Vector calculus

Vector calculus studies differentiation and integration of scalar fields and vector fields. Three operators answer different local questions: the gradient describes change of a scalar value, the divergence describes net outward flow, and the curl describes local circulation. Their associated integral theorems turn local derivatives into measurements along paths and across boundaries.

The difference between divergence and curl is especially visible in two smooth vector fields on three-dimensional Euclidean space:At each point they have the same Euclidean norm $\sqrt{x^2+y^2}$, but their directions differ. For $F$, $\nabla \cdot F=2$ and $\nabla \times F=0$. For $G$, $\nabla \cdot G=0$ and $\nabla \times G=(0,0,2)$. One expands radially in each horizontal plane; the other describes rigid rotation.

Around the counterclockwise unit circle in the plane $z=0$, $F$ is normal to the boundary and has zero tangential component. Its outward planar flux integral is $2\pi$, while its line integral around the circle is zero. For $G$ the values are reversed: outward planar flux is zero, and circulation is $2\pi$. The divergence and curl predict exactly which boundary measurement survives.

Why is this useful? A velocity field can move material through a boundary or circulate it within a region. Conservation of mass connects its divergence with density changes through the continuity equation; its curl is vorticity. In electromagnetism, the Maxwell equations use the same operators to connect electric fields, magnetic fields, and their sources. The common mathematical language lets the same integral calculations work across different physical models. ^[ref].

The geometric framework behind these operators is developed in differential forms. The Generalized Stokes theorem unifies the fundamental theorem for line integrals, Green theorem, Stokes theorem, and divergence theorem. In coordinates their formulas look different, but each relates accumulation over a region to a measurement on its boundary.

Wilson's Vector Analysis, published in 1901, was based on J. Willard Gibbs's lectures at Yale. Wilson's preface explains that examples from geometry, mechanics, and physics are interspersed with the mathematical development, so readers encounter applications as soon as they have the tools to understand them. That is a useful way to approach vector calculus: learn each operation through a measurement it makes possible. ^[ref].

For a twice continuously differentiable vector field in Cartesian coordinates, expand the curl and then its divergence. Every mixed partial derivative occurs twice with opposite signs, proving the identity. In electromagnetism this forces the right side of the Ampère-Maxwell equation to be divergence-free, yielding local charge conservation.

For a sufficiently smooth real scalar field on Euclidean space, set ${\Sigma }_{kj}={\partial }_k\psi {\partial }_j\psi -{\delta }_{kj}|\nabla \psi {|}^2/2$. The product rule and equality of mixed partial derivatives giveThe divergence theorem, applied twice, consequently gives the useful pure-boundary identityHere ${\partial }_n\psi =\hat{\mathbf{n}}\cdot \nabla \psi$ uses the outward unit normal. This is the Euclidean zero-potential counterpart of the scalar stress-energy divergence identity.

For continuously differentiable vector fields, the gradient of their dot product expands as displayed. The contraction of two Levi-Civita symbols gives $[F\times (\nabla \times G){]}_i=F_j{\partial }_i{G}_j-F_j{\partial }_j{G}_i$; adding the directional derivative cancels the second term. Interchanging the fields and using the product rule proves the identity. It does not require either field to be irrotational.

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.

For differentiable vector fields $A$ and $B$,

A vector field assigns a vector to every point of its domain.

A vector field directed along the radius from a fixed origin, with magnitude depending only on radius. In three-dimensional Euclidean space and away from the origin, $\mathbf{F}=f(r)\mathbf{r}=rf(r)\hat{\mathbf{r}}$. The product rule gives $\nabla \cdot \mathbf{F}=3f+r{f}^{\prime }$, and the curl vanishes because the derivative matrix is symmetric. These identities alone do not establish regularity at the origin.

A vector field on an open subset of Euclidean space whose component functions have continuous first partial derivatives. Its Jacobian matrix is continuous; this is a standard regularity assumption for the divergence theorem and Stokes theorem.

The local flow of a smooth vector field is the family of solutions of its ordinary differential equation, with ${\Phi }_0(p)=p$. Uniqueness gives ${\Phi }_{s+t}={\Phi }_s\circ {\Phi }_t$ wherever defined. A compactly supported smooth vector field on a manifold without boundary is complete, so its flow exists for all real times.

