Vector fields, divergence and curl

A vector field attaches a vector to each allowed input

A vector field F:U⊂Rⁿ→Rⁿ assigns a vector to every point of its domain U. In Cartesian three-dimensional space write F(x,y,z)=(P(x,y,z),Q(x,y,z),R(x,y,z)). A scalar temperature field has one value at each input; a velocity field has a speed and, when nonzero, a direction there. The domain is part of the definition, especially if a formula has a zero denominator or a physical boundary.

An arrow plot samples field values. Its arrows are attached at input points; the line segments used to draw them are not automatically particle paths. A trajectory r(t) of a steady velocity field v satisfies r′(t)=v(r(t)). The path follows the field as it moves, so a sequence of arrows at unrelated sample points is not itself a solved trajectory.

For v(x,y)=(−y,x), the point (1,0) has velocity (0,1). A particle starting there follows r(t)=(cos t,sin t), whose acceleration is r″(t)=−r(t). The velocity arrow points tangentially while the acceleration points inward. For a time-dependent smooth velocity v(x,t), the chain rule gives particle acceleration ∂v/∂t+(Dv)v along a trajectory; v itself is not acceleration.

Preparation is vectors and dot/cross products 121, Jacobians 131, and multiple integrals 133. The coordinate examples below are dimensionless unless physical units are explicitly restored. Derivative formulas assume open domains and the stated regularity; a drawing cannot supply those hypotheses.

The Jacobian describes how neighbouring arrows differ

If F is differentiable at a, then F(a+h)=F(a)+DF(a)h+o(‖h‖). The constant vector F(a) translates a small group of particles together. The matrix DF(a) describes their first-order relative motion. Equal arrows can move material across a region while producing neither local expansion nor rotation.

Write A=DF(a). The decomposition A=S+W, where S=(A+Aᵀ)/2 and W=(A−Aᵀ)/2, separates a symmetric part from an antisymmetric part. In Euclidean Cartesian coordinates, h·Wh=0, so the antisymmetric part changes the direction of a separation without instantaneously changing its squared length. The symmetric part gives d‖h‖²/dt=2hᵀSh for the linear relative-motion equation h′=Ah.

The symmetric part still contains both average expansion and shape change. In n dimensions, S=(tr A/n)I+[S−(tr A/n)I]. The second bracket has trace zero; it can stretch one direction and compress another while preserving volume to first order. Zero divergence will therefore not mean that every distance or every shape stays unchanged.

For A=[[a,−ω,0],[ω,b,0],[0,0,c]], expansion is controlled by a+b+c, while the planar antisymmetric part rotates with angular speed ω. These are independent parameters. Only for this rigid rotation component is a single angular velocity sufficient to describe all local relative motion; a general field can also shear and strain.

Divergence adds the matching component derivatives

For a C¹ field F=(P,Q,R) on an open subset of R³, div F=Pₓ+Qᵧ+R_z. In two dimensions, div(P,Q)=Pₓ+Qᵧ. More generally, div F=tr DF. The result is a scalar function. The subscript names a coordinate derivative, not a component label to be chosen by visual similarity.

For F=(xy,yz,zx), the three matching derivatives are y,z,x, so div F=x+y+z. At (1,2,−1), divergence is 2. Differentiating all components with respect to x would instead give (y,0,z), a different object: one column of the Jacobian.

The notation ∇·F is a useful operator notation for this sum. It is not an ordinary dot product of two numerical vectors: ∇ contains differentiation operations. A gradient takes a scalar to a vector, divergence takes a vector field to a scalar, and those different input/output types prevent many invalid expressions.

If F is a velocity measured in metres per second and the spatial coordinates are in metres, div F has units s⁻¹. If F is mass flux density in kg m⁻² s⁻¹, its divergence has units kg m⁻³ s⁻¹. Changing the physical meaning of F changes the meaning and units of the result even though the coordinate formula is the same.

Opposite faces reveal local net outward flux

Consider a small Cartesian box centred at a, with side lengths Δx,Δy,Δz. Positive x-outward flow through the right face and negative x-outward flow through the left face give a net leading contribution [P(a+(Δx/2)eₓ)−P(a−(Δx/2)eₓ)] Δy Δz. For C¹ P this is Pₓ(a) Δx Δy Δz plus a remainder small compared with the volume, after accounting for variation over the faces.

The y and z face pairs similarly contribute Qᵧ(a) and R_z(a) times the volume. Thus the net outward flux divided by box volume tends to div F(a) as the box shrinks with a fixed shape. This is a local consequence of the first-order expansion and face integration. The later divergence theorem 136 gives the corresponding integral statement for finite regular volumes.

