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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.

Stein's lemma relates Gaussian multiplication to differentiation. Let $Z\sim N(0,1)$. One sufficient set of conditions is that $g$ is locally absolutely continuous, both $\mathbb{E}|Zg(Z)|$ and $\mathbb{E}|g^{\prime }(Z)|$ are finite, and $g(x)\varphi (x)\to 0$ as $x\to \pm \infty$, where $\varphi$ is the standard normal density. ThenSince ${\varphi }^{\prime }(x)=-x\varphi (x)$, integration by parts on a finite interval moves the derivative from $\varphi$ to $g$; the boundary condition and integrability justify the infinite-interval limit.

For $g(z)=z$, the identity gives $\mathbb{E}{Z}^2=1$. For $g(z)=z^3$, it gives $\mathbb{E}{Z}^4=3\mathbb{E}{Z}^2=3$. More generally, for $X\sim N(\mu ,{\sigma }^2)$ with $\sigma >0$, standardization yieldsunder the corresponding conditions. The factor ${\sigma }^2$ is essential.

This integration-by-parts rule converts Gaussian error cross terms into derivatives of an estimator, which is the mechanism of the Stein risk identity for normal mean estimators. Boundary behavior matters for singular estimators: formal differentiation alone does not justify applying the identity across a nonintegrable singularity.

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.

Stochastic heating increases random particle kinetic energy through decorrelating electromagnetic kicks. A simple one-dimensional model makes the normalization explicit. Let $f(v,t)$ be a velocity distribution with number density $n=\int fdv$, and suppose constant-coefficient velocity-space diffusion givesThe coefficient has dimensions of squared velocity per time, and a freely diffusing component has variance $2Dt$. For sufficiently decaying tails, two applications of integration by parts yieldThe first velocity moment is unchanged, so this is growth of random motion rather than acceleration of the mean flow. In three independent isotropic components with coefficient $D$ in each, the heating rate per volume is instead $3mnD$. More generally each diagonal component contributes its own diffusion coefficient.

This idealization describes externally supplied, rapidly decorrelating forcing; its energy budget is not that of an isolated particle population. Screening, resonant self-consistent fields, collisions and velocity-dependent diffusion can change the result. In other plasma settings, stochastic heating also describes irregular orbits driven by spatial structure or overlapping resonances; that mechanism must be specified separately.

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.

Sturm-Liouville theory studies weighted eigenvalue problems of the formwith boundary conditions making the operator self-adjoint. For a regular finite-interval problem, $p,w$ are positive and appropriate coefficient regularity is assumed. Integration by parts cancels the boundary terms and proves that eigenfunctions at distinct eigenvalues are orthogonal in the weighted inner product $\int w{y}_1\overline{y_2}$.

For $p=w=1$, $q=0$, $y(0)=y(L)=0$, the eigenfunctions are $\sin (n\pi x/L)$ with eigenvalues $(n\pi /L{)}^2$, $n\ge 1$. Their expansions solve wave equations and heat equations by reducing spatial behavior to independent modes. Singular endpoints require additional domain and boundary analysis; regular-problem conclusions cannot simply be transferred without those hypotheses.

The Swift–Hohenberg equation is a partial differential equation for pattern formation. In one spatial dimension, a cubic-quintic version isLinearizing at $w=0$ and substituting a Fourier mode $e^{\sigma t+ikx}$ gives $\sigma (k)=r-(1-k^2{)}^2$. The first unstable modes therefore have wavenumber near one when $r$ becomes positive. Replacing one by $q_0^2$ inside the differential operator selects wavenumbers near $q_0$.

There is also a useful Lyapunov functional. On a periodic interval, or with sufficient decay at infinity, defineTwice using integration by parts gives the functional derivative $\delta E/\delta w=(1+{\partial }_x^2{)}^{2}w-rw-s{w}^3+w^5$. The equation is therefore a gradient flow, andThis dissipative structure constrains the evolution without requiring the final spatial pattern to be uniform. For $s>0$, the cubic term enhances finite amplitudes while the quintic term limits growth. Near onset, a slowly varying envelope can satisfy a quintic real Ginzburg-Landau equation.

For a period-$L$ absolutely continuous function with integrable derivative, integration by parts gives the coefficient identityThe endpoint term vanishes because the periodic values agree. This identity by itself does not assert pointwise convergence of the differentiated series.

A concrete sufficient condition for classical termwise differentiation isThe differentiated series then converges uniformly by the Weierstrass M-test, as does the original Fourier series. The uniform derivative convergence with an anchored value shows that its sum is continuously differentiable and has the displayed derivative series. For example, analytic strip decay satisfies this condition. A twice continuously differentiable periodic function also suffices, since absolute Fourier convergence from a square-integrable derivative applied to $f\prime$ proves absolute summability of the derivative coefficients.

The Fourier series of a periodic sawtooth illustrates the limitation: its formal derivative has nondecaying Fourier amplitudes and is not an ordinary convergent series. Its correct periodic derivative exists as a distributional derivative and includes the jump point masses. Coefficient differentiation and classical termwise summation are distinct claims.

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.

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 sets a specified boundary quantity to zero. The quantity must be named: zero boundary value is a homogeneous Dirichlet boundary condition, while zero outward normal derivative is a homogeneous Neumann boundary condition. These are different restrictions.

Which boundary terms disappear under integration by parts depends on the operator and the permitted test functions. For a second-order weak problem, zero test-function trace eliminates ${\int }_{\partial U}({\partial }_{n}u)v$. A fourth-order problem can also involve derivative traces, so a zero value alone does not remove every boundary term. On $[0,1]$, $u(x)=x(1-x)$ vanishes at both endpoints but has nonzero endpoint first derivatives; the clamped boundary condition would require those slopes to vanish as well.

For the real-coefficient differential operator $Mv=(p{v}^{\prime }{)}^{\prime }+qv$, integration by parts givesIf $p$ tends to zero at both endpoints and $u,v,u^{\prime },v^{\prime }$ remain bounded there, the boundary term vanishes without prescribing endpoint values. Thus regular endpoint behaviour can supply the domain condition that makes the expression symmetric. Endpoint degeneracy alone is insufficient for arbitrary singular functions: for $p=x(1-x)$ and $v=\log x$, one has $p{v}^{\prime }=1-x\to 1$ at zero. A self-adjoint differential operator requires a specified admissible domain, rather than only a formal differential expression.

Let $M$ be a connected noncompact Riemannian manifold without boundary, and let $u$ be a smooth real harmonic function. Assume every set $\{|u|\ge \epsilon \}$ is compact for $\epsilon >0$, expressing decay to zero at infinity. Then $u=0$.

For a positive regular value $\epsilon$, the superlevel set $D=\{u>\epsilon \}$ has compact closure and smooth boundary. Integration by parts givesHence the gradient vanishes on $D$. A nonempty component would then be constant with value greater than $\epsilon$, incompatible with boundary value $\epsilon$. Such a component cannot be all of the connected noncompact $M$, since its closure is compact. Thus no positive value occurs; applying the argument to $-u$ excludes negative values.

The noncompactness and decay hypotheses matter. On a compact connected manifold without boundary, every constant has compact superlevel sets, and a nonzero constant disproves the vanishing conclusion. Alternatively, under the stated noncompact decay hypotheses, the strong maximum principle for harmonic functions proves vanishing by an attained positive maximum or negative minimum.