Identities: div(curl) and curl(grad)

One-line statement: The divergence of any curl is zero,

\nabla \cdot (\nabla \times F) = 0

, and the curl of any gradient is zero,

\nabla \times (\nabla ϕ) = 0

, for sufficiently smooth fields. Key result in words: Curls are always divergence-free (solenoidal); gradients are always curl-free (irrotational). These two identities are exact algebraic consequences of the equality of mixed partial derivatives. Where it applies: Vector calculus on

R^{3}

(and beyond via differential forms), electromagnetism, fluid dynamics, the Helmholtz decomposition, and the Poincaré lemma. Who/when: Implicit in the work of Gauss, Green, and Stokes (early–mid 19th century); crystallized in Maxwell's electromagnetic theory (1860s) and the vector-analysis notation of Gibbs and Heaviside (1880s).

The two "second-derivative" vector identities state that applying divergence to a curl, or curl to a gradient, annihilates the field: $\nabla \cdot (\nabla \times F) = 0$ and $\nabla \times (\nabla ϕ) = 0$ . Each holds whenever the relevant second partial derivatives exist and are continuous (class $C^{2}$ ), because both reduce to the symmetry of mixed partial derivatives (Clairaut's/Schwarz's theorem). They matter enormously because they tell us which vector fields can be curls and which can be gradients — the necessary conditions behind potentials and the Helmholtz decomposition. The surprising fact is their depth: while they are trivial algebraically, their converses (a curl-free field is a gradient; a divergence-free field is a curl) are not automatic — they hinge on the topology of the domain, which is exactly the content of the Poincaré lemma and de Rham cohomology.

The Core Insight

Both identities are nothing more than the statement

\frac{\partial ^{2}}{\partial x _{i} \partial x _{j}} = \frac{\partial ^{2}}{\partial x _{j} \partial x _{i}}

dressed in vector notation. The curl and divergence operators combine partial derivatives with antisymmetric coefficients (the cross product, the alternating symbol

ε_{ij k}

). When you compose two such operators, you contract a quantity that is symmetric in two indices (the mixed second derivative) against one that is antisymmetric in those same indices — and any symmetric-times-antisymmetric contraction is identically zero. That single observation proves both identities at once.

Formal Statement

The three first-order operators

For a scalar field $ϕ : R^{3} \to R$ and a vector field $F = (F_{1}, F_{2}, F_{3})$ on $R^{3}$ , with $\nabla = (\partial_{x}, \partial_{y}, \partial_{z})$ :

\nabla ϕ = (\frac{\partial ϕ}{\partial x}, \frac{\partial ϕ}{\partial y}, \frac{\partial ϕ}{\partial z})

⟳the gradient stacks the three first partials of a scalar into a vector pointing in the direction of steepest ascent.

\nabla \cdot F = \frac{\partial F _{1}}{\partial x} + \frac{\partial F _{2}}{\partial y} + \frac{\partial F _{3}}{\partial z}

⟳the divergence is the trace of the Jacobian

\partial F_{i} / \partial x_{i}

, summed over the diagonal.

\nabla \times F = (\frac{\partial F _{3}}{\partial y} - \frac{\partial F _{2}}{\partial z}, \frac{\partial F _{1}}{\partial z} - \frac{\partial F _{3}}{\partial x}, \frac{\partial F _{2}}{\partial x} - \frac{\partial F _{1}}{\partial y})

⟳expand the symbolic determinant

det [\overset{e}{^}_{i}, \partial_{j}, F_{k}]

; each component is an antisymmetric difference of cross-partials.

Here $ϕ$ is assumed $C^{2}$ and each $F_{i}$ is assumed $C^{2}$ so that all second partials are continuous.

