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A scalar potential whose negative spatial gradient is the material acceleration. For a smooth global potential, the spatial curl of the material acceleration vanishes. On a simply connected domain the converse holds under the usual smoothness assumptions. Its physical dimension is squared length divided by squared time.

A homogeneous motion has an acceleration potential exactly when $A=\ddot{F}{F}^{-1}$ is a symmetric matrix. Necessity follows because $A$ is the spatial gradient of the material acceleration, which must be a negative Hessian matrix. For sufficiency, put $\mathbf{y}=\mathbf{x}-\mathbf{c}(t)$ and take $\xi =-{\mathbf{y}}^{T}A\mathbf{y}/2-\ddot{\mathbf{c}}\cdot \mathbf{y}+g(t)$. Direct spatial differentiation gives $-\nabla \xi =A\mathbf{y}+\ddot{\mathbf{c}}$. Equivalently $F^T\dot{F}-{\dot{F}}^{T}F$ is constant: its time derivative is $F^T\ddot{F}-{\ddot{F}}^{T}F$.

For smooth level surfaces $f=0$, $g=0$ meeting at a point where both gradients are nonzero, their acute angle is the angle between two planes tangent to the surfaces there. Each gradient is a normal vector to its tangent plane, proving the displayed formula. Vanishing gradients require further analysis because the level-set equation then need not specify a unique tangent plane.

For a Gaussian kernel $K{q}_{\parallel }^2+L{q}_{\perp }^4+t$, keeping both gradient coefficients fixed gives $q_{\parallel }^{\prime }=b{q}_{\parallel }$ and $q_{\perp }^{\prime }=b^{1/2}{q}_{\perp }$. The volume rescales by $b^{(d+1)/2}$. Fourier-field normalization is $m^{\prime }(q^{\prime })=m(q)/b^{(d+5)/4}$; the real-space normalization instead divides by $b^{(3-d)/4}$. The thermal coefficient scales by $b^2$ and the uniform source by $b^{(d+5)/4}$. Thus ${\xi }_{\parallel }\sim t^{-1/2}$, ${\xi }_{\perp }\sim t^{-1/4}$ and the formal Gaussian free-energy powers are $\alpha =(7-d)/4$, $\Delta =(d+5)/8$.

For a weak solution of a uniformly elliptic homogeneous divergence-form elliptic operator, testing with ${\eta }^2(u-c)$ and choosing the gradient of the cutoff function in $B_{2R}\setminus B_R$ gives ${\int }_{B_R}|\nabla u{|}^2\le C{R}^{-2}{\int }_{B_{2R}\setminus B_R}|u-c{|}^2$. The arbitrary constant $c$ can be chosen as an annular average, which makes the estimate suitable for a hole-filling argument.

For a $C^1$ flux $F$ and a weak solution of $\mathrm{div}F(Du)=0$, discrete integration by parts and the fundamental theorem of calculus along a line segment show that $w_h={\delta }_{\ell ,-h}u$ solves $\mathrm{div}(A_{h}D{w}_h)=0$, with $p_{-}=Du(x-h{e}_{\ell })$ and $p_{+}=Du(x)$. Coercive quadratic-form bounds and operator norm bounds for $DF$ on all segments between these gradients pass to $A_h$. This produces a divergence-form elliptic operator without initially assuming second partial derivatives of $u$.

Suppose a fixed-step size stochastic gradient descent iteration satisfiesand $f$ has a global minimizer $x^{\ast }$. Taking expectations and forming a telescoping sum proves the displayed bound with ${\Delta }_0=f(X^0)-f(x^{\ast })$ and deterministic $X^0$. The Cauchy-Schwarz inequality then bounds the averaged, or the best, expected gradient Euclidean norm by $\sqrt{M+2L{\Delta }_0/k}$. This is a guarantee of stationarity in the indicated senses, rather than a last-iterate convergence assertion. The additive $M$ is a stochastic gradient noise floor.

Target: real-analysis.bigb, under Strongly convex function.