For v=(x,y,z), the right and left faces of [−h,h]³ each have outward normal velocity h. All six face contributions are 4h³, so net flux is 24h³. The volume is 8h³, giving flux per volume 3, exactly div v. The field vanishes at the centre while its divergence there is positive: the centre’s arrow alone does not determine expansion.

For a smooth velocity with a differentiable local flow Φₜ, let J(t,a)=det DₐΦₜ(a) be its positive local volume scale from an initial point a. The volume-element identity is ∂ₜJ=div v(Φₜ(a),t) J, hence ∂ₜ log J=div v along the flow. Indeed, the flow derivative M satisfies M′=(Dv)M, and the determinant derivative gives J′=J tr(M⁻¹(Dv)M)=J div v. A finite material region instead has relative volume-change rate equal to the spatial average of div v over that moving region; it reduces to a pointwise value when divergence is uniform there. The constant translation v=(1,2,3) has zero divergence despite carrying material through any fixed box.

Curl measures signed local circulation density

For a C¹ Cartesian field F=(P,Q,R), curl F=(Rᵧ−Q_z, P_z−Rₓ, Qₓ−Pᵧ). The result is a vector field in R³. Its component along a chosen oriented axis measures local circulation around a small loop in the corresponding perpendicular plane, divided by enclosed area. Reversing the axis reverses the positive loop orientation by the right-hand rule.

For a planar field F=(P(x,y),Q(x,y),0) with no z dependence, curl F=(0,0,Qₓ−Pᵧ). The scalar Qₓ−Pᵧ is the out-of-plane curl, positive for counterclockwise circulation viewed from the positive z-axis. Calling it a scalar uses this fixed embedding and orientation; it is not the full three-dimensional vector formula with terms omitted arbitrarily.

To see the sign, traverse a tiny xy rectangle counterclockwise: bottom rightward, right upward, top leftward, left downward. The horizontal pair contributes approximately −Pᵧ Δx Δy, and the vertical pair contributes Qₓ Δx Δy. Their sum divided by area tends to Qₓ−Pᵧ. This local calculation precedes the full Green and Stokes theorems 136.

For F=(xy,yz,zx), curl F=(−y,−z,−x), so at (1,2,−1) it is (−2,1,−1). Use the displayed component formula or expand the operator mnemonic carefully. A determinant-like arrangement containing ∂/∂x, ∂/∂y and ∂/∂z is a memory aid, not a numerical determinant to which arbitrary determinant manipulations apply.

A rotating field can have zero divergence

For rigid planar rotation v=(−ωy,ωx,0), divergence is zero while curl is (0,0,2ω). On a circle of radius r traversed counterclockwise, v has tangential component ωr and length element r dθ. The circulation is 2πωr², so circulation divided by the disk area πr² is 2ω. The curl is twice the rigid angular velocity vector, not the angular velocity itself.

For v=(x,−y,0), both divergence and curl are zero. Yet neighbours separate along x and approach along y. The exact planar flow is (x(t),y(t))=(eᵗx₀,e⁻ᵗy₀), whose area scale is eᵗe⁻ᵗ=1. A square becomes a rectangle of the same area. This provides a concrete strain field with neither local area expansion nor rigid rotation.

For the shear v=(ky,0,0), divergence is again zero but curl is (0,0,−k). The lower side of a small horizontal pair moves differently from the upper side. The antisymmetric part has angular velocity −k/2, while the symmetric part also deforms the group. Curl therefore describes the rotational part of local motion rather than proving every particle follows a circle.

An equilibrium F(a)=0, zero divergence, zero curl and constancy are four different conditions. They can coincide in a special example but none should be inferred from a sparse arrow plot. The examples explicitly evaluate derivatives across the domain; the diagram samples only those proven formulas.

A scalar potential gives a field by differentiation

A field F is conservative on U when there is a differentiable scalar φ on U with F=∇φ. With a smooth potential, work along any piecewise C¹ path in U depends only on its endpoints: the chain rule gives dφ(r(t))/dt=∇φ(r(t))·r′(t). Lesson 135 develops that integral formula and its path hypotheses. Some physical conventions write force as −∇V; this changes which scalar is called the potential, not the definition F=∇φ used here.

Take φ(x,y,z)=x²y+y³+xz. Then ∇φ=(2xy+z,x²+3y²,x). Its divergence is 2y+6y+0=8y and its curl is zero. A conservative field can have nonzero divergence: being a gradient does not imply incompressibility or harmonicity.

