is a linear map when its domain and codomain are regarded as vector spaces over the rational numbers. An additive map on a rational vector subspace of the real numbers is automatically rational-linear: integer homogeneity follows by repeated addition, and division by a positive integer gives rational homogeneity of an additive function.

A real-linear map is a linear map over the real numbers. For a map from the real line to itself, real linearity means multiplication by a fixed real number, its value at one. Rational-linear maps need not be real-linear without an additional regularity condition.

In the one-sorted vector-space language naming each scalar multiplication, a first-order formula reduces by quantifier elimination to finitely many linear equations. An independent pair satisfies only the identically zero equations among them. A dependent pair $(x,ax)$ with nonzero $x$ satisfies the same listed equations if the scalar $a$ avoids their finitely many forbidden ratios, which is possible over an infinite field. Thus no formula defines pair independence. The complete independent-pair type is still consistent in an infinite-dimensional model; it is nonisolated.

A vanishing highest-derivative coefficient can make the two initial values of a second-order differential equation dependent. Suppose $h$ is continuous at zero, $a$ is constant, and a solution of $[t(y^{\prime }+ay){]}^{\prime }=h(t)$ has finite continuous $y,y^{\prime }$ at zero. Integration givesRegularity forces $C=0$. Dividing by $t>0$ and taking the limit gives $y^{\prime }(0)+ay(0)=h(0)$, since the integral average tends to $h(0)$. The remaining equation is first order and therefore has only one free regular initial value. For $a=1$ and $h(t)=t$, it yields $y^{\prime }+y=t/2$ and $y=A{e}^{-t}+(t-1)/2$. This explains the reduced constant count without treating the singular endpoint as an ordinary second-order initial point.

For a point vortex of circulation $\Gamma$ in the wedge $0<\theta <\pi /n$, with positive integer $n$, the conformal map $z\mapsto z^n$ sends the domain to the upper half-plane. An opposite image vortex gives the complex potentialRemove the singular self-induced velocity before setting $z=c$. The resulting complex velocity isThe finite first term follows from the expansion of $(z^n-c^n)/(z-c)$ at $c$. In polar coordinates this becomes $\dot{r}=n\Gamma \cot (n\theta )/(4\pi r)$ and $\dot{\theta }=-\Gamma /(4\pi r^2)$. Therefore $r\sin (n\theta )$ is constant along a vortex trajectory. The case $n=1$ is motion parallel to a plane wall, while $n=3$ describes a sixty-degree wedge.

A vortex of circulation $\Gamma$ distance $d$ from a plane wall has an opposite-sign image across the wall. The wall velocity has magnitude $|\Gamma |d/[\pi (d^2+y^2)]$, giving a pressure reduction $\rho {\Gamma }^2{d}^2/[2{\pi }^2(d^2+y^2{)}^2]$ by the Bernoulli equation. Integrating this deficit gives the displayed force per unit width, directed from the wall toward the fluid and vortex. The uniform ambient pressure is subtracted to isolate the finite vortex-induced force.

In the impermeable strip $x>0$, $0<y<a$, a line source of strength $Q$ at an interior point $c$ has the displayed complex potential of a two-dimensional potential flow, with locally consistent complex logarithm and $f(z)=\cosh (\pi z/a)$. The hyperbolic-cosine map of a half-strip sends the walls to the real axis. An equal image line source at $\overline{f(c)}$ makes that axis a streamline. Since $f^{\prime }(c)\ne 0$, the local singular part remains $Q\log (z-c)/(2\pi )$, proving preservation of strength. With no extra imposed flow, the complex velocity tends to $Q/a$ as $x\to \infty$, so the outgoing volume flux is $Q$. The source position $c$ has positive imaginary part; assuming it real would place it on a wall.