A smooth family $Y_t$ assigns a tangent vector at each space-time point. Its flow maps solve ${\partial }_t{\rho }_t=Y_t\circ {\rho }_t$, with ${\rho }_0$ the identity. For a varying differential form, differentiation gives ${\partial }_t({\rho }_t^{\ast }{\alpha }_t)={\rho }_t^{\ast }({\dot{\alpha }}_t+{\mathcal{L}}_{Y_t}{\alpha }_t)$.

A smooth time-dependent vector field on a compact manifold without boundary has flow maps throughout each compact time interval on which the field is defined. Finite-time escape is impossible on the compact state space. Uniqueness and the backward-time equation supply smooth inverses, so the flow is a smooth isotopy.

Differentiate the displayed homogeneity relation in $\lambda$ at $1$. On every regular point of $H=0$, the radial vector field $x$ is therefore perpendicular to the normal gradient and tangent to the level surface. Its flow $x(t)=e^{t}x(0)$ preserves that zero set. At a singular vertex this flow formulation remains meaningful even if there is no tangent plane.

For $r=|x|>0$, the gradient satisfies ${\partial }_{i}r=x_i/r$. The product rule and Einstein notation then give the displayed divergence formula in dimension $n$. In dimension three, its derivative matrix is symmetric, so the curl vanishes. Zero divergence forces $f(r)=C{r}^{-n}$ on each positive radial interval. This formula does not assert regularity at the origin.

Let $V=\{ϵ<r<R\}$ with $ϵ>0$ and $\mathbf{F}=\hat{\mathbf{r}}/r^{\gamma }$. Direct differentiation gives $\nabla \cdot \mathbf{F}=(2-\gamma )r^{-\gamma -1}$. For $\gamma \ne 2$, integration against $4\pi r^{2}dr$ gives the displayed expression. Independently, the two boundary spheres contribute $4\pi R^{2-\gamma }$ and $-4\pi {ϵ}^{2-\gamma }$, since the outward normal of the shell at its inner boundary points toward the origin. For $\gamma =2$, the divergence is zero in the shell and the surface contributions cancel exactly. The excluded origin must not be silently added to this regular-domain calculation.

The vector field $\mathbf{G}=(-y,x,0)/(x^2+y^2)$ has zero curl away from its excluded axis, but has circulation $2\pi$ around a positively oriented circle enclosing that axis. Stokes theorem gives zero total circulation for a smooth annular surface avoiding the axis, whose inner and outer boundary circles have opposite orientations. It cannot be applied directly to a disk meeting the axis, because the field is undefined there.

An integral curve follows a vector field by solving the displayed ordinary differential equation. For a smooth vector field, integral curves through initial points assemble into its local flow.

For a continuously differentiable nonzero vector field $F$, the curvature of a space curve following its direction isTake the unit tangent vector $t=\pm F/|F|$ and differentiate with respect to arc length. The derivative of $|F{|}^{-1}$ is parallel to $F$ and disappears from $t\times t^{\prime }$. Since $t\cdot t^{\prime }=0$, we have $|t\times t^{\prime }|=|t^{\prime }|=\kappa$, proving the formula. It remains valid with zero curvature, even where the usual Frenet frame is undefined.

An axisymmetric vector field is invariant under rotations about a chosen axis. In cylindrical coordinates, its components are independent of the azimuthal angle.

A solenoidal vector field has zero divergence. Magnetic fields are solenoidal because $\nabla \cdot \mathbf{B}=0$.

For a solenoidal vector field $J$ with compact support and suitable integrability, apply the divergence theorem to $x_i{x}_{j}J$. It gives zero boundary term and hence the displayed antisymmetry. Equivalently, for any constant vector $C$, the vector triple product identity yields $\int (C\cdot r)JdV=\frac{1}{2}C\times \int (J\times r)dV$. A piecewise smooth field must have no normal jump at its support boundary; otherwise its weak divergence contains a surface contribution and the argument fails.

