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Partial summation is the discrete analogue of integration by parts. If $A(x)={\sum }_{n\le x}{a}_n$ and $g$ is continuously differentiable, then

For a periodic absolutely continuous function with derivative in $L^2$, integration by parts gives $\widehat{f^{\prime }}(n)=in\hat{f}(n)$. Bessel's inequality and the Cauchy-Schwarz inequality prove the displayed bound in the normalized circle norm. Adding the constant coefficient gives an absolutely and uniformly convergent Fourier series. Merely bounding individual coefficients by $O(1/|n|)$ would not establish absolute convergence.

An adjoint eigenfunction is a nonzero eigenfunction of the adjoint operator on its adjoint domain. For an inner product conjugate-linear in its first argument, $L^{\ast }g=\overline{\lambda }g$ implies $⟨g,(L-\lambda )f⟩=0$ for every $f$ in the domain of $L$. Consequently a necessary solvability condition for $(L-\lambda )f=h$ is $⟨g,h⟩=0$. If $L-\lambda I$ is a Fredholm operator, orthogonality to every element of its adjoint kernel is also sufficient. Boundary conditions and weights in a coupled system enter the adjoint domain and must be derived by integration by parts; copying the original eigenfunction without checking them can give a wrong projection.

An adjoint Green's function solves the corresponding adjoint point-source problem, with boundary or terminal conditions chosen so that integration by parts gives a representation for a solution of the original operator. For $L={\partial }_t-{\partial }_x^2+c$ on the real line, the formal adjoint is $L^{\ast }=-{\partial }_t-{\partial }_x^2+c$. The backward kernel with terminal Dirac delta at $(\xi ,\tau )$ is the translated heat kernel with constant absorption, defined for $t<\tau$. Its pairing with a homogeneous forward solution is constant in $t$; taking the terminal limit evaluates that solution at $(\xi ,\tau )$.

Take $D_{\mu }X={\partial }_{\mu }X+[A_{\mu },X]$, signature $(+---)$ and the trace Lagrangian density $\mathrm{Tr}(F_{\mu \nu }{F}^{\mu \nu })/4-\mathrm{Tr}(D_{\mu }\Phi D^{\mu }\Phi )/2+m^2\mathrm{Tr}({\Phi }^2)/2$. The variation of the gauge curvature is $D_{\mu }\delta A_{\nu }-D_{\nu }\delta A_{\mu }$. Cyclicity of the trace and covariant integration by parts give gauge coefficient $-D_{\mu }{F}^{\mu \nu }+[D^{\nu }\Phi ,\Phi ]$ and scalar coefficient $D_{\mu }{D}^{\mu }\Phi +m^2\Phi$. Their vanishing gives the displayed Euler-Lagrange field equations. In static temporal gauge at zero mass they reduce to $D_j{F}_{ji}=[\Phi ,D_i\Phi ]$ and $D_i{D}_i\Phi =0$.

// Destination: integrable-systems.bigb

Integrate the formal pseudodifferential residue over a period, or over the real line when the integral converges and the relevant boundary terms vanish. More algebraically, apply a linear functional on the coefficient algebra that annihilates derivatives. For $A=a{\partial }^p$, $B=b{\partial }^q$ and $r=p+q+1\ge 0$, the two residues are $\left(\genfrac{}{}{0ex}{}{p}{r}\right)a{b}^{(r)}$ and $\left(\genfrac{}{}{0ex}{}{q}{r}\right)b{a}^{(r)}$. Since $\left(\genfrac{}{}{0ex}{}{q}{r}\right)=(-1{)}^r(\genfrac{}{}{0ex}{}{p}{r})$, integration by parts proves $\mathrm{Tr}(AB)=\mathrm{Tr}(BA)$. For $r<0$ both residues vanish. Bounded orders ensure only finitely many monomial pairs contribute to the residue of a product.

An oscillatory integral amplitude multiplies $e^{i\Phi }$ in an oscillatory integral with phase function $\Phi$. A finite symbol class order controls its frequency growth and the order of a distribution obtained by integration by parts.