Apply the chain rule to a recurrent network unrolled over its input sequence, adding contributions to parameters reused at multiple time steps. Products of state Jacobians convey effects from earlier states to later losses. This can compute exact gradients for the chosen finite unrolling, while physiological availability of the same history and error signals is a separate question.

For an inviscid barotropic fluid with only potential body forces, the vorticity $\mathbf{\omega }=\nabla \times \mathbf{u}$ obeysThe pressure force is a gradient because ${\rho }^{-1}\nabla p=\nabla {\int }^{\rho }{p}^{\prime }(s)ds/s$. A spatially uniform specific entropy makes a fixed-composition fluid barotropic. Merely having $Ds/Dt=0$ does not exclude baroclinic vorticity generation from spatial entropy gradients.

The Beckmann potential integrates each link's delay function. Its gradient with respect to route flows is the corresponding vector of route delays, making Wardrop equilibrium a convex optimization problem.

The extraordinary wave's Poynting vector can differ in direction from its wavevector. The latter specifies wave phase propagation, while the former specifies energy flux. This angular separation is walk-off.^[ref]

In the lossless, nondispersive model above, take the gradient of $\omega =c\sqrt{k_x^2/n_e^2+k_z^2/n_o^2}$ to obtain the group velocity, parallel to the Poynting vector. If its angle to the optic axis is $\varphi$, this gives $\tan \varphi =(n_o^2/n_e^2)\tan \theta$. Consequently the extraordinary wave generally carries energy away from its wavevector direction. An internal wavevector normal to a slab face therefore does not guarantee a normal energy path. Birefringent walk-off vanishes at $\theta =0$ and $\theta =\pi /2$; the latter direction still has unequal refractive indices. This is a useful waveplate geometry: overlapping components can acquire phase retardation without spatial splitting. A thin specimen, insufficient imaging resolution, or excitation of only one linear polarization can also hide a double image, for different reasons.

The orthogonal projection of the gradient onto the coordinate subspace indexed by $\mathcal{B}$. It is kept as a vector in the original Euclidean space, with zero coordinates outside the block. For a singleton block it is ${\partial }_{i}f(x)e_i$; for the full block it is $\nabla f(x)$.

Target: gradient-descent.bigb, under Lipschitz gradient.

In the smooth zero-winding long-wave sector, integrate the density Gaussian integral by completing $g(\delta \rho +i{\partial }_{\tau }\varphi /g{)}^2/2$. Its field-independent normalization leaves the displayed action. The real-frequency pole is ${\omega }^2=g{\rho }_0{k}^2/m$, so the phonon speed is $\sqrt{g{\rho }_0/m}$. Keeping density gradients yields ${\omega }^2=g{\rho }_0{k}^2/m+k^4/(4m^2)$, matching the Bogoliubov spectrum.

Although $1/r$ is harmonic away from the origin in three dimensions, it is not a regular harmonic function on a ball containing the origin. Its gradient is $-\hat{r}/r^2$, so the displayed improper volume integral follows by canceling the $r^2$ in the spherical volume element. On the annulus $\epsilon <r<a$, Green's first identity gives outer boundary contribution $-4\pi$ and inner contribution $4\pi \epsilon /a$, recovering the same limit. Its global Laplacian contains a Dirac delta distribution, so the regular-harmonic gradient-pairing theorem is inapplicable.

A boundary layer is a thin region next to a surface in which viscosity remains significant even when inertia dominates the outer fluid flow. Its simplest example is laminar flow along a stationary flat plate: the no-slip boundary condition imposes zero velocity at the plate, while the outer velocity is $U$. The resulting thin velocity profile contains a large normal velocity gradient, so small dynamic viscosity can still produce appreciable wall shear stress. The Blasius boundary layer makes this competition quantitative and explains how a plate aligned with the incoming fluid flow acquires drag.

The same balance appears in a singular perturbation of a differential equation: large derivatives in a thin region can keep a term with a small coefficient significant. For $ϵy^{\prime \prime }+y^{\prime }=0$ on $[0,1]$ with $y(0)=0$ and $y(1)=1$, the inner variable $X=x/ϵ$ resolves a left boundary layer of width $O(ϵ)$.