The two second-order identities

Let $ϕ \in C^{2} (U)$ and $F \in C^{2} (U; R^{3})$ on an open set $U \subseteq R^{3}$ . Then:

\nabla \times (\nabla ϕ) = 0 (curl of a gradient vanishes)

⟳the

k

-th component is

ε_{k ij} \partial_{i} \partial_{j} ϕ

; the symmetric Hessian contracts to zero against

ε

\nabla \cdot (\nabla \times F) = 0 (divergence of a curl vanishes)

⟳write it as

ε_{ij k} \partial_{i} \partial_{j} F_{k}

; antisymmetry in

i, j

kills the symmetric

\partial_{i} \partial_{j}

In index notation with the Einstein summation convention and the Levi-Civita symbol $ε_{ij k}$ :

(\nabla \times \nabla ϕ)_{k} = ε_{k ij} \partial_{i} \partial_{j} ϕ = 0, \nabla \cdot (\nabla \times F) = ε_{ij k} \partial_{i} \partial_{j} F_{k} = 0

⟳

ε

is antisymmetric under

i \leftrightarrow j

while

\partial_{i} \partial_{j}

is symmetric; swapping dummy indices shows each sum equals its own negative.

Why It Holds / Derivation

:::proof[Identity [4]: curl(grad) = 0] Step 1. Compute the first component of $\nabla \times (\nabla ϕ)$ directly from [3], substituting $F_{i} = \partial_{i} ϕ$ :

(\nabla \times \nabla ϕ)_{1} = \frac{\partial}{\partial y} (\frac{\partial ϕ}{\partial z}) - \frac{\partial}{\partial z} (\frac{\partial ϕ}{\partial y}) = \frac{\partial ^{2} ϕ}{\partial y \partial z} - \frac{\partial ^{2} ϕ}{\partial z \partial y} .

⟳replace

F_{3} \to \partial_{z} ϕ

and

F_{2} \to \partial_{y} ϕ

in the first curl component.

Step 2. Invoke Clairaut–Schwarz: since

ϕ \in C^{2}

, the mixed partials are equal,

\partial_{y} \partial_{z} ϕ = \partial_{z} \partial_{y} ϕ

. Hence the first component is

0

. Step 3. The second and third components,

(\nabla \times \nabla ϕ)_{2} = \partial_{z} \partial_{x} ϕ - \partial_{x} \partial_{z} ϕ, (\nabla \times \nabla ϕ)_{3} = \partial_{x} \partial_{y} ϕ - \partial_{y} \partial_{x} ϕ,

vanish identically by the same theorem.

⟳each component is a difference of two orderings of the same mixed partial.

Step 4 (index form). All three components are summarized by

(\nabla \times \nabla ϕ)_{k} = ε_{k ij} \partial_{i} \partial_{j} ϕ

. Rename dummies

i \leftrightarrow j

ε_{k ij} \partial_{i} \partial_{j} ϕ = ε_{k j i} \partial_{j} \partial_{i} ϕ = - ε_{k ij} \partial_{i} \partial_{j} ϕ

. A quantity equal to its own negative is zero.

■

:::proof[Identity [5]: div(curl) = 0] Step 1. Write the curl from [3] and apply the divergence [2]:

\nabla \cdot (\nabla \times F) = \frac{\partial}{\partial x} (\frac{\partial F _{3}}{\partial y} - \frac{\partial F _{2}}{\partial z}) + \frac{\partial}{\partial y} (\frac{\partial F _{1}}{\partial z} - \frac{\partial F _{3}}{\partial x}) + \frac{\partial}{\partial z} (\frac{\partial F _{2}}{\partial x} - \frac{\partial F _{1}}{\partial y}) .

⟳take

\partial_{x}

of component 1,

\partial_{y}

of component 2,

\partial_{z}

of component 3, then add.

Step 2. Expand and group by field component:

(\partial_{x} \partial_{y} F_{3} - \partial_{y} \partial_{x} F_{3}) + (\partial_{y} \partial_{z} F_{1} - \partial_{z} \partial_{y} F_{1}) + (\partial_{z} \partial_{x} F_{2} - \partial_{x} \partial_{z} F_{2}) .

⟳collect the two terms in

F_{1}

, the two in

F_{2}

, and the two in

F_{3}

Step 3. Each parenthesized pair is a difference of equal mixed partials (Clairaut–Schwarz, since

F_{i} \in C^{2}

), so each pair vanishes; the total is

0

. Step 4 (index form).

