Curl Calculator
The curl of a vector field F = (P, Q, R) measures how much the field rotates, or curls, around a point. In 3D it's a vector: curl F = (∂R/∂y − ∂Q/∂z, ∂P/∂z − ∂R/∂x, ∂Q/∂x − ∂P/∂y). In 2D, where F = (P, Q) has no z-component, only the k-component survives: curl F = ∂Q/∂x − ∂P/∂y. Enter the partial derivatives evaluated at your point of interest to get the curl.
The Curl Formula for a 3D Vector Field
For F = (P, Q, R), curl is a vector built from all six partial derivatives, pairing two components not matching its own axis. This calculator takes those six values directly, evaluated at your point of interest, rather than symbolically differentiating a typed-in function.
Worked Example: A Rotation Field
Take F = (−y, x, 0), the standard counterclockwise rotation field around the z-axis. Its partial derivatives are ∂P/∂y = −1, ∂Q/∂x = 1, and the four others involving z are 0. The curl's k-component is 1 − (−1) = 2, with the other two components 0. A curl of (0, 0, 2) confirms the field spins counterclockwise at a constant rate, exactly what the rotation field should produce everywhere.
When Curl Is Zero: Conservative Fields
If curl F = 0 at every point in a simply connected region, F is conservative: there's a scalar potential function such that F is its gradient, and the line integral of F between two points is path-independent. Gradient fields always have zero curl, a useful check before assuming a field has a potential function.
Frequently Asked Questions
What does curl measure?
Curl measures the tendency of a vector field to rotate around a point, how much a tiny paddle wheel dropped into the field would spin. A curl of zero everywhere means the field is irrotational; a nonzero curl means local rotation at that point.
Why does this calculator ask for partial derivatives instead of a formula for F?
Parsing an arbitrary symbolic vector field and differentiating it automatically is a much larger problem than most calculator tools attempt. This tool asks for the six partial derivatives evaluated at your point of interest, values you get by differentiating P, Q, R yourself, then combines them with the curl formula.
What's the difference between 2D and 3D curl here?
In 2D, F = (P, Q) has no z-component, so only the k-component survives: curl F = ∂Q/∂x − ∂P/∂y, a single number. In 3D, F = (P, Q, R) produces a full vector using all six partial derivatives.
What does a curl of zero mean?
A curl of zero at every point means the field is irrotational, also called conservative: it can be written as the gradient of some scalar potential function, and the line integral of F between two points doesn't depend on the path taken.
How is curl related to divergence?
Curl and divergence both come from the same del operator but measure different things: divergence is a scalar describing outward flow from a point, while curl is a vector describing rotation around a point. A field can have one, both, or neither.