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The squared L2 norm of a probability density function derivative measures roughness and appears in optimal smoothing constants. For $r=2$, its directional derivative along a smooth compactly supported perturbation $a$ is $2\int f^{\prime \prime }{a}^{\prime \prime }$. When integration by parts is valid twice, this equals $2\int f^{(4)}a$. Thus regular root-$n$ estimation requires substantially more regularity than merely estimating a smooth probability density function.

Let $L_{i}g$ be the average of $g$ on successive knot intervals. Minimizing the displayed loss over $C^1[a,b]$ gives a quadratic spline with constant tails outside the first and last knots. To prove the reduction, choose the unique constant-tail quadratic spline $q$ with the same interval averages as a candidate $g$, and put $v=g-q$. Since $q^{\prime \prime }$ is constant on every knot interval, $q^{\prime }$ is continuous and vanishes on both tails, integration by parts gives $\int q^{\prime }{v}^{\prime }=-{\sum }_i{q}^{\prime \prime }{|}_{I_i}{\int }_{I_i}v=0$. Consequently the data loss is unchanged and $\int (g^{\prime }{)}^2=\int (q^{\prime }{)}^2+\int (v^{\prime }{)}^2$. The penalty decreases strictly unless $v$ is constant; its zero interval averages then force $v=0$. The finite-dimensional quadratic optimization on this spline space also gives a unique minimizer because the interval-average map is injective.

The quantum variational principle estimates the lowest energy by choosing a trial wavefunction and computing its expectation value, without solving the Schrödinger equation exactly. Minimizing over a manageable family of trial wavefunctions gives an upper bound; enlarging that family can improve the estimate.

Let $H$ be a self-adjoint operator bounded below, and put $E_{\ast }=\inf \sigma (H)$. For $\Vert \psi \Vert =1$ in its operator domain $D(H)$,Trial states may also lie in the form domain $Q(H)=D((H-bI{)}^{1/2})$, where $b\le E_{\ast }$. Then use the quadratic form of a positive quantum Hamiltonian shifted by $b$:This extends the expectation value beyond $D(H)$ without requiring $H\psi$ to exist. Equality for a normalized state holds exactly in the energy eigenspace at $E_{\ast }$, if that eigenvalue exists. ^[ref].

A short proof makes the direction of the bound memorable. If $H$ has a complete orthonormal basis of energy eigenstates ${\varphi }_n$, writeThe expectation value is a weighted average of allowed energies:Equality requires every occupied energy eigenstate to have energy $E_{\ast }$. ^[ref]. With continuous spectrum, the spectral theorem replaces the sum by an integral against the scalar spectral measure; the same nonnegative-difference argument works.

For an unnormalized trial wavefunction, divide by its squared norm: $R_H[\psi ]=q_H[\psi ]/\Vert \psi {\Vert }^2$ is the form version of the Rayleigh quotient. The quantum variational principle supplies an upper bound on $E_{\ast }$; it supplies neither a lower bound on $E_{\ast }$ nor, by itself, a guarantee that a bound state exists.

To see the method at work, choose the real Gaussian wave packetThe Gaussian integral gives wavefunction normalization and the two useful position averages:Thus its position probability density is a normal distribution with standard deviation $1/\sqrt{2a}$: large $a$ means narrow confinement. In units $\hslash =m=1$, integration by parts and ${\psi }_a^{\prime }=-ax{\psi }_a$ give the kinetic energyThese smooth, rapidly decaying trial wavefunctions belong to the operator domains of both polynomial Quantum Hamiltonians below. Narrowing the Gaussian wave packet increases its kinetic energy; widening it increases the potential energy in either confining potential.

For the quantum harmonic oscillator, take $\hslash =m=\omega =1$:The trial values at $a=1/2,1,2$ are respectively $0.625,0.5,0.625$. The derivative $E_{\mathrm{ho}}^{\prime }(a)=(1-a^{-2})/4$ changes sign at $a=1$, where kinetic energy and potential energy are both $1/4$. Here the trial family contains an exact ground state:The zero-point energy remains positive even at the best width. For comparison, removing the potential leaves $E(a)=a/4\to 0$ as $a\to 0$: the free Quantum Hamiltonian on the line has spectral bottom zero but no normalizable zero-energy energy eigenstate. A sequence of ever broader trial wavefunctions need not approach an actual bound state.