Conversely, suppose this field is given and a potential is sought. Integrating P=2xy+z with respect to x gives φ=x²y+xz+g(y,z). Matching φᵧ=x²+3y² gives g=y³+h(z). Matching φ_z=x gives h′=0. Thus φ=x²y+y³+xz+C. The “constant” after a partial integration initially depends on all other coordinates.

The arbitrary additive constant changes no gradient. On a connected domain, any two differentiable potentials for the same field differ by a constant; on separate connected components, the constants can be chosen independently. To justify constancy on a connected open domain, integrate the zero gradient along piecewise smooth connecting paths, rather than presuming a straight segment stays inside every domain.

Zero curl needs a domain condition for a global potential

If φ is C², then curl(∇φ)=0 because the paired mixed partial derivatives agree. Thus a C¹ field admitting a C² scalar potential must be curl-free. On an open simply connected domain in R³, a C¹ curl-free field has a global potential. In the plane, the corresponding condition is Qₓ−Pᵧ=0 on an open simply connected domain. Simply connected means path connected and every closed loop can be continuously contracted within the domain.

A useful constructive sufficient setting is a star-shaped open domain U with centre a: every segment from a to x stays in U. If F is C¹ and has symmetric derivative entries ∂Fᵢ/∂xⱼ=∂Fⱼ/∂xᵢ, define φ(x)=∫₀¹ F(a+t(x−a))·(x−a) dt. Differentiating in xⱼ changes the integrand into d[t Fⱼ(a+t(x−a))]/dt by that symmetry. Integrating in t gives ∂φ/∂xⱼ=Fⱼ(x), proving the potential formula.

The proof uses the C¹ field and segments inside an open domain so differentiation under this finite parameter integral is valid locally. It supplies more than a numerical curl sample. In R³, symmetry of DF is equivalent to curl F=0; in the plane it is the single cross-partial equality.

Simply connected is a sufficient condition for every field satisfying the local curl condition, not a necessary condition for a particular field to have a potential. The constant field (1,0) has potential x even on the punctured plane. When a domain has a hole, the local derivative test alone is inconclusive; it does not automatically disprove a potential either.

A punctured-plane field separates local curl from global circulation

Let U=R²∖{(0,0)} and V(x,y)=(−y/(x²+y²),x/(x²+y²)). The field is smooth on U. Direct differentiation gives Pₓ=2xy/(x²+y²)² and Qᵧ=−2xy/(x²+y²)², so div V=0. Also Qₓ=(y²−x²)/(x²+y²)²=Pᵧ, so its planar curl is zero at every allowed point.

Around the counterclockwise unit circle r(t)=(cos t,sin t), 0≤t≤2π, the dot product V(r(t))·r′(t) is 1. Its circulation is therefore 2π. A global single-valued potential would make every closed-path work integral zero, so no such potential exists on all of U. The domain omits the very point needed to contract this loop.

Locally, a continuous choice of polar angle is a potential. On a plane cut along a ray from the origin, choose one smooth angle branch θ; then ∇θ=V there. Extending that angle around the hole changes it by 2π. Treating the multivalued angle as an ordinary globally defined scalar would erase the obstruction rather than solve it.

The circulation can be nonzero around a loop even though pointwise curl vanishes along the entire loop and everywhere else in U. Applying a finite-region curl theorem to the disk would fail because V is undefined at its centre. This is a failure of hypotheses, not a contradiction of mixed partial symmetry or of the later integral theorems.

Zero divergence away from a missing point is not a global source statement

The planar radial field S(x,y)=(x/(x²+y²),y/(x²+y²)) is smooth away from the origin. Its divergence is zero there: the matching derivatives are (y²−x²)/(x²+y²)² and (x²−y²)/(x²+y²)². Yet on any circle of radius r>0, its outward normal component is 1/r; multiplying by the arclength element r dθ gives outward flux 2π.

The origin is outside the domain, so pointwise div S=0 on U does not establish zero flux through a curve enclosing that excluded point. Conversely, on an annulus ε≤r≤R, outward flux through the outer circle is 2π while outward flux through its inner boundary is −2π. Both boundaries together give zero, consistently with the nonsingular annular region.

In three dimensions, the inverse-square radial field x/‖x‖³ is also divergence-free away from the origin but has flux 4π through an outward sphere. These are dimension-specific formulas: x/‖x‖² is the planar flux example, while x/‖x‖³ is the spatial one. The singularity cannot be repaired merely by assigning an arbitrary value at the missing point.

