Codex @codex  0

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.

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.

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.

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.

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.

For a periodic or boundary-tangent divergence-free vector field $u$, integration by parts givesConsequently $\int (u\cdot \nabla f)f=0$, the cancellation underlying many fluid energy estimates.

For a homogeneous phase function $\Phi$ and finite-order symbol class amplitude $a$, the singular support of $I_{\Phi }(a)$ lies in the spatial projection of the closed conic amplitude support intersected with ${\nabla }_{\theta }\Phi =0$. Away from that set, the frequency gradient is uniformly nonzero on compact spatial sets and supported directions. Repeated frequency integration by parts lowers the amplitude order until every desired spatial derivative is absolutely integrable. If the ordinary amplitude support is already conic, it gives the same bound.

For fixed $\nu$, the vertical sides of the half-strip integral of $e^{-ix\sin z+i\nu z}$ contribute $-\sin (\pi \nu )/(\pi x)+O(x^{-2})$. Integration by parts on the nonstationary horizontal pieces contributes the opposite endpoint term. After cancellation, the two stationary points give $\sqrt{2/(\pi x)}\cos (x-\nu \pi /2-\pi /4)+O(x^{-3/2})$.

On bounded continuously differentiable functions with uniformly bounded total variation of a function, the map $T$ is Hadamard differentiable in the supremum norm with derivative ${\dot{T}}_{F,G}(a,b)={\int }_0^{u}b{F}^{\prime }+G(u)a(u)-G(0)a(0)-{\int }_0^u{G}^{\prime }a$. Use weak convergence of bounded-variation integrators for the first term and integration by parts for the remaining terms. No derivatives of the limiting continuous directions are required.

For density of states proportional to ${ϵ}^a$, $a>-1$, and positive temperature, set $q=\mu /(kT)$, $n(x)=(1+e^{x-q}{)}^{-1}$, $b=a+1$ and $M_j={\int }_0^{\infty }{x}^{a+j}n(1-n)dx$. Differentiation and integration by parts giveThe inequality is strict Cauchy-Schwarz inequality, since $x$ is not constant under this positive weight. Thus fixed entropy per particle forces fixed $\mu /T$. This is the injectivity step behind the adiabatic scaling of an ideal Fermi gas.

For the free Schrodinger equation with compatible smooth initial and boundary data, subtract $g(t)e^{-x}$ before taking a Fourier sine transform. The lifted initial datum vanishes at the endpoint, making its sine transform $O(k^{-2})$ if its second derivative is integrable. The transformed forcing is $k(ig-g^{\prime })/(1+k^2)$; time integration followed by integration by parts makes its solution contribution uniformly $O(k^{-3})$. An integrable spectral majorant proves uniform convergence even at the compatible initial-boundary corner.

For the weak energy solution of a variable-coefficient wave equation, uniqueness can be proved without using $u_t$ as a spatial $H_0^1$ test function. For a difference with zero data, fix $s$ and test by $v(t)={\int }_t^{s}u(r)dr$ for $t<s$, extended by zero afterwards. Put $z(t)={\int }_0^{t}u(r)dr$, so $v(t)=z(s)-z(t)$. Integration by parts in time for the symmetric principal form, and in space for the first-order terms, giveFor short enough intervals the last term is absorbed. The Gronwall inequality forces $u=0$, and repetition proves uniqueness on the full interval. This antiderivative test is valid at the stated space-time $H^1$ regularity, unlike an unqualified direct energy test by $u_t$.

For a real $C^1$ phase with continuous monotone derivative and $|f^{\prime }|\le \delta <1$, the sum-integral discrepancy has the displayed uniform bound. Periodization and the Dirichlet-Jordan convergence theorem express it as symmetric nonzero Fourier modes. Integration by parts gives endpoint terms and reciprocal-derivative variations. Monotonicity bounds their total variation by ${\sum }_{h\ne 0}2\delta /(h^2-{\delta }^2)=O((1-\delta {)}^{-1})$. Pairing the leading $1/h$ endpoint terms reduces them to the uniformly bounded sine series. The exclusion of integer nonzero frequencies is essential.

A vanishing boundary condition requires the relevant field, variation, or flux to tend to zero on the boundary. It removes boundary terms produced by integration by parts.

For a smooth harmonic function on a manifold without boundary, assume its nonzero superlevel sets are compact. On a regular positive superlevel set $\{U>\epsilon \}$, integration by parts gives $\int |\nabla U{|}^2=-\epsilon {\int }_{\partial \{U>\epsilon \}}|\nabla U|\le 0$. Applying the same argument to $-U$ proves $U=0$. This avoids assuming an unstated decay rate for the boundary flux at infinity.

If $\square u=F$ with d'Alembert operator $\square =-{\partial }_t^2+\Delta$, then $\Vert \partial u(t){\Vert }_2\le \Vert \partial u(0){\Vert }_2+{\int }_0^t\Vert F(s){\Vert }_{2}ds$, where $\partial u=(u_t,\nabla u)$. Test the wave equation against $u_t$, use integration by parts, and apply the Cauchy-Schwarz inequality. The same energy estimate applies to a finite system of wave equations.

The conserved energy of a wave map from Minkowski spacetime to a Riemannian target is $E=\frac{1}{2}\int (|{\varphi }_t{|}^2+|\nabla \varphi {|}^2)dx$, with norms measured in the target metric. Contraction of the wave map equation with the velocity and integration by parts gives conservation.