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Fick's first law says that diffusive flux points down the concentration gradient:Combining it with local conservation gives
$C_t=\nabla \cdot (D(C)\nabla C)$.

At an interior local extremum of a differentiable function, its gradient is zero: restricting the function to every line through the point shows that all directional derivatives vanish. This is only a necessary condition in general. For a strictly concave function, an interior stationary point is the unique global maximizer on its convex domain. Constrained or boundary optima instead need conditions adapted to their feasible directions.

For a differentiable objective on an admissible real Hilbert space $H$, with $\Vert {\mathbf{u}}_0{\Vert }^2/2=E_0>0$, stationarity on the sphere in a normed vector space of fixed kinetic energy means that the projected gradient is normal to the sphere in a normed vector space. Thus $P_H\nabla J=\lambda {\mathbf{u}}_0$ for a real Lagrange multiplier. It follows by testing all tangent variations orthogonal to ${\mathbf{u}}_0$. The multiplier need not be positive; second-order conditions are still needed for a local minimum.

In vacuum analysis, an exact flat direction of a scalar potential is a nonconstant continuous family of degenerate stationary vacua. The scalar potential stays constant along this family and its gradient vanishes there. An ordinary level set does not suffice: $V(x,y)=x^2+y^2$ is constant on every circle, but its vacuum at the origin has no flat direction. A vanishing eigenvalue of the quadratic Hessian matrix also does not suffice: $V(x)=x^4$ has zero quadratic mass at zero but no family of nearby vacua. Exact flat directions can be lifted by corrections absent from the classical scalar potential.

The transformation $A\mapsto A+\nabla \chi$ leaves $B=\nabla \times A$ unchanged because the curl of a gradient vanishes.

After integrating a free-field momentum shell, restore the cutoff with $x^{\prime }=x/b$ and ${\varphi }^{\prime }(x^{\prime })=b^{(D-2)/2}{\varphi }_{<}(b{x}^{\prime })$. The gradient coefficient is invariant, the mass changes as $r^{\prime }=b^{2}r$, and a uniform conjugate field changes as $h^{\prime }=b^{(D+2)/2}h$. This follows by counting measure, derivative and field factors in the quadratic Hamiltonian. A nonuniform source instead obeys $h^{\prime }(x^{\prime })=b^{(D+2)/2}h(b{x}^{\prime })$, with the source on the right projected onto the retained Fourier modes. The field has engineering dimension $(D-2)/2$ and zero anomalous dimension.

For independent and identically distributed random variables $X,Y$ with a centered multivariate normal distribution, a continuously differentiable function $f$ with bounded gradient, and a convex function $\Psi$ for which the expressions are integrable, the displayed inequality holds. First apply Jensen inequality conditionally to $f(X)-f(Y)$. For the Gaussian rotation of independent copies, the chain rule gives $f(X)-f(Y)={\int }_0^{\pi /2}⟨\nabla f(U_{\theta }),V_{\theta }⟩d\theta$. A second application of Jensen inequality to the uniform measure on $[0,\pi /2]$, followed by the Gaussian rotation of independent copies, proves the inequality. The useful choice $\Psi (v)=e^{\lambda |v|}$ yields a Gaussian concentration inequality through an exponential moment of an absolute standard normal variable.

For a discrete image signal operator $A$, minimize $\alpha \Vert Au{\Vert }_2+\frac{1}{2}\Vert u-g{\Vert }_2^2$ with one Euclidean norm on the full gradient array. This differs from discrete isotropic total variation, which sums pixelwise Euclidean lengths. The global norm's dual ball is a single ball; its convex projection normalizes the whole vector array.

For a harmonic function $w$ smooth inside a bounded domain and $C^1$ on its closure, $\Delta |Dw{|}^2=2|D^{2}w{|}^2\ge 0$. The weak maximum principle for elliptic operators and continuity of the gradient imply ${\sup }_{\Omega }|Dw|={\sup }_{\partial \Omega }|Dw|$.

For an assumed $W^{1,1}$ minimizer of $\alpha A(u)+\frac{1}{2}\Vert u-g{\Vert }_2^2$, compactly supported variations give $u-g-\alpha \mathrm{div}(\nabla u/\sqrt{1+|\nabla u{|}^2})=0$ in distributions. If the regularizer is instead the total variation seminorm, a bounded calibration field replaces the quotient and handles zero gradients.

The Helmholtz decomposition separates a vector field into a gradient field and a field with zero divergence. The precise function spaces, boundary conditions and decay hypotheses determine existence and uniqueness. In the square-integrable bounded-domain formulation, the solenoidal component is selected by the Leray-Helmholtz projection.