\nabla \cdot (\nabla \times F) = \partial_{i} ε_{ij k} \partial_{j} F_{k} = ε_{ij k} \partial_{i} \partial_{j} F_{k}

. Swapping dummies

i \leftrightarrow j

gives

ε_{j ik} \partial_{j} \partial_{i} F_{k} = - ε_{ij k} \partial_{i} \partial_{j} F_{k}

, so the sum equals its negative and is therefore

0

■

The hypothesis is essential

Both identities require continuity of the second partials (class $C^{2}$ ). If $ϕ$ or $F$ has discontinuous or non-equal mixed partials at a point, the identities can fail at that point. The classic example $ϕ (x, y) = \frac{x y ( x ^{2} - y ^{2} )}{x ^{2} + y ^{2}}$ (with $ϕ (0, 0) = 0$ ) has $\partial_{x} \partial_{y} ϕ (0, 0) = - 1 \neq = 1 = \partial_{y} \partial_{x} ϕ (0, 0)$ , so $\nabla \times \nabla ϕ \neq = 0$ at the origin. The identities are theorems about smooth fields, not unconditional truths.

James Clerk Maxwell | 1831–1879

Maxwell did not "discover" these identities in isolation, but he wielded them as structural pillars of electromagnetic theory. Working partly in quaternion notation inherited from Hamilton, he recognized that the magnetic field's source-free nature ( $\nabla \cdot B = 0$ ) is guaranteed if $B = \nabla \times A$ for a vector potential $A$ — precisely identity [5]. Likewise, that a static electric field is curl-free ( $\nabla \times E = 0$ ) permits the scalar potential $E = - \nabla ϕ$ — the converse of identity [4]. His Treatise on Electricity and Magnetism (1873) embedded these relations into physics permanently, and Gibbs and Heaviside later recast them into the clean $\nabla$ -operator notation students use today.

Building Intuition

Think of the three operators as a chain of "differentiation flavors": gradient turns a scalar into a vector, curl turns a vector into a vector, divergence turns a vector into a scalar. The identities say two consecutive links in a particular order produce nothing.

curl(grad)=0: A gradient field always flows "downhill" along a single potential surface stack. Such a field can't circulate — there is no loop along which the potential keeps increasing and returns higher than it started. No circulation means no curl.
div(curl)=0: A curl field is built from rotation. Rotation neither creates nor destroys material — what swirls in one side of any tiny box swirls out the other. No net outflow means no divergence; a pure swirl has no sources or sinks.

Geometrically, both reduce to the same picture: the boundary of a boundary is empty. The "boundary" of a closed surface is nothing; the "boundary" of a closed loop is nothing. Curl integrates over loops bounding surfaces (Stokes), divergence over surfaces bounding volumes (Gauss), and applying one after the other tries to integrate over an empty boundary.

The operator chain: each composition of two consecutive arrows collapses to zero

Worked Examples

Verifying curl(grad)=0 for a concrete potential

Given:

ϕ (x, y, z) = x^{2} y + y z^{3} + 5 x

. Step 1 — gradient:

\nabla ϕ = (2 x y + 5, x^{2} + z^{3}, 3 y z^{2}) .

⟳differentiate

ϕ

separately with respect to

x

y

z

Step 2 — curl, component 1:

\partial_{y} (3 y z^{2}) - \partial_{z} (x^{2} + z^{3}) = 3 z^{2} - 3 z^{2} = 0.

Step 3 — component 2:

\partial_{z} (2 x y + 5) - \partial_{x} (3 y z^{2}) = 0 - 0 = 0.

Step 4 — component 3:

\partial_{x} (x^{2} + z^{3}) - \partial_{y} (2 x y + 5) = 2 x - 2 x = 0.

\nabla \times (\nabla ϕ) = (0, 0, 0)

⟳each component cancels because the relevant cross-partials of

ϕ

are equal.

Units check: if

ϕ

has units of (say) electric potential V, then

\nabla ϕ

has V/m and its curl has V/m² — and that quantity is identically the zero vector. ✓

Verifying div(curl)=0 for a concrete vector field

Given:

F = (x y z, x^{2} - y^{2}, z e^{x})

. Step 1 — curl:

\nabla \times F = (\partial_{y} (z e^{x}) - \partial_{z} (x^{2} - y^{2}), \partial_{z} (x y z) - \partial_{x} (z e^{x}), \partial_{x} (x^{2} - y^{2}) - \partial_{y} (x y z)) .

