Curl (Vector Calculus)

What curl is

For a smooth vector field F = ⟨P, Q, R⟩ in 3-space, the curl is another vector field, defined by the symbolic determinant

        | î   ĵ   k̂  |
∇×F  =  | ∂_x ∂_y ∂_z |
        | P   Q   R   |

Expanded out:

∇×F = ⟨ ∂R/∂y − ∂Q/∂z ,  ∂P/∂z − ∂R/∂x ,  ∂Q/∂x − ∂P/∂y ⟩.

The vector ∇×F at a point is the axis of an infinitesimal rotation that the flow induces locally; its magnitude is twice the angular speed of that rotation. Drop a tiny paddle wheel into the field and orient its axle along ∇×F — that is the orientation in which it spins fastest, and ½|∇×F| is the angular velocity it picks up.

Curl in 2D vs 3D

In a 2D field F = ⟨P(x,y), Q(x,y)⟩, the only swirling motion possible is in the plane, so all but the z-component of curl is zero. The "scalar 2D curl" is just

curl₂ F = ∂Q/∂x − ∂P/∂y.

That single number is what shows up on the right-hand side of Green's theorem.

	2D curl	3D curl
Output type	Scalar	Vector (3 components)
Formula	∂Q/∂x − ∂P/∂y	∇×F (full determinant)
Geometric reading	Signed angular speed in plane	Axis + rate of rotation
Theorem that uses it	Green's theorem	Stokes' theorem (Green's = Stokes' on flat surface)
Sign convention	Counterclockwise positive	Right-hand rule about ∇×F
Conservative ⇔	∂Q/∂x − ∂P/∂y = 0 (simply conn.)	∇×F = 0 (simply conn.)

Worked examples

Example 1 — the rigid rotation field

Take F = ⟨−y, x, 0⟩ — the velocity field of a body rotating about the z-axis with angular speed 1. Compute

∇×F = ⟨ ∂(0)/∂y − ∂(x)/∂z ,  ∂(−y)/∂z − ∂(0)/∂x ,  ∂(x)/∂x − ∂(−y)/∂y ⟩
     = ⟨ 0 , 0 , 1 − (−1) ⟩
     = ⟨ 0 , 0 , 2 ⟩  =  2 k̂.

The curl points along +z (the axis of rotation) with magnitude 2 — exactly twice the angular speed, as advertised. Reverse the rotation (F = ⟨y, −x, 0⟩) and curl flips to −2 k̂.

Example 2 — radial field has no curl

For F = ⟨x, y, z⟩ (purely outward), every off-diagonal partial of the determinant cancels — ∂R/∂y = 0 = ∂Q/∂z, etc. So ∇×F = 0. A purely radial flow has no swirl. The same field has nonzero divergence (∇·F = 3), reminding us that curl and divergence are independent measurements.

Example 3 — shear has curl even though nothing "spins"

F = ⟨y, 0, 0⟩ is a horizontal shear: water at height y moves at speed y in the x-direction. Compute curl:

∇×F = ⟨ 0 − 0 , 0 − 0 , 0 − 1 ⟩ = ⟨0, 0, −1⟩.

Even though no streamline curves, an immersed paddle wheel spins clockwise — the top of the wheel is dragged faster than the bottom. Curl detects this differential, not just visible looping.

Example 4 — verifying ∇×(∇φ) = 0

Take φ = x²y + yz². Then ∇φ = ⟨2xy, x²+z², 2yz⟩. Compute ∇×(∇φ):

i-comp: ∂(2yz)/∂y − ∂(x²+z²)/∂z = 2z − 2z = 0.
j-comp: ∂(2xy)/∂z − ∂(2yz)/∂x  = 0 − 0 = 0.
k-comp: ∂(x²+z²)/∂x − ∂(2xy)/∂y = 2x − 2x = 0.

All three components vanish — gradients are always curl-free, by Clairaut's theorem on equality of mixed partials.

Conservative vs path-dependent fields

	Conservative (gradient) field	Path-dependent field
Curl	∇×F = 0 everywhere	∇×F ≠ 0 somewhere
Has potential?	Yes (locally always; globally on simply-connected domains)	No
∮ F·dr around closed loop	Always 0	Can be nonzero
Line integral A→B	Depends only on endpoints	Depends on path
Example	F = ∇(x²+y²) = ⟨2x, 2y⟩	F = ⟨−y, x⟩ (rigid rotation)
Physics analogue	Gravitational, electrostatic force	Magnetic field around a wire
Energy interpretation	Work depends only on heights	Loop work pumps energy in/out

The slogan to remember: curl-free + simply-connected ⟹ conservative. The simply-connected hypothesis is essential. The classic counterexample is F = ⟨−y, x⟩ / (x²+y²) on ℝ²\{0}: it has zero curl everywhere it is defined, yet ∮ F·dr around the unit circle equals 2π. The hole in the domain wrecks the existence of a global potential.