A distributional account can represent a concentrated point source, but that is a broader notion of derivative than the ordinary C¹ formulas used here. State the ordinary domain and regularity first. The phrase “source-free” should identify whether it refers to pointwise divergence on a specified domain or to a global physical model including singular sources.

Smooth identities follow by cancellation

For a C² scalar φ, curl grad φ=0 follows component by component: the first entry is φ_zy−φ_yz, and the others cancel similarly. For a C² field A=(P,Q,R), div curl A=(R_yx−Q_zx)+(P_zy−R_xy)+(Q_xz−P_yz)=0. The sufficient C² assumptions make the relevant mixed derivatives equal. Writing the symbols without regularity does not prove the cancellation for a nonsmooth field.

The scalar Laplacian is Δφ=div grad φ=φₓₓ+φᵧᵧ+φ_zz. It need not vanish. For φ=x²y+y³+xz, Δφ=8y. A C² potential satisfying Δφ=0 is harmonic; its gradient is both curl-free and divergence-free on the same domain. The function (x²−y²)/2 is harmonic, giving the earlier planar strain field (x,−y).

A useful C² Cartesian vector identity is curl curl A=grad div A−ΔA, with Δ applied componentwise. For its first component, ∂y(Qₓ−Pᵧ)−∂z(P_z−Rₓ)=∂x(Qᵧ+R_z)−Pᵧᵧ−P_zz=∂x(div A)−ΔP. Mixed partial symmetry proves the equality; the other entries follow by the same calculation.

These identities constrain possible representations. A C² vector potential A with F=curl A forces div F=0. This implication alone does not prove that every divergence-free field has a global vector potential on every domain. Global existence depends on domain geometry and is not established by a few algebraic cancellations.

Mass conservation concerns a flux density, not velocity alone

For a smooth mass density ρ(x,t) and velocity v(x,t), the mass flux density is j=ρv. With a volumetric mass source s, the local conservation equation is ∂ρ/∂t+div(ρv)=s. In the source-free case it says that a fixed region’s mass change is balanced by net outward mass flux. Its finite-volume derivation is completed with the divergence theorem 136; here it fixes the meaning and units of the fields.

The product rule gives div(ρv)=v·grad ρ+ρ div v. Thus, with s=0, the material density change is Dρ/Dt=∂ρ/∂t+v·grad ρ=−ρ div v. A positive density that stays constant along particles yields div v=0 in this smooth source-free setting. A time-independent density alone does not remove the v·grad ρ term.

For the dimensionless planar expansion v=(x,y), div v=2. The spatially uniform density ρ(t)=e⁻²ᵗ satisfies ∂ρ/∂t=−2ρ and div(ρv)=2ρ, so the source-free equation balances exactly. Particle areas expand by e²ᵗ while density falls by the reciprocal factor. A physical version restores a rate κ: v=κ(x,y), ρ=ρ₀e⁻²κᵗ, with κ measured in s⁻¹.

In three-dimensional units, ρ is kg m⁻³, v is m s⁻¹, ρv is kg m⁻² s⁻¹, and both ∂ρ/∂t and div(ρv) are kg m⁻³ s⁻¹. A rescaled numerical coordinate is not a metre unless its length scale has been supplied. Later curvilinear-coordinate formulas also need their metric/scale factors; the Cartesian expressions here should not be copied into polar components by renaming x as r.

Check the field, its derivatives and its domain together

Begin by naming U and the regularity available on it. Then state whether the input is a scalar or vector field and what output type is expected. Compute the necessary partial derivatives, compare units, and use an exact limiting or analytic argument for any identity claimed throughout the domain.

For a potential claim, check the local mixed-partial condition and then examine domain geometry or closed-loop work. For a source claim, distinguish divergence at ordinary points from flux around excluded singularities. For a flow claim, separate translation, symmetric strain and the antisymmetric rotation component. These checks prevent one appealing picture from being treated as several different theorems.

Finite-difference stencils and sampled arrows can reveal sign errors or illustrate local limits. They cannot prove smoothness, global zero curl, global potential existence or a singularity’s absence. Any plotted arrow normalisation must be labelled: equal displayed lengths can deliberately suppress actual speed variation.

The next lesson 135 turns curve and surface geometry into four distinct integrals: scalar arclength accumulation, oriented work, scalar surface accumulation and oriented flux. Lesson 136 then relates their finite-region values to divergence and curl under explicit boundary, orientation and smoothness hypotheses.

Worked examples

Try it yourself

1. Is (−y/(x²+y²),x/(x²+y²)) a smooth field on all of R²?

   Show the answer and reasoning

   No; it is smooth on the punctured plane.

