For differentiable endpoints $a,b$ and a function $f$ with suitable continuous partial derivative in the parameter, the moving-endpoint rule isTo prove it locally, put $H(x,u)={\int }_c^{u}f(x,t)dt$ for a fixed reference endpoint $c$. The original integral is $H(x,b(x))-H(x,a(x))$. Apply the chain rule, use the fundamental theorem of calculus for $H_u=f$, and use differentiation under the integral sign for $H_x$. The two boundary terms account for the moving endpoints, while the remaining integral accounts for variation of the integrand with the parameter. For improper integrals, convergence and a suitable domination condition must be verified before interchanging the derivative and integral.

For every real parameter $a$,where $F(0)=0$. For $a>0$, differentiation under the integral sign and a partial fraction decomposition giveSolving this first-order linear differential equation yields $F(a)=a[\pi \ln (1+a)+C]$. To fix the constant, dominated convergence gives ${\lim }_{a\to 0^{+}}F(a)/a={\int }_0^{\infty }\ln (x^2)/(1+x^2)dx=0$; the last absolutely convergent integral changes sign under $x=1/t$. Thus $C=0$. Oddness in $a$ supplies the result for negative parameters. The initial value $F(0)=0$ alone does not determine $C$, since every solution $a[\pi \ln (1+a)+C]$ tends to zero as $a\to 0^{+}$.

For real $\alpha$, the displayed improper integral converges and is an even function of $\alpha$. For $\alpha >0$, differentiation under the integral sign and partial fraction decomposition giveFor $\alpha \ne 1$, use $x^2/[(1+{\alpha }^2{x}^2)(1+x^2)]=[1/(1+x^2)-1/(1+{\alpha }^2{x}^2)]/({\alpha }^2-1)$. Integrate the resulting rational functions and extend to $\alpha =1$ by continuity. Integrating in $\alpha$ and using $E(0)=0$ gives $E(\alpha )=\pi \ln (1+\alpha )$ for $\alpha \ge 0$; evenness gives the formula for negative parameters. Thus $E^{\prime }(\alpha )=\pi \mathrm{sgn}(\alpha )/(1+|\alpha |)$ away from zero. At zero the one-sided derivatives are opposite, so $E$ is continuous but not differentiable. Substituting $\alpha =0$ into the pointwise differentiated integrand would therefore be an invalid interchange.

For every real $a$, define $I(a)={\int }_0^1\sin (a\log x)/\log xdx$. The apparent endpoint singularity at $x=1$ is removable, with limit $a$, and the bound $|\sin (a\log x)/\log x|\le |a|$ ensures integrability. Differentiation under the integral sign is justified by the integrable bound $|\cos (a\log x)|\le 1$ on the parameter derivative. Thuswhere $x=e^{-t}$ and the last integral is a Laplace transform. Since $I(0)=0$, integrating with respect to $a$ gives $I(a)=\arctan a$. This obtains an integral with a logarithmic denominator by differentiating away that denominator, then restoring the parameter dependence with one elementary antiderivative.

For real $m>-1$ and integer $n\ge 0$, start from ${\int }_0^1{x}^{m}dx=1/(m+1)$ and differentiate $n$ times with respect to $m$. Each derivative adds one factor $\log x$, givingTo justify differentiation under the integral sign, restrict $m$ to a compact interval with lower endpoint $m_0>-1$. The derivatives are dominated by $x^{m_0}|\log x{|}^n$, which is integrable: $x=e^{-t}$ turns its integral into ${\int }_0^{\infty }{e}^{-(m_0+1)t}{t}^{n}dt$. A power of $t$ is integrable against this decaying exponential. This also explains why $m=-1$ is the convergence threshold.

Alternating sample and reference readings can suppress instrument drift. In this original four-reading model, measure objects $A,B,B,A$ at equally spaced times $0,\Delta t,2\Delta t,3\Delta t$. Assume the objects are stable during that short sequence and each reading is its object's length plus the same affine function of time, $a+bt$. Thenbecause both pairs have mean time $3\Delta t/2$. The constant offset and linear drift cancel. With a further drift term $c{t}^2$, direct substitution instead leaves bias $-2c\Delta t^2$. A shorter sequence reduces that term, but neither this ordering nor averaging corrects unequal temperatures, sample changes or reference measurement uncertainty. This is a modern illustrative protocol; the historical account below establishes alternation, without attributing this particular four-reading sequence to Charles-Édouard Guillaume.

A stationary solution of an evolution differential equation is independent of time. Substituting ${\partial }_{t}u=0$ leaves the spatial boundary value problem that determines its profile. For an ordinary differential equation this is an equilibrium; for a partial differential equation it may vary in space. A stationary profile need not have zero flux, although zero flux is forced in a one-dimensional conserved system with reflecting endpoints.

