🎓 University Year 1 · Lesson 7 of 15
Curl: The Rotation Hidden in a Vector Field
Drop a tiny paddlewheel into a flow — whether it spins, and how fast, is exactly what curl measures.
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The big idea: Where divergence measures how much a flow spreads out, <strong>curl</strong> measures how much it <em>rotates</em>. Imagine anchoring a microscopic paddlewheel at a point: curl tells you the axis it spins about and how fast. In two dimensions it collapses to a single number, ∂Q/∂x − ∂P/∂y. Strikingly, even a flow whose arrows all point the same way can have curl — if the speed varies across the flow, the paddlewheel still turns.
Does the flow spin you?
Float a tiny paddlewheel in a flowing vector field F = (P, Q), holding its centre fixed. If the fluid pushes harder on one side than the other, the wheel rotates. The curl is the precise measure of that tendency to spin — the local circulation of the field.
In three dimensions rotation happens about an axis, so curl is a vector (pointing along the spin axis). In two dimensions there is only one possible axis — straight out of the plane — so curl reduces to a single signed number.
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Curl measures local rotation
The curl of a 2D field is ∂Q/∂x − ∂P/∂y. A positive value means counterclockwise spin, negative means clockwise, and zero means the paddlewheel does not turn at all (the field is irrotational there).
\[ (\operatorname{curl}\mathbf F)_z=\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y} \]
The 2D (scalar) curl of F = (P, Q): the out-of-plane component of the rotation.
🎮 The Curl Paddlewheel LIVE
Drop a paddlewheel in the field and watch curl spin it.
The full 3D curl
In three dimensions, curl is the cross product ∇×F. Each component measures rotation in one of the coordinate planes; together they give the vector along the axis the flow twists around, by the right-hand rule.
\[ \nabla\times\mathbf F=\left(\frac{\partial R}{\partial y}-\frac{\partial Q}{\partial z},\ \frac{\partial P}{\partial z}-\frac{\partial R}{\partial x},\ \frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right) \]
The curl of a 3D field F = (P, Q, R); its z-component is the 2D curl above.
📝 Worked example: Find the curl of the rotational field F = (−y, x).
- Identify P = −y and Q = x.
- Compute the partials: \( \partial Q/\partial x = 1 \) and \( \partial P/\partial y = -1 \).
- Curl = \( \partial Q/\partial x - \partial P/\partial y = 1 - (-1) \).
✓ curl = <strong>2</strong> (positive) — this field circulates counterclockwise, so a paddlewheel spins the same way everywhere.
📝 Worked example: Find the curl of the radial field F = (x, y).
- P = x and Q = y.
- \( \partial Q/\partial x = \partial (y)/\partial x = 0 \) and \( \partial P/\partial y = \partial (x)/\partial y = 0 \).
- Curl = \( 0 - 0 \).
✓ curl = <strong>0</strong> — the flow pushes straight out with no twist, so the paddlewheel does not rotate (it is irrotational).
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Straight flow can still have curl
Consider the shear field F = (y, 0): every arrow points horizontally, so it looks like there is no rotation. But the flow is faster higher up and slower lower down. Curl = ∂(0)/∂x − ∂(y)/∂y = 0 − 1 = −1. The uneven speeds push the top and bottom of a paddlewheel in opposite senses, spinning it clockwise. Rotation is about differences in the flow, not about the arrows curving.
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Curl vs divergence — don't mix the signs
The 2D curl is ∂Q/∂x − ∂P/∂y (a difference of cross terms), while divergence is ∂P/∂x + ∂Q/∂y (a sum of matching terms). Swapping the two, or dropping the minus sign in curl, is the classic error. Remember: curl uses the opposite partials with a minus; divergence uses the matching partials with a plus.
Check your understanding
1. The 2D curl of F = (P, Q) is:
The scalar curl is ∂Q/∂x − ∂P/∂y — the cross partials, subtracted. The first option is the divergence.
2. Physically, curl measures a field's tendency to…
Curl captures local rotation/circulation — how a tiny paddlewheel placed in the flow would spin. Spreading out is divergence.
3. What is the curl of the rotational field F = (−y, x)?
∂(x)/∂x − ∂(−y)/∂y = 1 − (−1) = 2. Positive curl means counterclockwise circulation.
4. What is the curl of the radial field F = (x, y)?
∂(y)/∂x − ∂(x)/∂y = 0 − 0 = 0. A purely outward field is irrotational — no spin.
5. For the shear field F = (y, 0), whose arrows are all horizontal, the curl is:
∂(0)/∂x − ∂(y)/∂y = 0 − 1 = −1. The changing speed across the flow spins a paddlewheel even though no arrow curves.
✅ Key takeaways
- Curl measures the local rotation of a vector field — how a tiny paddlewheel would spin.
- In 2D it is the scalar ∂Q/∂x − ∂P/∂y; in 3D it is the vector ∇×F pointing along the spin axis.
- F = (−y, x) has curl 2 (counterclockwise); the radial field F = (x, y) has curl 0 (irrotational).
- Shear flow F = (y, 0) has curl −1 even though all arrows are parallel — varying speed causes rotation.
- Don't confuse curl (difference of cross partials, with a minus) with divergence (sum of matching partials).