Suppose two oriented surfaces have the same oriented boundary and $S_1\cup (-S_2)$ bounds a volume inside a domain where the vector field $\mathbf{F}$ is smooth and solenoidal. The divergence theorem gives zero flux through their closed union, proving the displayed equality. If a global vector potential $\mathbf{F}=\nabla \times \mathbf{G}$ exists on the relevant surfaces, the Stokes theorem proves the equality directly from their common boundary. The domain condition matters: a singularity inside the joining volume can obstruct the divergence argument.

A spherical poloidal-toroidal decomposition represents a solenoidal vector field with zero flux through each sphere by scalar potentials $S,T$. The second term is tangent to concentric spheres; the first carries the radial component. Requiring zero angular means for the potentials fixes their otherwise irrelevant radial additions. In a plane layer, the analogous fluctuating components are $\nabla \times \nabla \times (\varphi \hat{\mathbf{z}})$ and $\nabla \times (\psi \hat{\mathbf{z}})$, with any horizontally uniform components handled separately. These constructions enforce the divergence-free condition because the divergence of a curl vanishes.

// Target: fluid-mechanics.bigb

For an isolated solenoidal magnetic field, zero flux through every sphere removes the degree-zero spherical harmonic of $P=\mathbf{B}\cdot \mathbf{x}$. In a poloidal-toroidal decomposition, let ${\mathcal{L}}^2=-{\Delta }_{S^2}$. The spherical poloidal component obeysIf $P={\sum }_{l\ge 1,m}{p}_{lm}(r)Y_{lm}$, put $D_{lm}={\int }_0^{\infty }[r^2|p_{lm}^{\prime }{|}^2+l(l+1)|p_{lm}{|}^2]dr$. The two sides of the bound are respectively $\sum D_{lm}/[l(l+1)]$ and $\frac{1}{2}\sum D_{lm}$. Since $l(l+1)\ge 2$, the bound follows, with equality for degree-one radial scalars. Together with the radial-flow dynamo energy bound it gives $(q/\eta {)}^2\ge 2E_P/E_{\mathrm{total}}$ for a marginal or growing dynamo field.

// Target: fluid-mechanics.bigb

A scalar function on Euclidean space is spherically symmetric when it depends only on the distance $r=|x|$ from a fixed center.

If a helicity density evolves as a sum of directional derivatives along divergence-free fields tangent to the boundary, its volume integral is conserved because both terms become vanishing boundary fluxes.

For $F=(F_1,F_2,F_3)$ in Cartesian coordinates,

For a continuously differentiable vector field, the divergence is the limiting outward flux per unit volume through a small closed surface. Opposite faces of a small rectangular box almost cancel; the remaining differences in the three components give the sum of partial derivatives above. The divergence theorem extends this local balance to a whole region.

For $F=(x,y,0)$, the divergence is $1+1+0=2$. In a planar unit square, the right edge contributes outward flux $1$ and the top contributes $1$, while the left and bottom contribute zero. The total $2$ equals the integral of the planar divergence over area $1$.

If $F$ is a velocity field, positive divergence means local volume expansion, not automatically creation of mass. The continuity equation gives $D\rho /Dt=-\rho \nabla \cdot F$, where $D/Dt$ is the material derivative. For this field, a uniform initial density evolves as $\rho (t)=\rho (0)e^{-2t}$ while material volumes grow as $e^{2t}$. Their product stays constant.

Horizontal flow convergence is the negative horizontal divergence of a velocity field. Positive convergence reduces the horizontal area of a material patch; for constant-depth linear shallow water equations, the surface equation is ${\eta }_t=HC$, so convergence raises the free surface and divergence lowers it. This is a fluid-flow notion, distinct from convergence of a numerical sequence.

The divergence defined through weak derivatives: pair a locally integrable vector field $J$ with a smooth compactly supported test function $\phi$ using the displayed integration-by-parts identity. A piecewise smooth field can have a surface contribution where its normal component jumps. Thus zero classical divergence on each side does not imply zero weak divergence across the interface.

The divergence with respect to velocity coordinates, with position held fixed. In $d$ dimensions, $\mathbf{A}(\mathbf{x},\mathbf{v})=\mathbf{B}(\mathbf{x})-a(\mathbf{x})\mathbf{v}$ has ${\nabla }_{\mathbf{v}}\cdot \mathbf{A}=-da$. A kinetic conservation law must retain this term for velocity-dependent forces.

