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For \(\mathbf{F} = \langle P, Q, R\rangle\), the curl is \(\nabla \times \mathbf{F} = \left\langle R_y – Q_z,\ P_z – R_x,\ Q_x – P_y \right\rangle\), where each subscript is a partial derivative. For example, the curl of \(\langle y, -x, 0\rangle\) is \(\langle 0, 0, -2\rangle\): the field spins clockwise around the z-axis.
The Curl Formula
Expanding that “determinant” the same way you would for a cross product gives three components:
| Component | Formula | Memory trick |
|---|---|---|
| \(\mathbf{i}\) | \(\frac{\partial R}{\partial y} – \frac{\partial Q}{\partial z}\) | Skip x: use y and z |
| \(\mathbf{j}\) | \(\frac{\partial P}{\partial z} – \frac{\partial R}{\partial x}\) | Skip y: use z and x (watch the order) |
| \(\mathbf{k}\) | \(\frac{\partial Q}{\partial x} – \frac{\partial P}{\partial y}\) | Skip z: use x and y |
Each piece is a partial derivative: you treat the other variables as constants.
What Curl Means: The Paddle Wheel Picture
Imagine the vector field is the flow of water, and you drop a tiny paddle wheel into it. If the water pushes harder on one side of the wheel than the other, the wheel spins. The curl is a vector that points along the wheel’s axle, and its length tells you how fast the wheel spins. If the curl is zero, the wheel does not spin anywhere. Such a field is called irrotational.
Worked Example: \(\mathbf{F} = \langle x^2y,\ yz,\ xz^2\rangle\)
Here \(P = x^2y\), \(Q = yz\), \(R = xz^2\).
- \(\mathbf{i}\): \(R_y – Q_z = 0 – y = -y\)
- \(\mathbf{j}\): \(P_z – R_x = 0 – z^2 = -z^2\)
- \(\mathbf{k}\): \(Q_x – P_y = 0 – x^2 = -x^2\)
So \(\nabla \times \mathbf{F} = \langle -y,\ -z^2,\ -x^2 \rangle\). At the point \((1, 2, 3)\) the curl is \(\langle -2, -9, -1\rangle\).
Curl vs. Divergence
Curl \(\nabla \times \mathbf{F}\)
- Gives a vector
- Measures spinning
- Zero curl means no rotation
- Used in Stokes’ theorem
Divergence \(\nabla \cdot \mathbf{F}\)
- Gives a number (scalar)
- Measures spreading out
- Zero divergence means no sources or sinks
- Used in the divergence theorem
The calculator shows the divergence too, as a bonus, so you get both at once.
Testing Whether a Field Is Conservative
A vector field on all of 3D space is conservative (it is the gradient of some potential function) exactly when its curl is the zero vector. For example, \(\mathbf{F} = \langle yz,\ xz,\ xy\rangle\) has curl \(\langle 0, 0, 0\rangle\), and indeed \(\mathbf{F} = \nabla(xyz)\). Conservative fields make line integrals easy, because the answer only depends on the start and end points.
Working in 2D? Enter \(R = 0\) and use only x and y. The curl then points straight up or down, and its k-component \(Q_x – P_y\) is the “scalar curl” used in Green’s theorem.
Learn the Math Behind the Curl Calculator
Curl measures how a vector field spins, and divergence measures how it spreads out. Both are built from partial derivatives.
Frequently Asked Questions
Is curl the same as a cross product?
It uses the same pattern. You “cross” the del operator \(\nabla = \langle \partial_x, \partial_y, \partial_z\rangle\) with the field. But the top-row entries are derivatives, not numbers, so the order inside each term matters.
What does a negative curl component mean?
It means the rotation is clockwise when you look from the positive end of that axis. For \(\langle y, -x, 0\rangle\), the k-component is \(-2\), so the field turns clockwise when you look down from above.
What is the curl of a gradient?
It is always the zero vector, for any smooth function \(f\): \(\nabla \times (\nabla f) = \mathbf{0}\). This is why gradient fields are irrotational.
Can I evaluate the curl at a point?
Yes. Type the point in the “At point” box, like 1, 2, 3. The calculator shows the general curl and its value at that point.