For a pure quartic oscillator, keep $\hslash =m=1$ and set the quartic coefficient to one:The trial values at $a=1,2,3$ are respectively $1,0.6875,0.833333\dots$. Setting the derivative to zero givesThe sign of this derivative changes from negative to positive at the optimum, giving the smallest energy within this Gaussian wave packet family:At this width, $⟨T⟩\approx 0.454280$ and $⟨V⟩\approx 0.227140$, so $2⟨T⟩=4⟨V⟩$, as expected from the power-law quantum virial theorem for a scale-optimized trial wavefunction. The number $0.681420$ is an upper estimate of the ground-state energy, not an asserted exact energy eigenvalue. Indeed,depends on $x$ for every $a>0$, so no member of this trial family is an energy eigenstate. Nevertheless, a single width parameter already turns a difficult Schrödinger equation into a useful, explicit energy estimate.

For axisymmetric heat conduction in a unit disk, separated modes satisfy $-(r\kappa R^{\prime }{)}^{\prime }=\nu rR$, with weight $r$, outer Dirichlet boundary condition $R(1)=0$, and zero radial flux at the origin. Integration by parts gives the displayed Rayleigh quotient. Its Euler-Lagrange equation under fixed $\int r{R}^2$ recovers the mode equation; the natural endpoint term is $r\kappa R^{\prime }\delta R$. The Rayleigh-Ritz variational principle says that every admissible nonzero trial profile gives an upper bound on the bottom of the spectrum, and on the lowest eigenvalue when this bottom is attained by a fundamental diffusion mode. Finite weighted energy, rather than pointwise existence of a second derivative at the origin, is the relevant trial admissibility requirement.

Let a solenoidal flow be confined to a sphere $V$, with the insulating boundary condition for the radial magnetic scalar. Define $N={\int }_V{P}^2$, $D={\int }_{{\mathbb{R}}^3}|\nabla P{|}^2$, $M={\int }_{{\mathbb{R}}^3}|\mathbf{B}{|}^2$ and $q={\max }_V|\mathbf{u}\cdot \mathbf{x}|$. Integration by parts inside and outside the sphere givesThe Cauchy-Schwarz inequality therefore makes $q^2\ge {\eta }^{2}D/M$ necessary whenever $N^{\prime }\ge 0$ and $D>0$. A uniform strict violation with a positive gap forces decay, using a Sobolev inequality to bound $N$ by a constant times $D$. A bounded statistically steady field instead requires $q_{\ast }^2\ge {\eta }^2⟨D⟩/⟨M⟩$, with $q_{\ast }={\sup }_{t}q(t)$. This is a necessary condition for dynamo action, not a sufficient criterion or a pointwise assertion at every instant of an arbitrary time-dependent solution.

// Target: analysis.bigb

For $a>0$, polar-coordinate reduction and integration by parts give $m\sin (m||x{|}^2-a|)=2\delta (|x{|}^2-a)-\pi \cos (ma){\delta }_0+o_{{\mathcal{D}}^{\prime }}(1)$ in two dimensions. The circle term comes from the cusp in the folded phase, while the origin term is a radial endpoint contribution. A punctured-domain limit therefore need not extend across the quadratic critical point.

If $f\in L^1(0,1)$ has mean zero, $j$ is a positive integer, and $\varphi \in \mathcal{D}(0,1)$, the displayed estimate holds for every integer $N\ge 1$. Let $h_N=P^{N}f$ use periodic primitives with zero mean. Repeated integration by parts givesThere are no boundary terms, and integration over exactly $j$ periods gives $\Vert {\overline{h}}_N(j\cdot ){\Vert }_{L^1(0,1)}=\Vert h_N{\Vert }_{L^1}\le \Vert f{\Vert }_{L^1}$. This proves the estimate. In particular, if zero-mean $g_j$ satisfy $\Vert g_j{\Vert }_{L^1}=O(j^m)$, their periodic extensions satisfy ${\overline{g}}_j(j\cdot )\to 0$ in weak convergence of distributions by choosing $N=m+1$.

Target: Essential closure. Parent: Lebesgue measure. Intended file: measure-theory.bigb.

