What Is a Curl Calculator?
A curl calculator computes the curl of a three-dimensional vector field — a vector that describes the rotation, or circulation, of the field at each point. Curl is a core operation in vector calculus, used heavily in electromagnetism, fluid dynamics, and physics. Enter the three components of your vector field and the calculator returns the curl vector, computed as the cross product of the del operator with the field.
How to Use the Curl Calculator
- Enter the field components — P, Q, and R as functions of x, y, and z.
- Optionally specify a point to evaluate the curl numerically.
- Calculate — see the curl vector and its component expressions.
The Curl Formula
For a vector field F = (P, Q, R), the curl is:
curl F = ∇ × F = (∂R/∂y − ∂Q/∂z, ∂P/∂z − ∂R/∂x, ∂Q/∂x − ∂P/∂y)
Each component is a difference of partial derivatives, capturing how the field rotates around each axis.
What the Curl Tells You
| Curl Result | Interpretation |
|---|---|
| Zero everywhere | Irrotational field (conservative, no circulation) |
| Nonzero | The field has rotation at that point |
A field with zero curl is called irrotational and is often conservative, meaning it can be written as the gradient of a scalar potential.
Where Curl Is Used
- Electromagnetism: Maxwell's equations relate curl of electric and magnetic fields.
- Fluid dynamics: curl of velocity measures vorticity, the local spinning of fluid.
- Physics: describing rotational forces and fields.
Curl vs Divergence
Curl measures rotation (a vector), while divergence measures how much a field spreads out or converges at a point (a scalar). Both come from the del operator: curl is the cross product ∇ × F, and divergence is the dot product ∇ · F.
Frequently Asked Questions
What is the curl of a vector field?
The curl is a vector that measures the rotation or circulation of a field at each point, computed as the cross product of the del operator with the field.
What is the curl formula?
curl F = (∂R/∂y − ∂Q/∂z, ∂P/∂z − ∂R/∂x, ∂Q/∂x − ∂P/∂y) for a field F = (P, Q, R).
What does zero curl mean?
A field with zero curl everywhere is irrotational, meaning there is no local rotation. Such fields are often conservative and can be derived from a scalar potential.
What is the difference between curl and divergence?
Curl measures rotation and is a vector (∇ × F), while divergence measures outward flux at a point and is a scalar (∇ · F).
Is this curl calculator free?
Yes — it is completely free, requires no signup, and returns the full curl vector.