For the Levi-Civita connection of a Riemannian metric, the divergence of $X$ is ${\mathrm{div}}_{g}X=\mathrm{tr}(Y\mapsto {\nabla }_{Y}X)$. In coordinates it equals $(\det g{)}^{-1/2}{\partial }_i((\det g{)}^{1/2}{X}^i)$. On an oriented manifold it satisfies ${\mathcal{L}}_X{\omega }_g=({\mathrm{div}}_{g}X){\omega }_g$. The codifferential on a one-form satisfies $\delta \theta =-{\mathrm{div}}_g({\theta }^{♯})$, where $♯$ is the musical isomorphism; this follows by integration by parts or the Hodge star formula.

For a vector field directed radially from a fixed axis and depending only on distance $\rho$ from that axis, cylindrical coordinates give $\nabla \cdot [f(\rho ){\mathbf{e}}_{\rho }]=f\prime +f/\rho$. Zero divergence gives $f=A/\rho$ on any positive radial interval. Unless $A=0$, this field is singular on the axis, so applications of the divergence theorem must use a domain avoiding it or account separately for that singularity.

For a locally integrable vector field, this formula defines its divergence as a distribution. For a piecewise smooth field, a jump of its normal component across an interface adds a surface measure to the smooth divergences. A continuous normal component eliminates that measure, as follows by integration by parts on the two sides.

The inverse-square radial vector field has zero ordinary divergence at every nonzero radius but flux $4\pi$ through every centred sphere. It is locally integrable in three dimensions. For a smooth compactly supported test function $\eta$, integrate by parts outside a ball of radius $ϵ$. The inward-oriented inner boundary gives $-\int \mathbf{F}\cdot \nabla \eta ={\int }_{S^2}\eta (ϵ\hat{\mathbf{r}})d\Omega$, which tends to $4\pi \eta (0)$. This proves the displayed distributional divergence, with the three-dimensional Dirac delta function representing the point source.

For $\mathbf{u}=u_r{\mathbf{e}}_r+u_{\theta }{\mathbf{e}}_{\theta }$,

For a scalar field $\varphi$ and vector field $F$,

A total divergence has the form $\nabla \cdot \mathbf{F}$. The divergence theorem converts its volume integral into a boundary flux, which vanishes when the field has suitable decay or boundary conditions.

Adding $dF(q,t)/dt$ to a mechanical Lagrangian changes its action by $F(q_f,t_f)-F(q_i,t_i)$. Fixed endpoint variations therefore leave its bulk Euler-Lagrange equations unchanged. For fields, adding ${\partial }_{\mu }{K}^{\mu }$ changes the action by a boundary flux. The same conclusion holds for compactly supported variations or appropriate fixed boundary data. If $K$ depends on field derivatives, those boundary data must also control their variations. Such terms can still change canonical momenta, boundary observables, and representatives of Noether currents; equivalence of bulk equations does not mean that every boundary-value problem is identical.

For a twice continuously differentiable vector field, commuting partial derivatives and antisymmetry of the Levi-Civita symbol give

The curl of $F$ has components

For a continuously differentiable vector field in three-dimensional Euclidean space, the component of its curl along a unit normal vector is the limiting circulation per unit area around a small loop in the perpendicular plane. The loop direction and the normal obey the right-hand rule. Stokes theorem gives the corresponding relation for a finite surface.

For the rigidly rotating velocity field $G=(-\Omega y,\Omega x,0)$, direct differentiation gives $\nabla \times G=(0,0,2\Omega )$. A circle of radius $R$ has tangential speed $\Omega R$ and circumference $2\pi R$, so its circulation is $2\pi \Omega R^2$, twice the angular velocity times the enclosed area. The factor of two distinguishes vorticity from rigid-body angular velocity.

Curved integral curves alone do not demonstrate nonzero curl. The field $H=(x,-y,0)$ has zero curl, yet its trajectories satisfy $x=x_0{e}^t$, $y=y_0{e}^{-t}$ and hence $xy=x_0{y}_0$: away from the axes they are hyperbolas. The curl measures local circulation of the field, not the curvature of an individual trajectory.