For $\mathbf{C}=\mathbf{u}\times (\mathbf{B}\cdot \nabla )\mathbf{u}$, the cross product identity $(\mathbf{u}\times \nabla \times \mathbf{C}{)}_i=u_j{\partial }_i{C}_j-u_j{\partial }_j{C}_i$ and integration by parts give ${\mathcal{E}}_i^{(2)}=-⟨{\partial }_i{u}_j{C}_j⟩$. Its alpha tensor isInterchanging $i,j$ reverses the cross product, showing that the coefficient is an antisymmetric matrix. Such a coefficient represents a turbulent magnetic pumping contribution rather than the symmetric part of the alpha effect.

// Target: fluid-mechanics.bigb

Let ${\sigma }^2=\hslash /(2m\omega )$ and $g(x)=(2\pi {\sigma }^2{)}^{-1/2}{e}^{-x^2/(2{\sigma }^2)}$ be the quantum harmonic oscillator ground-state probability density. Put $p(a)={\int }_{-a}^{a}g(x)dx$. The first excited-state density is $x^{2}g(x)/{\sigma }^2$. Integration by parts, using $g^{\prime }=-xg/{\sigma }^2$, gives the first-order differenceThe bracket grows from zero to one, since its derivative is $2a^{2}g(a)/{\sigma }^2$. Thus the first-order predicted level gap for $H_0+\lambda H^{\prime }$ is $\hslash \omega [1-\lambda \{p(a)-2ag(a)\}/2]$. For every $\lambda >2$ it becomes negative at sufficiently large $a$.

For $\lambda >1$ the full potential has a central barrier and symmetric minima at $\pm a$. The unperturbed even ground state samples the central barrier more strongly than the odd excited state, whose node reduces its shift. This explains the perturbative tendency, but the negative predicted gap is a failure diagnostic of the first-order truncation. The exact one-dimensional ground state remains nodeless and below the first excited state; in a high symmetric barrier the actual pair has a small positive double-well tunneling splitting.

An axisymmetric Stokes streamfunction disturbance $\varphi (r)e^{ik(z-ct)}$ of an axial basic flow obeys the displayed ordinary differential equation. Regularity requires $\varphi =O(r^2)$ at the axis, and an impermeable wall requires $\varphi (a)=0$ for nonzero $k$. Multiplication by ${\varphi }^{\ast }/r$ and integration by parts shows that a growing real-wavenumber mode requires $(U^{\prime }/r{)}^{\prime }$ to change sign; if it vanishes identically, the positive energy integral rules out such a mode. The condition concerns this cylindrical vorticity gradient rather than $U^{\prime \prime }$ alone.

For a nonanomalous Dirac current $j^{\mu }$, introduce $Z[B]=\int \mathcal{D}\bar{\psi }\mathcal{D}\psi e^{iS+i\int B_{\mu }{j}^{\mu }}$. The local phase change of variables has unit functional Jacobian, while its kinetic variation cancels the shift of the source. Thus $Z[B+\partial \alpha ]=Z[B]$. At first order, integration by parts yields ${\partial }_{\mu }\delta Z/\delta B_{\mu }=0$. Charged sources transform too when deriving the full Ward-Takahashi identity with Ward identity contact terms.

With constant density divided out, a decaying or periodic geostrophic balance on an f-plane haswhere $q=\zeta -f\eta /H$ is the signed linearized shallow-water potential-vorticity anomaly. The stationary height equation and integration by parts prove the identity. Replacing $q$ by $|q|$ destroys it: an odd height paired with symmetric opposite-sign sheets would then misleadingly give zero.

Black-body radiation is the electromagnetic radiation in thermal equilibrium with an ideal blackbody. It can be described as a photon gas: a cavity whose walls are all at the same temperature contains radiation with the universal Planck's law spectrum. A small opening in such a cavity approximates a blackbody emitter because incoming radiation undergoes repeated absorption inside. The radiation's equilibrium spectrum depends on temperature, rather than the wall material. ^[ref]

There are three useful quantities to keep distinct: directional spectral radiance $B_{\nu }$ or $B_{\lambda }$, total energy density $u$ in the cavity, and outward radiant exitance $j^{\ast }$ from its opening. Planck's law supplies the first; integrals over frequency and directions supply the others. For isotropy, every direction has the same $B_{\nu }$, so the full $4\pi$ solid angle givesHere $\hslash =h/(2\pi )$ is the reduced Planck constant, and the radiation constant is $a_{\mathrm{rad}}\approx 7.565733250\times 10^{-16}\mathrm{J}{\mathrm{m}}^{-3}{\mathrm{K}}^{-4}$. The explicit Boltzmann constant makes this an SI formula; adopting natural units with $k_B=1$ recovers $u={\pi }^2{T}^4/(15{\hslash }^3{c}^3)$.

