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For $y(0)=y(1)=0$, write $A={\int }_0^{1}ydx$ and $w=x(1-x)$. Integration by parts gives $2A={\int }_0^1{y}^{\prime }{w}^{\prime }dx$, while ${\int }_0^1{w}^{\prime 2}dx=1/3$. The Cauchy-Schwarz inequality therefore gives the displayed sharp bound, with equality exactly when $y=Kw$. A prescribed positive budget $B={\int }_0^1(y+y^{\prime 2})dx$ consequently givesThis proves the unique global optimum, rather than only stationarity from a Lagrange multiplier.

For real $x>0$, the sine integral approaches $\pi /2$. Exponential damping gives ${\int }_0^{\infty }{e}^{-at}\sin t/tdt=\arctan (1/a)$, so the limiting constant follows by letting $a\downarrow 0$. Integration by parts of the tail gives $\mathrm{Si}(x)=\pi /2-\cos x/x-\sin x/x^2+O(x^{-3})$.

Define the definite integral $F(k)={\int }_0^{\pi }{x}^k\sin xdx$. For real $k>1$, two applications of integration by parts giveThe boundary term $x^{k-1}\sin x$ vanishes at both endpoints. At $k=1$, direct integration gives $F(1)=\pi$, consistent with the displayed relation; $F(-1)$ is finite because $\sin x/x\to 1$ at zero. This recurrence and $F(0)=2$ evaluate every nonnegative integer moment, with separate even and odd chains.

For a divergence-free drift $\mathbf{b}$ and a function $v$ with zero boundary Sobolev trace, integration by parts gives ${\int }_{\Omega }v\mathbf{b}\cdot \nabla v=\frac{1}{2}{\int }_{\partial \Omega }{v}^2\mathbf{b}\cdot \mathbf{n}-\frac{1}{2}{\int }_{\Omega }(\nabla \cdot \mathbf{b})v^2=0$. Consequently the advection matrix in a conforming Galerkin method has zero quadratic form; its symmetric part vanishes. Adding a diffusion stiffness matrix gives a nonpositive spatial generator and a nonincreasing L2 norm in the semidiscrete advection-diffusion equation. The same identity holds with a suitable zero normal-flux boundary term instead of zero trace.

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 light along ${\mathbf{e}}_x$, vorticity along ${\mathbf{e}}_y$ and $\epsilon =\omega /(2D_r)$, spherical coordinates $\mathbf{p}=(\cos \theta ,\sin \theta \cos \varphi ,\sin \theta \sin \varphi )$ give a first orientation-density correction proportional to $\sin \varphi g(\theta )$. The linearized alignment-diffusion operator preserves azimuthal harmonics and the rotation of the axisymmetric equilibrium forces only this sine harmonic. Angular integration makes the first correction to the mean purely vertical. Uniqueness follows from the weighted identity $\mathcal{A}(f_{0}h)={\nabla }_p\cdot (f_0{\nabla }_{p}h)$: integration by parts shows the only homogeneous smooth solution is a multiple of $f_0$, removed by normalization.

The weak form of $(L^{\ast }-\lambda )u=f$ includes the same scalar shift as the differential equation. The unshifted identity $B^{\ast }[u,v]=(f,v)$ instead represents $L^{\ast }u=f$ and cannot imply the shifted equation for nonzero $\lambda$ unless $u=0$. On a smooth bounded domain choose any nonzero compactly supported smooth $u$ and take $L=-\Delta +1$, $f=Lu$ for a direct counterexample. For constant drift coefficients, integration by parts reverses the drift sign in $L^{\ast }$ without changing this spectral term.

A symmetric stress tensor is in static elastic equilibrium when its divergence balances the prescribed body force. Its boundary traction is $t_i={\sigma }_{ij}{n}_j$. With zero body force, integration by parts gives ${\int }_{\Omega }\sigma :\mathrm{sym}\nabla v={\int }_{\partial \Omega }t\cdot v$. This is the weak principle of virtual work for a static elastic body.

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 $\sqrt{2/(\pi x)}\cos (x-\nu \pi /2-\pi /4)+O(x^{-3/2})$.

Impose zero Dirichlet boundary conditions on an interval. The isoperimetric constraint $I[y]=1$ is regular because $D{I}_y[y]=2I[y]=2\ne 0$. The Lagrange multiplier rule applied to $J-\lambda (I-1)$ gives the Sturm-Liouville equationMultiplying by $y$ and using integration by parts gives $J[y]=\lambda I[y]=\lambda$. Thus the multiplier is the stationary value of the original functional, or its Rayleigh quotient. This calculation does not require $w$ to have one sign; positive-weight spectral conclusions require additional hypotheses.

If two stationary functions have different multipliers, multiplying their equations crosswise and subtracting gives $({\lambda }_n-{\lambda }_m)\int w{y}_n{y}_m=0$. The endpoint terms vanish. This proves the weighted orthogonality of Sturm-Liouville eigenfunctions. Higher stationary values need not be local minima of the constrained functional.