The diagrams solve an explicitly bounded Dirichlet problem in a uniform charge-free medium of constant permittivity. Take a width scale $W=1$ and terminal electric potential difference $V=1$. Two zero-thickness ideal plate segments occupy $|x|\le W/2$ at $y=\pm d/2$, with electric potentials $\pm V/2$. Fix $\varphi =0$ on the entire rectangle $x=\pm 2W$, $y=\pm 1.5W$. This boundary is an additional enclosing equipotential, rather than a condition at infinity. The model is invariant along the omitted third direction. The connection between the electric potential, its gradient, and the charge-free Laplace equation follows from electrostatics.^[ref]

On a square mesh of spacing $h=W/160$, the finite difference method imposesat free nodes, while plate and outer nodes retain their prescribed electric potentials. There are $641\times 481$ nodes. Each plotted gap aligns exactly with the mesh; the smallest gap has 16 mesh intervals. The generators solve the resulting linear system by the conjugate gradient method, stopping when the maximum dimensionless five-point residual is below $2\times 10^{-10}$, and use central finite differences to approximate $\mathbf{E}=-\nabla \varphi$ away from plate nodes.

The discrete maximum principle keeps the computed electric potential between $-V/2$ and $+V/2$. Reflection checks require evenness in $x$ and oddness in $y$; halving $h$ tests discretization effects, while enlarging the outer rectangle tests sensitivity to its artificial boundary conditions. These are different checks: a small residual only establishes that the discrete equations were solved. Differentiating near zero-thickness endpoints is especially sensitive to the mesh, so the plots do not report a converged edge peak. No total electric charge, exact open-domain three-dimensional capacitance, or measured material property is extracted from this model.

For a bounded numerical check, compare $h=W/80$ with the plotted $h=W/160$ at midplane positions $x/W=0,0.25,0.40,0.50,0.60,0.80$. At $d/W=0.10$, the largest change in $-E_{y}d/V$ is approximately $0.0265$, at $x/W=0.50$; the other five sampled changes are below $0.00472$. At $d/W=0.25$, moving the outer rectangle from $\pm 2W\times \pm 1.5W$ to $\pm 3W\times \pm 2W$ while retaining $h=W/80$ changes these samples by at most approximately $0.00622$. These calculated sensitivities support the qualitative comparison and expose its limits; they are not experimental uncertainties or evidence that the artificial boundary has converged to infinity.

On a bounded Lipschitz domain, a sequence with bounded $L^1$ norms and bounded total variation seminorm has a subsequence converging strongly in $L^1$. The limit is in the BV space, and its variation does not exceed the limit inferior. This supplies the compactness step of the direct method in the calculus of variations for linearly growing gradient energies.

For probability measures on ${\mathbb{R}}^d$ with finite second moments and a source satisfying absolute continuity of measures with respect to Lebesgue measure, the quadratic Kantorovich optimal transport problem has a unique optimal transport plan. It is induced by the gradient of a convex function, which also uniquely solves the Monge optimal transport problem up to a source-null set.

The spatial gradient of buoyancy determines the buoyancy anomaly produced by a small parcel displacement: $b^{\prime }=-\mathbf{\xi }\cdot \nabla \bar{b}$. Its vertical component is the squared buoyancy frequency in a stably stratified Boussinesq approximation. A horizontal component enters thermal wind and the propagation of a Boundary Rossby wave.

In the upper half-plane $y>0$, the hyperbolic length element is $ds/y$. The function $\Phi =\mathrm{arsinh}(x/y)$ has gradient $(1,-x/y)/\sqrt{x^2+y^2}$ and norm $1/y$. The Cauchy-Schwarz inequality gives $\int ds/y\ge \Phi (B)-\Phi (A)$ for any oriented path from $A$ to $B$. Equality holds along a semicircle $x^2+y^2=R^2$ traversed with increasing $x$, because its tangent is parallel to that gradient. This is a calibration proof of global minimality, beyond the stationary-path conclusion of the Beltrami identity. Translating $x$ gives the analogous proof for semicircles with other boundary centers.