The divergence theorem applied to $G\times c$ for every constant vector $c$ gives ${\int }_V\nabla \times GdV={\int }_{\partial V}n\times GdS$, with outward unit normal $n$. Indeed $\nabla \cdot (G\times c)=c\cdot (\nabla \times G)$, while $(G\times c)\cdot n=c\cdot (n\times G)$. Equality of the scalar projections for every $c$ proves the vector identity. Reversing the surface cross product reverses the sign.

For a continuously differentiable scalar field $\varphi$ and vector field $\mathbf{F}$, the curl satisfiesIndeed, with fixed Cartesian basis vectors ${\mathbf{e}}_i$, the Cartesian expression for the curl and the product rule giveThe first sum is the cross product of the gradient of $\varphi$ with $\mathbf{F}$; the second is $\varphi$ times the curl of $\mathbf{F}$. The formula is useful when applying the Stokes theorem to a weighted vector field.

For a twice continuously differentiable function, the curl of its gradient vanishes. In components, ${ϵ}_{ijk}{\partial }_j{\partial }_{k}f=0$ because the second derivatives are symmetric in $j,k$ while the Levi-Civita symbol is antisymmetric. Thus adding a gradient to a vector potential preserves the represented curl. No topological assumption is needed for this identity, unlike the converse assertion that every curl-free vector field has a global potential.

A continuously differentiable vector field on an open subset of three-dimensional Euclidean space is irrotational when its curl is zero. Under the Euclidean identification $\alpha =X_{i}d{x}^i$, this says that the differential one-form $\alpha$ is a closed differential form. The Poincaré lemma gives a potential of a conservative vector field locally. A global potential additionally requires zero line integrals around every closed curve; simply connected domains ensures this. On the punctured plane extended in the third direction, $X=(-y,x,0)/(x^2+y^2)$ has zero curl but integral $2\pi$ around a circle, so it is locally conservative and has no single-valued global potential.

A vector potential for a divergence-free vector field $F$ is a vector field $A$ satisfying $F=\nabla \times A$. It is not unique because adding a gradient leaves its curl unchanged.

Let $D$ be bounded, connected and simply connected, and let $u$ be a smooth divergence-free field. Suppose a sufficiently smooth vector field $A$ satisfies the displayed Poisson and boundary conditions. Divergence gives $\Delta \mathrm{div}A=0$ with zero boundary value, so harmonic uniqueness gives $\mathrm{div}A=0$. The vector Laplacian identity then implies $\nabla \times (\nabla \times A-u)=0$.

On a simply connected domain, define a scalar potential by integrating this curl-free difference along a path from a fixed point. Vanishing circulation around contractible loops proves path independence. Thus the difference is $\nabla \chi$; its divergence and boundary normal component give $\Delta \chi =0$ and ${\partial }_n\chi =0$. Green's identity yields ${\int }_D|\nabla \chi {|}^2=0$, proving the reconstruction. Additional conditions on $A$ can specify its own boundary problem; the argument assumes its existence and proves that its curl is exactly $u$. Simple connectivity is essential for eliminating possible non-gradient harmonic circulation fields.

Let $B$ be a continuously differentiable solenoidal vector field on an open star-shaped set containing the origin. Thenis a vector potential for $B$. For $D={\int }_0^{1}tB(tx)dt$, differentiation gives $\nabla \cdot D=0$ and $\nabla \times (D\times x)=2D+(x\cdot \nabla )D={\int }_0^1{\partial }_t(t^{2}B(tx))dt=B(x)$. Boundedness near the origin removes the lower endpoint. A field defined only on a punctured domain need not satisfy these hypotheses.