The Stefan–Boltzmann law gives the total hemispherical radiant exitance of an ideal blackbody at absolute temperature $T$:Here $\sigma$ is the Stefan-Boltzmann constant. Total means integrated over all wavelengths; hemispherical means integrated over all outward directions with the projected area weighting. Thus $M_b$ is emitted power per actual emitting area, rather than directional radiance or spectral radiance. Absolute temperature must be in kelvin; raising a Celsius reading to the fourth power gives the wrong result. A uniform surface of area $A$ emits radiant flux $P_{\mathrm{emit}}=A{M}_b$. A real surface instead has $M=ϵ(T)\sigma T^4$, where $ϵ(T)$ is its total hemispherical emissivity. The older symbol $F$ in the formula above denotes this emitted radiant exitance.^[ref]

For an ideal blackbody, doubling absolute temperature multiplies radiant exitance by $16$. With the approximate Stefan-Boltzmann constant $5.670374419\times 10^{-8}\mathrm{W}{\mathrm{m}}^{-2}{\mathrm{K}}^{-4}$, the radiant exitances at $300\mathrm{K}$ and $600\mathrm{K}$ are approximately $459.300$ and $7348.805\mathrm{W}{\mathrm{m}}^{-2}$. This steep response explains why thermal radiation becomes so consequential at high temperature. It also enables heat transfer through a vacuum, where thermal conduction through a gas and convection are absent.

A short derivation integrates Planck's law in its frequency form. For a blackbody, direction-independent spectral radiance is $B_{\nu }(T)=2h{\nu }^3/[c^2(e^{h\nu /(k_{B}T)}-1)]$, where $h$ is the Planck constant, $k_B$ the Boltzmann constant and $c$ the speed of light. The outward hemisphere contributes $\int \cos \theta d\Omega =\pi$, soExpanding $(e^x-1{)}^{-1}={\sum }_{n=1}^{\infty }{e}^{-nx}$, nonnegative terms allow termwise integration. Repeated integration by parts gives ${\int }_0^{\infty }{x}^3{e}^{-nx}dx=6/n^4$; the Riemann zeta function value $\zeta (4)={\pi }^4/90$ gives the remaining integral as ${\pi }^4/15$. Hence $M_b=\sigma T^4$ with the coefficient in Stefan-Boltzmann constant. The angular integral is essential: omitting it confuses radiance with radiant exitance.

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.

The three-dimensional quasi-geostrophic potential vorticity relation defines an anisotropic elliptic boundary value problem. Interior potential vorticity and boundary buoyancy perturbation $b=f_0{\psi }_z$, together with lateral and global conditions, determine the quasi-geostrophic streamfunction up to its allowed constant. If two solutions have the same data, their difference has zero elliptic source and homogeneous boundary data. Integration by parts gives zero integral of $|{\nabla }_h\psi {|}^2+(f_0^2/N^2)|{\psi }_z{|}^2$, proving uniqueness modulo a constant in a connected domain when the boundary terms vanish. Eady edge waves show that zero interior source alone is insufficient.

A stress glut tensor describes a seismic source through an equivalent body-force density $\mathbf{f}=-\nabla \cdot \mathbf{\Gamma }$. For a symmetric stress glut tensor supported away from the boundary, integration by parts makes its coupling to a normal mode equal to $\int \mathbf{\Gamma }:{\mathbf{e}}^{\ast }dV$, where $\mathbf{e}$ is that mode's strain.

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 a uniform string with tension $T>0$, linear mass density $\rho >0$, left end fixed, and a freely transverse-moving point mass $M\ge 0$ at $x=L$, the wave equation and endpoint law areThe last equation is Newton's second law: the string pulls the mass back toward the interior, giving the minus sign. With wave speed $c=\sqrt{T/\rho }$, a nonzero normal mode has spatial part $X(x)=\sin (\omega x/c)$ and satisfiesFor $M>0$, its dimensionless roots $q=\omega L/c$ occur once in each $(n\pi ,n\pi +\pi /2)$, $n\ge 0$, because $\cot q$ decreases from infinity to zero while $Mq/(\rho L)$ increases. For $M=0$, $q=(n+1/2)\pi$: the right end has a homogeneous Neumann boundary condition. There is no nonzero zero-frequency mode, since a linear spatial profile vanishing at the left end and with zero right slope must vanish everywhere.

Multiplying the spatial equation by $X$ and using integration by parts givesFor two different modes, the same calculation gives the weighted inner product orthogonality $\rho {\int }_0^L{X}_m{X}_{n}dx+M{X}_m(L)X_n(L)=0$. The endpoint mass therefore contributes to both inertia and mode orthogonality.

For a real scalar height field, integration by parts givesThe bulk expression is even under time reversal. The original density is not pointwise even, but changes by a total divergence; its stationary equation is time-reversal invariant. The first-order diffusion equation selected by zero residual is not invariant under ordinary time reversal at fixed positive $\nu$. Boundary conditions or the probabilistic interpretation can distinguish the two time directions even when the higher-order stationary equation does not.

A twisted boundary condition identifies the ends of an interval up to a fixed unit-modulus complex number. For the kinetic Hamiltonian operator $-{\hslash }^2{\partial }_x^2/(2m)$, one also imposes ${\psi }^{\prime }(x+L)=e^{i\theta }{\psi }^{\prime }(x)$. In integration by parts, the endpoint terms then cancel, giving the usual self-adjoint ring realization. The allowed plane waves have momenta $\hslash (2\pi n+\theta )/L$.

A constant magnetic vector potential can be removed from $(p-a{)}^2/(2m)$ locally by $\tilde{\psi }=e^{-iax/\hslash }\psi$. If $\psi$ is periodic, $\tilde{\psi }$ has twist $\theta =-aL/\hslash$. Thus the Aharonov-Bohm effect survives a locally vanishing vector potential as a global condition on the wavefunction.

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.