An orthogonal trajectory intersects every regular member of a specified family of plane curves at a right angle. If eliminating the family parameter gives its tangent slope $dy/dx=F(x,y)$, the dot product of the two tangent directions gives the orthogonal slope $dy/dx=-1/F(x,y)$ wherever both slopes are finite and nonzero. Vertical and horizontal tangent directions are interpreted geometrically. For the family $y=c{x}^n$ with $n>0$, eliminating $c$ gives $F=ny/x$, and the orthogonal trajectories satisfy $x^2+n{y}^2=C$. For $n=2$ these are ellipses; for $n=1$ they are circles.

Suppose $h$ is bounded by one in absolute value on a punctured neighbourhood of zero. Suppose also that there are positive sequences $p_n,q_n\to 0$ with $h(p_n)=1$ and $h(q_n)=0$. Extend $F_a(x)=|x{|}^{a}h(x)$ by $F_a(0)=0$. ThenWhen the derivative exists, it equals zero. For continuity, the bound $|F_a(x)|\le |x{|}^a$ gives the epsilon-delta definition of continuity with $\delta ={\epsilon }^{1/a}$ when $a>0$. If $a\le 0$, the values $F_a(p_n)=p_n^a$ remain at least one for sufficiently large $n$, contradicting continuity at zero.

For differentiability, the difference quotient is bounded in absolute value by $|x{|}^{a-1}$, so it tends to zero when $a>1$. Conversely its values along $q_n$ are zero, forcing any possible derivative to be zero. Its values along $p_n$ equal $p_n^{a-1}$ and fail to tend to zero when $a\le 1$.

For example, $h(x)=\sin (\pi \sin (1/x))$ has the required positive sequences $p_n=1/(2\pi n+\pi /6)$ and $q_n=1/(2\pi n)$. This separates the suppression needed for continuity from the stronger suppression needed for differentiability.

Anomers differ only in configuration at the anomeric carbon of the same cyclic monosaccharide form.

The incremental capacitance associated with changes of electric charge around a p-n junction. Its depletion contribution is often estimated by $C_j\simeq {\epsilon }_{s}A/W$, so an independent junction capacitance measurement can constrain the width $W$ of the depletion region under the model assumptions. Other stored electric charge, including injected minority carriers, can also contribute to a measured terminal capacitance.

The depletion approximation replaces the junction's transition region by a region without mobile carriers, leaving only fixed ionized dopants. For a uniformly doped abrupt p-n junction, $\rho =-e{N}_A$ on the p side, $\rho =+e{N}_D$ on the n side and $\rho \simeq 0$ outside. Charge neutrality gives $N_A{x}_p=N_D{x}_n$. Integrating Poisson equation gives the electric field and the electrostatic band bending.

Thermal generation of electron-hole pairs within a reverse-biased depletion region can supply a leakage electric current in addition to neutral-region diffusion. For a simple midgap trap with equal electron and hole lifetime parameters, its scale is $e\mathcal{A}W{n}_i/(2{\tau }_g)$, where $W$ is depletion width and ${\tau }_g$ the lifetime parameter. The electric field separates generated carriers. Unlike ideal diode minority-carrier diffusion current, this contribution scales approximately with $n_i$ rather than $n_i^2$ and can vary with reverse voltage through $W$. Consequently a measured activation energy need not equal the full band gap.

The Asakura-Oosawa model treats dilute nonadsorbing polymer coils as mutually interpenetrable ideal particles excluded from a layer around each colloid. For colloid radius $a$ and polymer exclusion thickness $R_g$, two exclusion spheres of radius $R_e=a+R_g$ overlap when $2a\le R<2R_e$. Adding their two spherical cap volumes gives $V_{\mathrm{ov}}=\pi (4R_e+R)(2R_e-R{)}^2/12$. Multiplication by $-{\Pi }_p$ gives the displayed pair free energy, with zero depletion interaction beyond $2R_e$. The range is controlled by polymer size and the depth by reservoir polymer concentration. At larger polymer concentration, polymer interactions and many-body overlap effects require care.

A DNA lesion is physical or chemical damage to DNA, such as an altered nucleobase or strand break. DNA repair can restore the sequence before a lasting mutation occurs.

Gas density is the mass density of a gas. For an ideal gas sample of amount of substance $n$, mean molar mass $\overline{M}$ and volume $V$, combine $m=n\overline{M}$ with $pV=nRT$ to obtainHere $R$ is the molar gas constant. For a homogeneous mixture of gases, $\overline{M}={\sum }_i{x}_i{M}_i$, where $x_i$ is the mole fraction of component $i$. Equal pressure and temperature make a gas density ratio equal to a mean-molar mass ratio within this model. Argon detection in a binary gas mixture gives a worked example and a temperature confounder.

For a spherical surface of radius $R$ with exterior mass density $\rho (r)={\rho }_0{e}^{-(r-R)/h_0}$, integrate the spherical volume element to obtain $M=4\pi {\rho }_0{\int }_0^{\infty }(R+h{)}^2{e}^{-h/h_0}dh$. The moments ${\int }_0^{\infty }{h}^n{e}^{-h/h_0}dh=n!h_0^{n+1}$ give the displayed expression. Its leading thin-atmosphere approximation is surface area times column mass, $4\pi R^2{\rho }_0{h}_0$; the two positive correction terms account for the growth of shell area with altitude.