For a short integral outline, substitute $x=h\nu /(k_{B}T)$ and useThe expansion is a geometric series with nonnegative terms; repeated integration by parts evaluates each integral, and $\zeta (4)={\pi }^4/90$ is a value of the Riemann zeta function. This gives the coefficient above.

The photon gas also exerts radiation pressure, $p=u/3$. The factor $1/3$ comes from the angular average of the squared directional component of photon momentum, $⟨{\cos }^2\theta ⟩=1/3$. The thermodynamic derivation of blackbody radiation pressure gives the same result. In contrast, outgoing power involves one projected-direction factor over an outward hemisphere, yielding radiant exitanceThis is the Stefan–Boltzmann law with unit emissivity; $\sigma$ is the Stefan-Boltzmann constant. The factor $\pi$ follows from Lambert's cosine law, rather than the $4\pi$ full-sphere factor used for energy density.

For example, an ideal $6000\mathrm{K}$ cavity has $u\approx 0.980519\mathrm{J}{\mathrm{m}}^{-3}$, radiation pressure $p\approx 0.326840\mathrm{Pa}$, and outward radiant exitance $j^{\ast }\approx 7.34881\times 10^7\mathrm{W}{\mathrm{m}}^{-2}$. A stipulated $3000\mathrm{K}$ cavity has one-sixteenth each value because all three scale as $T^4$. These describe stored energy, mechanical pressure, and outgoing power per area, respectively. In a closed equilibrium cavity, equal opposing radiation balances the directional power flow even though $u$ and $p$ are nonzero.

The usefulness of black-body radiation is this common reference: Planck's law, Wien's displacement law, and the Stefan–Boltzmann law consistently connect spectral measurements, temperature, and total emission. The three illustrative curves in Planck's law are ideal equilibrium spectra; applying them to a material requires considering its emissivity and whether thermal equilibrium is a suitable approximation.

On a closed Riemannian manifold, a Killing vector field $X$ satisfies ${\nabla }^{\ast }\nabla X={\mathrm{Ric}}^{♯}X$ for the nonnegative rough Laplacian. This follows by differentiating ${\nabla }_i{X}_j+{\nabla }_j{X}_i=0$, contracting and commuting covariant derivatives; the contraction also gives $\mathrm{div}X=0$. Taking the inner product with $X$ and using integration by parts gives the displayed identity. If Ricci curvature is nonpositive, the nonnegative left side must vanish, so $X$ is a parallel vector field. In particular every Killing vector field on a closed flat Riemannian manifold is a parallel vector field.

For real smooth coefficients with $p,w>0$, the Rayleigh quotient $\Lambda [y]=\int [p(y^{\prime }{)}^2+q{y}^2]dx/\int w{y}^{2}dx$ has the displayed first variation. Its stationary points solve the Sturm-Liouville equation on an admissible class whose boundary variations make $[p{y}^{\prime }\eta {]}_a^b=0$. Homogeneous Dirichlet boundary conditions do this by setting $\eta =0$ at the ends; homogeneous Neumann boundary conditions do it through $y^{\prime }=0$. More general self-adjoint boundary conditions can require a boundary contribution to the quadratic form.