For a twice differentiable vector field $A$,

The Laplacian takes the divergence of a scalar field's gradient. For a real-valued function $f$ on an open region of Euclidean space, with continuous partial derivatives through order two, its expression in a Cartesian coordinate system isThus the Laplacian, also called the Laplace operator, adds the coordinate second derivatives; equivalently it is the matrix trace of the Hessian matrix. Its output is another scalar field.^[ref]

Three original examples distinguish shape from this sum. The bowl $f(x,y)=x^2+y^2$ has $\Delta f=2+2=4$. The saddle $g(x,y)=x^2-y^2$ instead has $\Delta g=2-2=0$: opposite second derivatives cancel. It is a harmonic function, although its graph of a function is not a plane. For the Gaussian function $h(x,y)=e^{-r^2}$, with $r^2=x^2+y^2$, differentiation givesIts Laplacian is negative inside the circle of radius one, zero on that circle, and positive outside it. The Gaussian function itself is positive throughout: its value and its Laplacian describe different properties.

The sum does not favor a particular orientation. Under a change of Cartesian coordinate system represented by an orthogonal matrix $Q$, the Hessian matrix becomes $Q^{\mathsf{T}}(D^{2}f)Q$. The cyclic property of the trace and $Q{Q}^{\mathsf{T}}=I$ preserve its matrix trace. This makes the Laplacian useful when the behavior depends on spatial differences equally in every direction, rather than on the arbitrary choice of axes.

For a vector field, the Euclidean Laplacian acts separately on components expressed in a fixed Cartesian coordinate system. Components in a position-dependent basis require differentiating that basis too; applying a scalar formula separately to polar components omits those terms.^[ref] The scalar formula in Laplacian in polar coordinates is valid away from $r=0$; smoothness at the origin must be handled in coordinates that remain regular there.

For a constant orthogonal matrix $R$, the Laplacian is unchanged by the coordinate transformation $\xi =Rx$. If $u$ is twice continuously differentiable, the chain rule givesbecause ${\sum }_i{R}_{ji}{R}_{ki}={\delta }_{jk}$ by orthogonality of the rows. Here ${\delta }_{jk}$ is the Kronecker delta. In two dimensions, the rotation matrix $\left(\begin{array}{cc}a& -b\\ b& a\end{array}\right)$ with $a^2+b^2=1$ makes the mixed partial derivatives cancel explicitly. The same argument works for reflections and translations, so the Laplacian depends on the Euclidean geometry rather than the orientation or origin of the coordinates.

The Laplacian detects whether a value lies below or above nearby values, to second order. For a scalar field whose partial derivatives through order four are continuous functions, take the arithmetic mean of four equally spaced neighbors:Applying Taylor's theorem in each coordinate cancels the first- and third-order terms between opposite neighbors. The two pairs leaveThis is an asymptotic expansion, rather than an equality for arbitrary $h$. Rearranging gives the familiar central finite difference approximationThis finite difference replaces partial derivatives by five sampled values. The arithmetic mean lies above the center for a sufficiently small $h$ if the Laplacian is strictly positive there, and below if it is strictly negative. A zero Laplacian cancels this second-order discrepancy without requiring equality of individual values.

At $(0,0)$, the bowl $x^2+y^2$ has center value zero and four neighbor values $h^2$, so the discrepancy is exactly $h^2$. The saddle $x^2-y^2$ has neighbor values $h^2,h^2,-h^2,-h^2$, whose arithmetic mean is exactly zero. For the Gaussian function $e^{-x^2-y^2}$ the discrepancy is $e^{-h^2}-1=-h^2+O(h^4)$, agreeing with $\Delta f(0,0)=-4$.

A harmonic function need not equal this four-neighbor arithmetic mean for a finite $h$. The polynomial $f(x,y)=x^4-6x^2{y}^2+y^4$ has $\Delta f=0$ everywhere, but $A_{h}f(0,0)=h^4$. Higher-order terms remain even when the second-order terms cancel.

At a local minimum, each coordinate second derivative is at least zero, hence the Laplacian is at least zero. At a local maximum the inequality reverses. The sum alone does not characterize an extremum: cancellation at a saddle point of a scalar function can also produce zero. The local-average interpretation explains why the heat equation smooths differences, and helps motivate the heat equation maximum principle.