Curl as circulation density (Stokes' theorem)

The cleanest definition of curl is via Stokes' theorem. Pick a small flat disk D of area ΔA at a point, with unit normal n̂. The circulation of F around its boundary divided by ΔA approaches the projection of curl onto n̂:

(∇×F)·n̂  =  lim_{ΔA→0}  (1/ΔA)  ∮_{∂D} F·dr.

So curl really is "circulation per unit area." Sum this density over an entire surface S and you recover Stokes' theorem,

∬_S (∇×F)·dS = ∮_{∂S} F·dr.

The 2D specialization, where S lies flat in the xy-plane, is Green's theorem: ∮(P dx + Q dy) = ∬(∂Q/∂x − ∂P/∂y) dA.

Curl in Maxwell's equations

Two of the four Maxwell equations are equations about curl:

• Faraday's law. ∇×E = −∂B/∂t. A changing magnetic flux induces a circulating electric field. Run a wire loop through this E field and you get a voltage — the principle behind every generator and transformer.
• Ampère–Maxwell law. ∇×B = μ₀ J + μ₀ε₀ ∂E/∂t. Currents and changing electric fields curl up magnetic fields. The displacement-current term (Maxwell's correction) is what makes electromagnetic waves possible.

The other two equations involve divergence (Gauss for E and B). Together — two divergence statements, two curl statements — they specify the electromagnetic field completely, by the Helmholtz decomposition theorem.

Curl in fluid dynamics — vorticity

In a fluid with velocity field v, the curl ω = ∇×v is called vorticity. It quantifies the local rotation a tiny dye drop would experience. Vorticity dominates the dynamics of:

• Vortex rings — smoke rings, dolphin bubbles. A torus of vorticity that propels itself by inducing flow through its center.
• Lift on a wing — the Kutta–Joukowski theorem says lift per unit span = ρ · v_∞ · Γ, where Γ is the circulation around the wing — a curl integral.
• Tornadoes and hurricanes — coherent vortices whose persistence is governed by vorticity-stretching.
• Turbulence — energy cascades from large-scale eddies to small ones along the vortex-stretching direction; modelling curl statistics is the Kolmogorov problem.

Useful identities

• Curl of a gradient is zero. ∇×(∇φ) = 0.
• Divergence of a curl is zero. ∇·(∇×F) = 0.
• Vector Laplacian. ∇×(∇×F) = ∇(∇·F) − ∇²F.
• Product rule. ∇×(φF) = φ(∇×F) + (∇φ)×F.
• Helmholtz decomposition. A nice enough field on ℝ³ splits uniquely as F = ∇φ + ∇×A — a curl-free piece plus a divergence-free piece.

Orientation and the right-hand rule

Curl carries a sign that depends on how you orient the world. The convention: curl of F at a point follows the right-hand rule — curl your fingers in the direction the flow rotates, your thumb is ∇×F. If the field is rotating counterclockwise as seen from above, ∇×F points up (+z). Reverse the rotation, reverse the curl.

This same convention drives the boundary orientation in Stokes' theorem: when n̂ points out of the page, ∂S is traversed counterclockwise. Get the orientation wrong and you get a sign error — the second-most-common bug in vector-calculus computations after dropping a partial.

Common mistakes

• Wrong boundary orientation. Stokes' theorem ties n̂ on the surface to the direction of traversal on ∂S by the right-hand rule. Mismatch flips the sign of the answer. Always pick the surface normal first, then orient the boundary to match.
• Confusing curl with divergence. Both are first-order differential operators on F, but curl detects rotation and produces a vector; divergence detects expansion/compression and produces a scalar. A radial field has nonzero divergence and zero curl; a rigid rotation has zero divergence and nonzero curl.
• Assuming curl-free ⟹ conservative on any domain. Only on simply-connected domains. Punctured planes, tori and any domain with handles can host curl-free fields with nonzero loop integrals. The 2π anomaly of F = ⟨−y, x⟩/(x²+y²) is the canonical warning.
• Computing ∇×F componentwise without the right ordering. The determinant expansion has a definite cyclic order (i, j, k from x, y, z). Easy to swap a sign. A safer mnemonic: write the 3×3 determinant with î ĵ k̂ on top and expand by minors.
• Forgetting "curl is local." ∇×F at a point depends only on the field's behaviour in a neighbourhood — it can be zero at one point and nonzero next door. Treating curl as a global property is wrong; only its integrals (Stokes') see global behaviour.
• Mixing scalar 2D curl with full 3D curl. The 2D version is a number. Embedding a 2D field in 3D as ⟨P, Q, 0⟩ and computing 3D curl gives ⟨0, 0, ∂Q/∂x − ∂P/∂y⟩. The two agree only after the embedding is done correctly.