A homogeneous Sobolev space controls derivatives without an independent low-frequency L2 norm: formally $\Vert f{\Vert }_{{\dot{H}}^s}^2=\int |\xi {|}^{2s}|\hat{f}(\xi ){|}^{2}d\xi$, with the normalization chosen according to the Fourier transform. Completions or quotient conventions must be specified when polynomial ambiguities occur. In three dimensions ${\dot{H}}^1$ is represented by $L^6$ functions with gradient in $L^2$, and $\Vert f{\Vert }_6\le C\Vert \nabla f{\Vert }_2$.

An image edge is a rapid grey-value transition, modeled as a large gradient or a jump discontinuity. It differs from an edge of a graph. Variational models can penalize image edge length, while diffusion image processing estimates image edge directions to limit image smoothing across them.

For a smooth uniformly elliptic homogeneous equation away from other boundaries, an interior gradient estimate on a ball of radius comparable to the distance $s$ from a flat side gives $|\nabla u|\le C{s}^{-1}{\sup }_{B_{cs}}|u|$. If $u$ extends continuously with value zero on that side, $s|\nabla u|\to 0$ locally along it. This justifies vanishing boundary terms such as $u_x\sin (nx)$ in a distributional Fourier sine projection.

For a finite support whose convex hull has full dimension, every interior target mean is realized by a unique natural parameter. Minimize the coercive function $\log {\sum }_n{e}^{\theta \cdot n}-\theta \cdot \mu$; its gradient is the mean discrepancy and its Hessian matrix is the positive-definite covariance matrix of the support.

For a stable homogeneous scalar quartic saddle with gradient coefficient one, the Ornstein--Zernike correlation function has denominator $q^2+r_0+u_0{m}_0^2/2$. At zero source and $u_0>0$, the correlation length is $r_0^{-1/2}$ in the disordered phase and $(-2r_0{)}^{-1/2}$ on either selected ordered branch. Linear thermal tuning of $r_0$ gives correlation-length critical exponent $\nu =1/2$ with different amplitudes on the two sides.

A harmonic function on all of ${\mathbb{R}}^n$ with a finite global Hölder seminorm of exponent $0<\mu <1$ is constant. Indeed $|w(x)-w(0)|\le C|x{|}^{\mu }$, and the interior derivative estimate for a harmonic function on balls of radius $R$ bounds $|\nabla w(0)|$ by $C{R}^{\mu -1}$. Letting $R\to \infty$ makes this gradient zero, and translating the centre proves the assertion everywhere.

With constant amplitude outside a core, the planar nematic divergence-elasticity identity reduces the elastic density to $K{\lambda }_0^2|\nabla \chi {|}^2/2$. A topological charge of a two-dimensional nematic disclination $q$ has $\chi =q\theta +{\chi }_0$ and $|\nabla \chi {|}^2=q^2/r^2$. Integrating over an annulus gives the displayed energy, plus finite core and boundary terms. The outer cutoff is the sample size, compensating-defect separation, or another screening length. An isolated defect in an infinite ordered plane has divergent energy. A neutral pair $q,-q$ separated by $R$ instead has $E_{\mathrm{pair}}=2\pi K{\lambda }_0^2{q}^2\log (R/r_0)$ plus core terms, since its far-field gradients cancel.

For the positive quadratic kernel $\tilde{\Delta }(p)={\kappa }^{-1}{p}^2+m^2$, $\kappa ,m>0$, the connected correlation function is its inverse. Fourier inversion gives $G(r)=\kappa (\sqrt{\kappa }m/r{)}^{D/2-1}{K}_{D/2-1}(\sqrt{\kappa }mr)/(2\pi {)}^{D/2}$. The Modified Bessel function of the second kind has an exponentially decaying large-argument tail, so the correlation length is $1/(\sqrt{\kappa }m)$ at fixed positive gradient coefficient. In three dimensions this becomes $\kappa e^{-\sqrt{\kappa }mr}/(4\pi r)$. The statement concerns the continuum long-distance theory; regulator-dependent contact terms do not define the physical correlation length.

If a smooth minimal surface equation for a graph solution on ${\mathbb{R}}^n\setminus \{0\}$ is homogeneous of degree one, with $n\ge 2$, then its graph extends to a plane through zero. Its gradient is homogeneous of degree zero, so $|Du{|}^2$ attains a global maximum on the unit sphere. The strong maximum principle for elliptic operators makes it constant. The differentiated equation in the gradient maximum principle for a minimal graph then forces $D^{2}u=0$, and degree-one homogeneity removes the affine constant. This argument applies to cones that are graphs of smooth functions away from their vertex; it is not a claim that every minimal surface cone is flat.