For a nonzero real solution of ${\varphi }^{\prime \prime }+(C-k^2)\varphi =0$ on a finite interval, integration by parts gives $\mathcal{R}=\int (C{\varphi }^2-{\varphi }^{\prime 2})dz/\int {\varphi }^{2}dz$ in the displayed form. A Neumann boundary condition only at the lower endpoint does not remove the upper boundary term. Equality with $k^2$ additionally requires $\varphi (z_2){\varphi }^{\prime }(z_2)=0$. With the lower condition, nonconstant oscillatory solutions satisfy this when their phase advance is an integer multiple of $\pi /2$; affine solutions must be constant, while nonzero hyperbolic-cosine solutions fail the condition. Infinite intervals additionally require convergence of both integrals.

Multiply the narrow-gap centrifugal stability equation by the complex conjugate mode and use integration by parts with zero endpoint radial velocities. This gives the displayed real Rayleigh quotient. Its denominator is positive. Positive Rayleigh discriminant throughout the gap gives positive ${\omega }^2$ for every nonzero mode, excluding exponential axisymmetric growth; nonnegativity permits marginal cases. In the alternative convention $\Phi =-{\kappa }_{\mathrm{ep}}^2$, the numerator is $-k^2\int \Phi |u{|}^{2}dr$, so the stable sign is $\Phi <0$. For rigid rotation, ${\kappa }_{\mathrm{ep}}^2=4{\Omega }^2$ gives ${\omega }_n^2=4{\Omega }^2{k}^2/[k^2+(n\pi /d{)}^2]$.

The lowest even quantum harmonic oscillator quantum state has density $\rho (y)=e^{-y^2/{\tau }^2}/(\sqrt{\pi }\tau )$, and the lowest odd quantum state has density $2y^2\rho (y)/{\tau }^2$. Since ${\rho }^{\prime }=-2y\rho /{\tau }^2$, integration by parts gives ${\mathbb{E}}_{\mathrm{F}}[f]={\mathbb{E}}_{\mathrm{B}}[f+y{f}^{\prime }]$ whenever the boundary terms vanish. For $f(y)=y^n$, this proves the displayed identity. Odd moments vanish in both quantum states by parity; even moments quantify the larger spatial spread in the odd quantum state. In particular the relative-coordinate variance is three times as large, and the odd density vanishes at particle coincidence.

For a magnetic vector potential $\mathbf{A}$, a solenoidal magnetic field $\mathbf{B}=\nabla \times \mathbf{A}$ and constant magnetic diffusivity ${\eta }_m$, write ${\partial }_t\mathbf{A}=\mathbf{u}\times \mathbf{B}-{\eta }_m\nabla \times \mathbf{B}-\nabla \varphi$. In a fixed closed volume with zero helicity surface flux, integration by parts makes the two differentiated factors in $H={\int }_V\mathbf{A}\cdot \mathbf{B}dV$ equal; the scalar-potential boundary term vanishes when $\mathbf{B}\cdot \mathbf{n}=0$. One sufficient boundary gauge is $\mathbf{n}\times \mathbf{A}=0$; another standard setting is integration over all space with sufficient decay. Orthogonality of $\mathbf{u}\times \mathbf{B}$ to $\mathbf{B}$ gives the displayed identity. Unlike the resistive magnetic energy loss, $\dot{H}$ is not sign definite. The zero normal magnetic field also makes $H$ gauge invariance under single-valued smooth gauge transformations.

Using the positive polar molecular field $h_i=\delta F/\delta p_i$ and velocity gradient ${\partial }_i{v}_j$, the reversible order-parameter stress consists of a distortion contribution and rotational and alignment contributions:For a local free-energy density, a representative is ${\Sigma }_{ij}^{(1)}=(f-p_k{h}_k){\delta }_{ij}-f_{{\partial }_i{p}_k}{\partial }_j{p}_k$. The Euler-Lagrange equation verifies its divergence. Integration by parts gives $\delta F=\int {\Sigma }_{ij}^p{\partial }_i{u}_j$ for an incompressible displacement without surface work; the corresponding mechanical power has the opposite sign. Pressure contributions may be reassigned in an incompressible flow.