For diffusion with a constant diffusion coefficient $D>0$ and no sources, the model flux is $J=-D\nabla u$. A continuity equation gives $u_t=-\nabla \cdot J=D\Delta u$. This proportionality is a constitutive assumption about transport; the continuity equation expresses conservation. If $D$ varies with position, it must remain inside the divergence, giving $u_t=\nabla \cdot (D\nabla u)$ instead.^[ref]

An exact constructed heat equation example on the whole plane uses dimensionless coordinates with $D=1$ and the smooth initial Gaussian function $u(x,y,0)=e^{-r^2}$. Set $s=1+4t$. For $t\ge 0$,Direct differentiation verifies both the initial condition andThe center decreases, while the region $r^2>s$ increases at that instant. The sign-change circle expands with radius $\sqrt{s}$. A fixed outer location can initially increase and later decrease as this circle passes it.

At time $t$, the peak is $1/s$, and the radius where the profile is $1/e$ times its peak is $\sqrt{s}$. The width grows as the height falls. Integration in polar coordinates givesThus a decreasing peak is compatible with a conserved spatial integral. Dividing $u$ by $\pi$ makes a probability density function with coordinate variances $s/2=1/2+2t$, connecting the same spreading to Brownian motion with diffusion coefficient one. Equivalently the profile is evolved by the Gaussian heat kernel with time parameter $2t$, since that article uses generator $\frac{1}{2}\Delta$.

For a twice differentiable scalar field on the plane,

For $x=r\cos \theta$, $y=r\sin \theta$ and $r>0$, set $c=\cos \theta$, $s=\sin \theta$. The differential operators are ${\partial }_x=c{\partial }_r-(s/r){\partial }_{\theta }$ and ${\partial }_y=s{\partial }_r+(c/r){\partial }_{\theta }$. Applying them twice givesTheir sum is the Laplacian in polar coordinates. The first-derivative terms come from differentiating the coordinate coefficients, as required for second derivatives under a two-variable coordinate change. The origin is excluded from the coordinate formulas, even when the underlying Cartesian function is smooth there.

For twice differentiable scalar fields,

The biharmonic operator applies the Laplacian twice. Under clamped boundary conditions, two integrations by parts give the positive quadratic form

The clamped problem uses the weak solution identity $\int \Delta u\Delta v=\int fv$ for every $v$ in the clamped second-order Sobolev space. For $f\in L^2(U)$, the clamped Hessian identity and the Poincaré inequality make this a bounded coercive bilinear form. The Lax-Milgram theorem gives a unique weak solution without a mean-zero condition on $f$.

The biharmonic equation is ${\nabla }^{4}f=0$, where the biharmonic operator is the square of the Laplacian. In planar Stokes flow, taking the curl of the momentum equation and using a Cartesian streamfunction produces this equation. Its decaying spatial Fourier series modes have form $(A+By)e^{-|k|y}{e}^{ikx}$.

For angular Fourier modes of integer order $n\ge 2$, the biharmonic equation in polar coordinates has radial basis $r^n,r^{-n},r^{n+2},r^{2-n}$. Apply the Laplacian twice to obtain the displayed indicial equation. For $n=0$, repeated roots give the radial basis $1,\log r,r^2,r^2\log r$; for $n=1$, the basis is $r^{-1},r,r\log r,r^3$. These are Cauchy-Euler equations. Regularity at the origin and boundary conditions select the admissible powers. For $n=2$, a constant radial factor still multiplies $\cos 2\theta$ or $\sin 2\theta$, so it is bounded but fails continuity at the origin.

If $H,G$ are holomorphic functions, both real and imaginary parts of $\bar{z}H(z)+G(z)$ solve the biharmonic equation. In Wirtinger derivatives, $\Delta =4{\partial }_z{\partial }_{\bar{z}}$, so $\Delta (\bar{z}H+G)=4H^{\prime }$ is harmonic and its next Laplacian is zero. This construction gives local representations of planar solutions of the biharmonic equation; it is useful for the biharmonic stream function for planar Stokes flow.

For a smooth three-dimensional biharmonic function regular throughout a ball, its surface average has exactly this form. Taylor expansion and rotationally invariant spherical moments turn each even term into a multiple of an iterated Laplacian; terms from the fourth derivative onward vanish when ${\Delta }^{2}f=0$. This is the mean-value identity that converts the reciprocal surface average into Faxén's first law.

For smooth vector fields $F,G$,andBoth follow by contracting two Levi-Civita symbols and applying the product rule.