⟳apply the determinant formula [3] to the three components of

F

Evaluating: $\nabla \times F = (0 - 0, x y - z e^{x}, 2 x - x z) = (0, x y - z e^{x}, 2 x - x z) .$

Step 2 — divergence of that curl:

\nabla \cdot (\nabla \times F) = \partial_{x} (0) + \partial_{y} (x y - z e^{x}) + \partial_{z} (2 x - x z) .

⟳differentiate each curl component by its matching coordinate.

Step 3 — evaluate:

\partial_{x} (0) = 0

\partial_{y} (x y - z e^{x}) = x

\partial_{z} (2 x - x z) = - x

. Sum

= 0 + x - x

\nabla \cdot (\nabla \times F) = 0

⟳the

+ x

and

- x

terms cancel exactly — a manifestation of the symmetry of mixed partials of

F

Units check: if

F

is a velocity (m/s), its curl is in s⁻¹ and the divergence of that is m⁻¹·s⁻¹ — and equals zero everywhere. ✓

Common Misconceptions

Six common errors

"The converse is automatic." $\nabla \times F = 0$ does not by itself imply $F = \nabla ϕ$ , and $\nabla \cdot G = 0$ does not by itself imply $G = \nabla \times A$ . The converses need the domain to be simply connected / have trivial topology (Poincaré lemma).
"Identities hold for any field." They require $C^{2}$ smoothness. Drop continuity of mixed partials and they can fail pointwise.
Confusing the two zeros. $\nabla \times (\nabla ϕ)$ is the zero vector $0$ ; $\nabla \cdot (\nabla \times F)$ is the scalar $0$ . Different objects.
Wrong composition order. $\nabla \cdot (\nabla ϕ) = \nabla^{2} ϕ$ (the Laplacian) is generally nonzero; only div-of-curl and curl-of-grad vanish.
Assuming $\nabla \times (\nabla \times F) = 0$ . It is not; instead $\nabla \times (\nabla \times F) = \nabla (\nabla \cdot F) - \nabla^{2} F$ , generally nonzero.
Forgetting dimension. Curl as a vector is special to 3D; in $n$ dimensions the right framework is the exterior derivative $d$ , where the identities become $d^{2} = 0$ .

A "potential" whose curl is nonzero at a point

Let $ϕ (x, y) = \frac{x y ( x ^{2} - y ^{2} )}{x ^{2} + y ^{2}}$ for $(x, y) \neq = (0, 0)$ and $ϕ (0, 0) = 0$ . The first partials exist everywhere, but

\frac{\partial ^{2} ϕ}{\partial x \partial y} (0, 0) = 1, \frac{\partial ^{2} ϕ}{\partial y \partial x} (0, 0) = - 1.

⟳compute

\partial_{y} ϕ

then differentiate in

x

along the axis, and vice versa; the two limits differ because the second mixed partials are discontinuous at the origin.

The third component of $\nabla \times (\nabla ϕ)$ at the origin is $\partial_{x} \partial_{y} ϕ - \partial_{y} \partial_{x} ϕ = 1 - (- 1) = 2 \neq = 0$ . The identity [4] fails precisely because $ϕ$ is not $C^{2}$ at the origin — confirming the hypothesis cannot be dropped.

Connections

These identities sit at the crossroads of analysis, geometry, and physics.

Helmholtz Decomposition — Splits a field into a curl-free (gradient) part plus a divergence-free (curl) part; the two identities guarantee each piece behaves as advertised.
Poincaré Lemma — Provides the converses: on a contractible domain, curl-free $\Rightarrow$ gradient and divergence-free $\Rightarrow$ curl. The identities are the "easy direction."
Exterior Derivative and Differential Forms — Both identities are the single statement $d^{2} = 0$ : $d (d ω) = 0$ for any form $ω$ . Gradient is $d$ on $0$ -forms, curl is $d$ on $1$ -forms, divergence is $d$ on $2$ -forms (up to Hodge stars).
Stokes' Theorem and Divergence Theorem — Integral counterparts; "boundary of a boundary is empty" ( $\partial^{2} = \emptyset$ ) is the integral shadow of $d^{2} = 0$ .
Maxwell's Equations — $\nabla \cdot B = 0$ permits $B = \nabla \times A$ ; $\nabla \times E = 0$ (statics) permits $E = - \nabla ϕ$ .
Vector Potential and Scalar Potential — Their very existence rests on these identities.