In the upper half-plane, a line source of outward volume flux $Q$ at height $a$ above an impermeable wall has an equal image source below it. On the wall $u=Qx/[\pi (x^2+a^2)]$ and the normal velocity vanishes. The steady Bernoulli equation gives $p-p_{\infty }=-\rho Q^2{x}^2/[2{\pi }^2(x^2+a^2{)}^2]$. Sincethe ambient-relative force per unit length pulls the wall toward the source with the displayed magnitude. The torque about the source's projection onto the wall is zero in the symmetric Cauchy principal value sense. Individual one-sided moments diverge logarithmically. This planar result differs from the three-dimensional hydrodynamic attraction of a plane wall to a point source.

Take formulas whose free variables lie among a fixed $n$-tuple and identify them when their universal equivalence follows from $T$. Conjunction, disjunction and negation induce a Boolean algebra; its order is implication modulo $T$. Complete types correspond to ultrafilters and isolated complete types to atoms. For a complete theory in a countable language, finiteness of these algebras for every finite $n$ characterizes countable categoricity.

A subset of the plane can have three successive strict changes under taking its set of accumulation points. For example letPut $B=\{(1/n+1/(n^{2}m),0):n,m\ge 1\}$ and $C=\{(1/n,0):n\ge 1\}$. Then $A^{\prime }=B\cup C\cup \{0\}$, $A^{\prime \prime }=C\cup \{0\}$ and $A^{\prime \prime \prime }=\{0\}$, where $0$ denotes the origin. For fixed $n,m$, increasing $k$ gives a point of $B$; increasing $m$ gives a point of $C$; increasing $n$ gives the origin. These are all possible limits: in any convergent sequence of distinct points, either $n$ is unbounded, or a subsequence fixes $n$ and has unbounded $m$, or a subsequence fixes both and has unbounded $k$. The first-coordinate cluster intervals $[1/n,1/n+1/n^2]$ are disjoint and separated away from the origin. Repeating this classification after each differentiation proves the displayed derived sets. Each derivative removes one layer of isolated points, and all three displayed changes are strict.

A small conducting sphere of radius $a$ has electric polarizability $\alpha =4\pi {ϵ}_0{a}^3=a^3/k$. In the field $E\simeq k{Q}_1/R^2$ of a much more strongly charged distant source, its leading induced contribution is $-\alpha E^2/2$, in addition to $k{Q}_1{Q}_2/R$ for its fixed net charge. Taking $-dU/dR$ gives $F_R\simeq k{Q}_1{Q}_2/R^2-2k{Q}_1^2{a}^3/R^5$. The approximation requires $a/R\ll 1$ and control of the source's finite-size corrections; the threshold is not an exact finite-sphere result.

Call a Lie algebra pre-supersoluble if it is abelian or its quotient by its centre has a one-dimensional ideal. Every nonzero such algebra has nonzero abelianization over every field. To prove this, choose $x$ outside the centre with $Z(L)+Fx$ an ideal and write $[a,x]=\lambda (a)x+z(a)$ with $z(a)$ central. The Jacobi identity gives $\lambda ([a,b])=0$. If $\lambda \ne 0$, the derived algebra lies in its proper kernel. If $\lambda =0$, Jacobi gives $[[a,b],x]=0$, so the derived algebra lies in the proper centralizer of an element of a Lie algebra of $x$. The abelian case is immediate. This condition does not imply solvability: ${\mathfrak{sl}}_2(\mathbb{C})$ direct sum the affine Lie algebra of the line has zero centre and a one-dimensional ideal in the affine factor, but is not solvable.

Strict positivity of the highest-derivative coefficient on the closed parameter interval is the strengthened Legendre condition. Continuity then gives a uniform positive lower bound and makes the Jacobi equation a regular second-order equation. It permits the Jacobi positivity criterion. It is not necessary for mere strict positivity of a second variation: $\int {\phi }^2$ is strictly positive on nonzero variations even though its derivative coefficient vanishes.

The sigma-algebra of all Lebesgue measurable sets, equivalently the completion of a measure of the Borel sigma-algebra under Lebesgue measure. It contains every subset of every Lebesgue null set. Applying Carathéodory's criterion to Lebesgue outer measure gives another construction of this same sigma-algebra.