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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 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.

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.

For the mean-field alignment equation, multiply by $|\mathbf{v}{|}^2/2$ and use integration by parts. Exchanging $z$ with $z_{\star }$ and averaging the two results gives the displayed nonpositive kinetic energy derivative. The factor $1/2$ accounts for counting ordered pairs twice. Total momentum is conserved, so the dissipated part is motion relative to the global mean velocity.

For the mean-field alignment equation in $d$ position and $d$ velocity coordinates, put $\rho =\int f{d}^{d}v$. Its force has velocity-space divergence $-d\int K{\rho }_{\star }{d}^d{x}_{\star }$. Consequently integration by parts gives the nonnegative derivative in the title, assuming suitable positivity, decay and integrability. The fine-grained Gibbs entropy decreases because these dynamics contract phase space; the interaction is dissipative.

For constant mass density $\rho$, kinematic viscosity $\nu >0$, ideal induction, zero boundary velocity and tangent magnetic field, put $E={\int }_D[\rho |\mathbf{v}{|}^2/2+|\mathbf{B}{|}^2/(2{\mu }_0)]dV$. Multiplying the momentum equation by $\mathbf{v}$ and integrating cancels pressure work and advective flux. The magnetic-energy derivative is minus the integral of Lorentz force density dotted with velocity, canceling the same work in the kinetic-energy equation. Integration by parts then gives the displayed identity. For incompressible flow with zero boundary velocity, $\int |\nabla \mathbf{v}{|}^2=\int |\nabla \times \mathbf{v}{|}^2$.

The Wallis integrals are $I_n={\int }_0^{\pi /2}{\sin }^{n}xdx$ for integers $n\ge 0$. Integration by parts gives $n{I}_n=(n-1)I_{n-2}$ for $n\ge 2$, with $I_0=\pi /2$, $I_1=1$. HenceBecause $0<\sin x<1$ in the integration interval, $0<I_n<I_{n-1}$ for $n\ge 1$. Combining this with the recurrence gives $2n/(2n+1)<I_{2n+1}/I_{2n}<1$, proving that the ratio tends to one and yielding the Wallis product.

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.

If continuously differentiable $F_j,F$ converge uniformly on a compact interval and have uniformly bounded total variation of a function, then $\int b{F}_j^{\prime }\to \int b{F}^{\prime }$ for every continuous $b$. For continuously differentiable test functions this is integration by parts. Uniform approximation by polynomials extends the conclusion to continuous tests using the variation bound.

A weak formulation multiplies a differential equation by a test function, integrates, and uses integration by parts to move derivatives away from the unknown function. Natural boundary conditions appear in the resulting boundary terms.

Convergence in the displayed sense combines strong function convergence with weak-star convergence of derivative vector measures against $C_0$ tests. Strong $L^1$ convergence and a uniform variation bound imply the derivative convergence by integration by parts and uniform test-function approximation. Conversely the Uniform boundedness principle bounds the derivative measures of a convergent sequence.

For a nonreal regular Rayleigh equation for inviscid shear flow mode vanishing at the walls, put $F=\hat{w}/(U-c)$. The equation becomes $[(U-c{)}^2{F}^{\prime }{]}^{\prime }-k^2(U-c{)}^{2}F=0$. Multiplication by $\overline{F}$ and integration by parts gives the displayed identity. With $Q=|F^{\prime }{|}^2+k^2|F{|}^2$, its imaginary and real parts give $\int UQ=c_r\int Q$ and $\int U^{2}Q=|c{|}^2\int Q$. Since $(U-U_{\min })(U_{\max }-U)\ge 0$, these imply Howard's semicircle theorem.

For a contraction semigroup with semigroup generator $L$, the bounded operators $L_{\lambda }$ approximate $L$ on its generator domain. The Laplace-transform formula for a semigroup resolvent and strong continuity give $\lambda R_{\lambda }g\to g$ for every vector $g$: split ${\int }_0^{\infty }\lambda e^{-\lambda t}(P_{t}g-g)dt$ into a short interval and an exponentially small tail. For $f\in D(L)$, integration by parts gives $R_{\lambda }Lf=\lambda R_{\lambda }f-f$, so $L_{\lambda }f=\lambda R_{\lambda }Lf\to Lf$. No operator norm convergence of $\lambda R_{\lambda }$ is implied.

For a stationary planar advected magnetic flux potential, average its advection-diffusion equation horizontally and split it into mean and fluctuation. The mean transport is $\overline{w{a}^{\prime }}=\eta {\bar{a}}_z+C$, with $C=⟨w{a}^{\prime }⟩$. Multiplying the fluctuation equation by $a^{\prime }$ and applying integration by parts gives $⟨\overline{w{a}^{\prime }}{\bar{a}}_z⟩+B_{0}C=-\eta ⟨|\nabla a^{\prime }{|}^2⟩$. A vanishing mean derivative removes the constant term in the first correlation, yielding the displayed dissipative identity. The boundary assumptions must eliminate surface terms; a periodic layer replaces, rather than accompanies, decay at vertical infinity.