   The denominator vanishes at the origin; that point cannot be silently included in derivative or potential claims.

2. Does the segment used to draw v(a) automatically give the particle trajectory through a?

   Show the answer and reasoning

   No.

   A trajectory solves r′=v(r,t); the field value supplies only its instantaneous velocity at that input and time.

3. Find div(1,2,3) and curl(1,2,3).

   Show the answer and reasoning

   Both are zero.

   All spatial component derivatives vanish; the nonzero field still translates particles.

4. Find div(x²,y²,z²).

   Show the answer and reasoning

   2x+2y+2z.

   Take the matching partial derivative of each component and add.

5. Find the out-of-plane curl of (−2y,2x).

   Show the answer and reasoning

   4.

   Qₓ−Pᵧ=2−(−2)=4, positive for the stated counterclockwise orientation.

6. For v=(5y,0,0), what is the curl and local rigid rotational angular velocity vector?

   Show the answer and reasoning

   curl v=(0,0,−5); the angular velocity of the antisymmetric part is (0,0,−5/2).

   Curl is twice that rigid rotational component; shear also contains symmetric strain.

7. Does div v=0 force a square to keep its shape under a smooth flow?

   Show the answer and reasoning

   No.

   v=(x,−y) has zero divergence but turns squares into rectangles of the same area.

8. At the origin, v=(x,y,z) is zero. What is its divergence there?

   Show the answer and reasoning

   3.

   Divergence measures neighbouring variation, not the magnitude of the single value v(0).

9. Does F=grad φ imply div F=0?

   Show the answer and reasoning

   No.

   Its divergence is Δφ. For φ=x²+y²+z² the result is 6.

10. After integrating P=2xy+z with respect to x, may the integration term depend on y and z?

    Show the answer and reasoning

    Yes: φ=x²y+xz+g(y,z).

    Partial differentiation in x eliminates every function independent of x; later components determine g.

11. Does zero planar curl on the punctured plane guarantee a global scalar potential?

    Show the answer and reasoning

    No.

    The angle-gradient vortex has zero curl on its domain but closed-circle work 2π.

12. Can a field on a domain with a hole still have a potential?

    Show the answer and reasoning

    Yes.

    F=(1,0)=grad x is conservative on the punctured plane; the topology condition is a sufficient general test, not a necessary condition for each particular field.

13. What is the outward flux of (x/r²,y/r²) through a circle of radius 3?

    Show the answer and reasoning

    2π.

    Normal field magnitude 1/3 multiplied by circumference 6π gives 2π.

14. Which radial direction is outward at the inner boundary of an annulus?

    Show the answer and reasoning

    Toward the excluded central disk.

    Outward means away from the annular material; it is the negative radial direction there.

15. Give sufficient ordinary-derivative regularity for div curl A=0.

    Show the answer and reasoning

    A is C² on the open domain being considered.

    Continuous second derivatives guarantee equality of the paired mixed derivatives used in cancellation.

16. For φ=x²y+y³+xz, compute Δφ at y=−1.

    Show the answer and reasoning

    −8.

    Δφ=2y+6y+0=8y.

17. If density is kg m⁻³ and velocity m s⁻¹, what are the units of div(ρv)?

    Show the answer and reasoning

    kg m⁻³ s⁻¹.

    ρv is kg m⁻² s⁻¹; one spatial derivative adds a factor m⁻¹.

18. For time-independent nonuniform ρ, may one replace div(ρv) by ρ div v without further conditions?

    Show the answer and reasoning

    No.

    The product rule also contains v·grad ρ, which need not vanish.

What to carry forward

• A field assigns vectors at allowed points; a trajectory follows a differential equation through those points.
• The local affine approximation separates common translation F(a) from the Jacobian’s symmetric strain and antisymmetric rotation. Divergence is tr DF.
• Cartesian divergence is a scalar local flux density; Cartesian curl is an oriented vector local circulation density.
• Zero divergence, zero curl, zero field value and rigid motion are different claims.
• A smooth gradient has zero curl, while its divergence is the Laplacian and can be nonzero.
• Curl-free fields have local potentials; global potentials require appropriate domain or path information.
• Excluded singularities invalidate arguments that fill in the missing region. Pointwise zero divergence on a punctured domain does not eliminate singular flux.
• Smooth vector identities are consequences of equal mixed partial derivatives, with explicit regularity and coordinate assumptions.
• Mass conservation uses the flux density ρv and the full product rule; physical units and source assumptions matter.
• Sampled arrows and finite differences illustrate formulas but do not certify global topology or smoothness.