Real-World Applications

Magnetic vector potential (electromagnetism). Because $\nabla \cdot B = 0$ always, one writes $B = \nabla \times A$ ; identity [5] then automatically satisfies the no-monopole law for any $A$ . Gauge freedom $A \to A + \nabla χ$ uses identity [4] to leave $B$ unchanged.
Electrostatic potential. Static fields obey $\nabla \times E = 0$ , so $E = - \nabla ϕ$ and circuit/voltage analysis becomes scalar bookkeeping (consistency guaranteed by the converse of [4]).
Incompressible fluid dynamics. Vorticity $ω = \nabla \times v$ is automatically divergence-free (identity [5]), so vortex lines never start or stop inside the fluid — central to aerodynamics and the stream-function/vector-potential formulations.
Computational electromagnetics (finite element methods). Edge ("Whitney") elements are designed so the discrete curl-grad and div-curl identities hold exactly on the mesh, preventing spurious modes — the field of "structure-preserving" or "compatible" discretization.
Gauge theories and the Aharonov–Bohm effect. The potentials' redundancy (rooted in these identities) becomes physically subtle: $A$ matters quantum-mechanically even where $B = 0$ .
Geophysics / potential-field methods. Gravitational and magnetic anomaly fields are modeled as gradients of potentials, exploiting irrotationality for inversion and data processing.
Computer graphics and simulation. Divergence-free velocity fields (via curl/streamfunction representations) keep fluids visually incompressible; identity [5] ensures the construction is exact.

Historical Development

1746 / 1740s — Clairaut & Euler: the equality of mixed partial derivatives (Clairaut's theorem) emerges in studies of differentials and exact equations — the analytic seed of both identities.
1813 — Gauss: formulates the divergence (flux) theorem in work on attraction of ellipsoids, foundational for the divergence operator.
1828 — George Green: An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism introduces potential functions, implicitly using irrotationality of gradients.
1831–1839 — Carl Friedrich Gauss / Mikhail Ostrogradsky: rigorous statements of the divergence theorem (Ostrogradsky 1826/1831).
1849 — George Stokes: poses the curl–circulation theorem (the "Stokes theorem") in a Cambridge exam, set by Kelvin (1850); ties curl to loop integrals.
1860s–1873 — James Clerk Maxwell: systematizes the operators in his electromagnetic work and Treatise; uses $B = \nabla \times A$ and $E = - \nabla ϕ$ , embedding both identities into physics.
1880s — J. Willard Gibbs & Oliver Heaviside: independently develop modern vector analysis and the $\nabla$ notation; the identities take the form taught today.
1899 — Henri Poincaré: establishes the Poincaré lemma, giving the topological converses and revealing the identities as one half of a deep cohomological duality.
1930s–1945 — Élie Cartan / Georges de Rham: the exterior calculus recasts everything as $d^{2} = 0$ , and de Rham cohomology measures exactly when the converses fail.

Subtleties & Limits

Where the easy identities meet hard mathematics

The identities $\nabla \times \nabla = 0$ and $\nabla \cdot (\nabla \times) = 0$ are elementary, but their failure of converse opens onto genuinely deep terrain. On a domain with holes, a curl-free field need not be a gradient (e.g., the angle-field $F = (- y, x, 0) / (x^{2} + y^{2})$ is curl-free off the $z$ -axis yet has nonzero circulation), and a divergence-free field need not be a global curl. Measuring this gap is the subject of de Rham cohomology: the dimensions of the cohomology groups count the independent obstructions. Open and active questions live in the regularity-limited setting (when $F$ is only weakly differentiable, as in PDE theory and weak solutions of Maxwell's and Navier–Stokes equations), in discrete settings (constructing finite-element complexes that preserve $d^{2} = 0$ for high-order and on general meshes — "finite element exterior calculus"), and in infinite-dimensional or singular geometries where the smooth Poincaré lemma must be replaced by sheaf-theoretic or distributional machinery.