Key takeaways
- 2D curl: ∂Q/∂x − ∂P/∂y, where F = (P, Q).
- Positive means counter-clockwise spin; negative means clockwise.
- Zero curl means no spin: the field is irrotational there.
- The order matters: ∂Q/∂x comes first. Swapping flips the sign.
In this guide
What is curl?
Picture a tiny paddle wheel floating in moving water, pinned in place but free to rotate. If the water pushes harder on one side of the wheel than the other, the wheel spins. Curl measures that spinning.
- Positive curl means counter-clockwise spin.
- Negative curl means clockwise spin.
- Zero curl means no spin. The field is called irrotational at that point.
In 2D, curl is a single number. In 3D it becomes a vector, but the 2D version is the same idea: it is the z-part of the 3D curl.
Why does curl matter?
Curl finds the swirl hidden in a flow. Whirlpools, tornadoes and the magnetic field around a wire all have strong curl. Engineers designing wings, pumps and pipes need to know where flow starts to rotate.
Curl also tells you something powerful about forces. If a force field has zero curl everywhere (on a region with no holes), then the work it does depends only on the start and end points, not on the path. Gravity is like that.
A flow can go round in circles yet have zero curl, and a flow in perfectly straight lines can have curl. Curl is about local spin, not the shape of the path.
The formula
| Symbol | What it means | Example |
|---|---|---|
| F = (P, Q) | The vector field. Here P = yⁿ and Q = xᵐ | (y, x²) |
| ∂Q/∂x | How the up-down push changes as you move right | 2x = 6 |
| ∂P/∂y | How the left-right push changes as you move up | 1 |
| (x₀, y₀) | The point where you measure | (3, 1) |
| curl F | The spin at that point | 5 |
This calculator uses the field F = (yⁿ, xᵐ), so ∂Q/∂x = m·xᵐ⁻¹ and ∂P/∂y = n·yⁿ⁻¹. You choose n, m and the point (x₀, y₀).
How to calculate curl step by step
- 1Identify P and Q. P is the x-component, Q is the y-component.
- 2Find ∂Q/∂x. Differentiate Q in x, treating y as a constant. See the partial derivative calculator.
- 3Find ∂P/∂y. Differentiate P in y, treating x as a constant.
- 4Subtract and evaluate. Work out
∂Q/∂x − ∂P/∂y, then put in the point.
Now try your own field below. The calculator shows both derivatives and the difference.
Try it yourself
Pre-filled with the example — change any value.
Common mistakes to avoid
- Using the same-letter derivatives. ∂P/∂x and ∂Q/∂y belong to divergence, not curl.
- Thinking curved flow must have curl. Look at the local spin, not the overall path.
- Forgetting to evaluate at the point. Curl can change from place to place.
- Expecting a vector in 2D. In 2D, curl is one number.
Tips and tricks
- Zero-curl surprise: with this calculator, F = (y, x) (n = 1, m = 1) gives
1 − 1 = 0. It looks twisted, but it doesn't spin. - Potential check: zero curl everywhere (on a region with no holes) means F is the gradient of some function.
- Green's theorem: adding up curl over a region equals the flow around its edge.
Where you'll use it in real life
- Weather: measuring the spin of storms, tornadoes and cyclones.
- Fluid engineering: finding whirlpools and eddies behind wings and in pipes.
- Electromagnetism: the magnetic field curls around a current-carrying wire.
- Physics: testing whether a force is conservative, like gravity.
Quick summary
2D curl, ∂Q/∂x − ∂P/∂y, measures how much a field makes things spin at a point: positive for counter-clockwise, negative for clockwise, zero for no spin. Take the two cross derivatives, subtract in the right order and evaluate. Use the calculator above for any field (yⁿ, xᵐ).
Worked example
A river model gives the water velocity as F = (y, x²). At the point (3, 1), would a small floating leaf start to spin, and which way? Here n = 1, m = 2, x₀ = 3 and y₀ = 1.
- 01Vector field F = (P, Q) = (y^1, x^2)
- 02∂Q/∂x = 2x^1 → at x=3: 6
- 03∂P/∂y = 1y^0 → at y=1: 1
- 04∇×F = ∂Q/∂x − ∂P/∂y
- 05= 6 − 1
- 06= 5
These numbers are pre-filled in the calculator above.
Frequently asked questions
What does it mean if the curl is zero?
There is no spinning at that point. If the curl is zero everywhere on a region with no holes, the field is conservative: it is the gradient of some function.
What is the curl of F = (−y, x)?
∂Q/∂x = 1 and ∂P/∂y = −1, so the curl is 1 − (−1) = 2. This field spins counter-clockwise everywhere.
Is 2D curl a vector or a number?
In 2D it is a number. It equals the z-component of the 3D curl vector for a field that lies flat in the xy-plane.
What does a positive curl of 5 mean?
A tiny paddle wheel at that point would spin counter-clockwise. A bigger number means faster spinning.