For $2<p<3$ and a continuous path $h$ of bounded variation, translate the first level by $h$ and add the mixed integrals $\int X_{s,r}\otimes d{h}_r$, $\int h_{s,r}\otimes d{X}_r$ and $\int h_{s,r}\otimes d{h}_r$ to the second level. Integration by parts defines the second mixed integral. The operation is continuous in the rough path metric and the variation norm of $h$. For canonical lifts it is precisely the enhancement of the sum of the paths.

Scalar variance measures spatial departure from the conserved mean. For zero mean, incompressible flow bounded by impermeable walls and homogeneous scalar Neumann boundary conditions, integration by parts gives $d\int {\Theta }^2/dt=-2\kappa \int |\nabla \Theta {|}^2$. Advection preserves the instantaneous quadratic integral but can sharpen gradients, allowing the diffusion equation to remove scalar variance faster.

For $a>0$, this integral is an entire function of $x$: the exponential series converges uniformly on bounded subsets after multiplication by the integrable bound $e^{|x|}{u}^{a-1}$. ThusFor positive real $x$, a change of variables gives $F_a(x)=x^{-a}\gamma (a,x)$, where here $\gamma (a,x)={\int }_0^x{e}^{-t}{t}^{a-1}dt$ denotes the lower incomplete gamma function. The singular-looking factor $x^{-a}$ is removed by the integral over $[0,1]$. Applying integration by parts to the derivative of $u^a{e}^{-xu}$ gives $x{F}_a^{\prime }+a{F}_a=e^{-x}$; differentiating and eliminating $e^{-x}$ proves $x{F}_a^{\prime \prime }+(a+1+x)F_a^{\prime }+a{F}_a=0$.

These seminorms measure all polynomially weighted derivatives and generate the Fréchet topology of the Schwartz space. Finitely many such bounds control each weighted derivative's integrable norm by adding a spatial decay factor stronger than $⟨x{⟩}^{-n}$. Continuity of the Fourier transform follows by applying those bounds after differentiation and integration by parts. Equivalent families use powers of $⟨x⟩$ instead of individual monomials.

The Schwarz integral formula reconstructs a holomorphic function from boundary values of its real part. For a real boundary function $f$ on the complex upper half-plane satisfying ${\int }_{\mathbb{R}}|f(s)|/(1+s^2)ds<\infty$, the regularized form isIts real part is the Poisson integral of $f$. The subtraction makes the integral converge and changes only the imaginary normalization. Integration by parts, when the boundary terms vanish, gives $W^{\prime }(w)=(\pi i{)}^{-1}\int f^{\prime }(s)/(s-w)ds$; jumps of $f$ are included as Dirac delta function terms in its distributional derivative.

For real $x$, the integral representation isUsing differentiation under the integral sign twice and then integration by parts shows ${\mathrm{Hi}}^{\prime \prime }-x\mathrm{Hi}=1/\pi$. The cubic decay makes the integral converge for every real $x$.

For the elastic filament operator $K=A{\partial }_x^4$, two integration by parts operations give boundary form $A[\overline{u}{v}^{\prime \prime \prime }-\overline{u^{\prime }}{v}^{\prime \prime }+\overline{u^{\prime \prime }}{v}^{\prime }-\overline{u^{\prime \prime \prime }}v]$. Four elementary homogeneous choices at each endpoint are $h=h^{\prime }=0$ (clamped), $h=h^{\prime \prime }=0$ (hinged), $h^{\prime \prime }=h^{\prime \prime \prime }=0$ (free), and $h^{\prime }=h^{\prime \prime \prime }=0$ (slope-constrained and transverse-force-free). Each defines a self-adjoint operator on a finite interval when imposed at both ends. More general real Robin boundary conditions also give self-adjointness; the four choices are not an exhaustive classification of all boundary subspaces.

Let $q$ be a real harmonic function in a semistrip, $g_0(x)=q(x,0)$, $g_l(x)=q(x,l)$ and $h(y)=q(0,y)$. Let $G_j,N_j$ be the negative-exponential Half-range Fourier transforms of $g_j$ and $q_y(x,j)$, and put $H(k)={\int }_0^l{e}^{ky}h(y)dy$, $W(k)={\int }_0^l{e}^{ky}{q}_x(0,y)dy$. The closed differential form $e^{-ikz}{q}_{z}dz$ has zero boundary integral. Integration by parts of tangential derivatives cancels all corner values and gives the displayed global relation, for $\mathrm{Im}k\le 0$.

// Target: analysis.bigb