For an eigenfunction, the weaker identity $[py{y}^{\prime }{]}_a^b=0$ suffices to give $\lambda =\Lambda [y]$ by integration by parts, but it does not by itself imply the condition on every variation. An explicit counterexample uses $p=q=w=1$ on $[0,1]$ and $y=\cos \pi x+\sin \pi x$. This solves $-y^{\prime \prime }+y=(1+{\pi }^2)y$ and has $[y{y}^{\prime }{]}_0^1=0$. Define a nearby familyIt keeps $y_{\epsilon }(0)=1$, $y_{\epsilon }^{\prime }(0)=\pi$, $y_{\epsilon }(1)=-1+\epsilon$, and $y_{\epsilon }^{\prime }(1)=\pi /(-1+\epsilon )$, so $[y_{\epsilon }{y}_{\epsilon }^{\prime }{]}_0^1=0$ for small $\epsilon$. Nevertheless, $\eta ={\partial }_{\epsilon }{y}_{\epsilon }{|}_0$ has $\eta (0)=0$, $\eta (1)=1$, and ${\int }_0^1{y}^{2}dx=1$, giving $\delta \Lambda =2[y^{\prime }\eta {]}_0^1=-2\pi \ne 0$. Boundary-energy cancellation alone therefore cannot replace a specified variational domain.

The anisotropic-drag contribution to axial force from a small-slope transverse motion is $F=(c_{\perp }-c_{\parallel })\int y_x{y}_{t}dx$. Using the small-slope elastohydrodynamic filament equation and integration by parts givesThus this propulsive contribution is determined by endpoint slope, bending moment, and shear. A separately imposed axial translation adds ordinary longitudinal drag. Exact second-order longitudinal motion required by inextensibility also contributes to instantaneous total drag, but its time-derivative contribution averages to zero over a deformation cycle.

A boundary term is an endpoint or boundary integral left when derivatives are transferred by integration by parts. It vanishes under some boundary conditions and otherwise supplies natural boundary conditions, conserved charges, or physical surface forces.

If the spatially integrated Lagrangian changes by $dB[\Phi ]/dt$, where $B$ depends on fields and their spatial derivatives but not velocities, the canonical momentum components change by ${\pi }_a\mapsto {\pi }_a+\delta B/\delta {\Phi }_a$. This follows by varying $dB/dt$ and using spatial integration by parts. The canonical one-form changes by the exact variation $\delta B$, so the symplectic form and canonical commutation relations are preserved. For the two Feynman gauge Maxwell densities, $B=\int A_0{\partial }_i{A}_i{d}^{3}x$ gives shifts $\Delta {\pi }^0={\partial }_i{A}_i$ and $\Delta {\pi }^i=-{\partial }_i{A}_0$. Boundary flux must vanish for this spatial functional calculation.

Among curves of fixed arc length $L>2a$ joining $(-a,0)$ to $(a,0)$, the hanging catenary globally minimizes $\int yds$. Parametrize a competitor by arc length $s\in [-L/2,L/2]$, so $x^{\prime 2}+y^{\prime 2}=1$. Integration by parts and the fixed horizontal endpoint separation give, for every $b>0$,The inequality is the Cauchy-Schwarz inequality. Equality requires $x^{\prime }=b/\sqrt{b^2+s^2}$ and $y^{\prime }=s/\sqrt{b^2+s^2}$. Integration gives the catenaryThe last equation has a unique positive $b$, since $\sinh q/q$ increases strictly for $q>0$. The equality curve is a single-valued graph below the chord, so the same global bound proves the result under those additional restrictions.

The locally integrable function $E(z)=1/(\pi z)$ on the plane is a fundamental solution for the Wirtinger derivative ${\partial }_{\bar{z}}=({\partial }_x+i{\partial }_y)/2$. Let $L=(4\pi {)}^{-1}\log (x^2+y^2)$ be the fundamental solution of the Laplace equation. Its first distributional derivatives agree with its ordinary ones: deleting a radius-$\epsilon$ disk in integration by parts gives an error $O(\epsilon |\log \epsilon |)$, tending to zero. Hence $E=4{\partial }_{z}L$ andThe analogous fundamental solution for ${\partial }_z$ is $1/(\pi \bar{z})$. The singular support of each kernel is exactly $\{0\}$: they are smooth elsewhere, and smoothness near zero would make their Wirtinger derivatives smooth there, contradicting the Dirac delta distribution identity.

Target: Distributional Cauchy-Riemann lemma. Parent: Cauchy-Riemann equations. Intended file: analysis.bigb.