Curl Calculator

Curl is a vector operator that measures the rotation or circulation of a vector field at a point. In 3D, it produces a vector that indicates the axis and magnitude of rotation. The curl operation is denoted as ∇ × F (del cross F) and is essential in fluid dynamics, electromagnetism, and other physics applications. A zero curl indicates an irrotational or conservative field.

For a 3D vector field F = (P, Q, R), the curl is calculated using the formula: curl F = (∂R/∂y - ∂Q/∂z)i + (∂P/∂z - ∂R/∂x)j + (∂Q/∂x - ∂P/∂y)k. Simply enter your P, Q, and R components into our calculator, and it will compute all partial derivatives and show you step-by-step solutions. You can also evaluate the curl at specific points to get numerical results.

In 3D, curl produces a vector result showing the axis and magnitude of rotation. In 2D (for F = (P, Q)), curl produces a scalar result: ∂Q/∂x - ∂P/∂y. This scalar represents the rotation magnitude perpendicular to the plane (in the z-direction). Both calculators are available on this page - use 3D for three-component vector fields and 2D for two-component planar fields.

When curl F = 0 everywhere in a region, the vector field is called irrotational or conservative. This means the field has no local rotation at any point. Conservative fields can be expressed as the gradient of a scalar potential function (F = ∇φ). Examples include gravitational fields and electrostatic fields. Our calculator can help you verify if a field is conservative by checking if the curl is zero.

Curl has numerous applications: In fluid dynamics, it measures vorticity (rotation of fluid particles). In electromagnetism, Maxwell's equations use curl to relate electric and magnetic fields (Faraday's law: ∇ × E = -∂B/∂t, Ampere's law: ∇ × B = μ₀J). In meteorology, it helps analyze atmospheric circulation patterns. Engineers use curl to study rotational forces in mechanical systems and vector field analysis.

Stokes' Theorem states that the line integral of a vector field around a closed curve equals the surface integral of the curl over the surface bounded by that curve: ∮C F · dr = ∫∫S (curl F) · dS. This fundamental theorem connects local rotation (curl) to circulation around boundaries. It's the 3D generalization of Green's Theorem and has important applications in electromagnetic theory and fluid mechanics.

Yes, our curl calculator handles both symbolic and numerical calculations. Enter your vector field components using variables (x, y, z) to get symbolic partial derivatives and the curl formula. Then use the "Evaluate at a Point" feature to get numerical results at specific coordinates. The calculator shows step-by-step symbolic derivations and provides precise numerical evaluations for any point in the field.

The calculator supports standard mathematical operations: addition (+), subtraction (-), multiplication (*), division (/), and exponentiation (^). You can use variables x, y, and z (for 3D) or x and y (for 2D). Examples: "2*x + y", "x^2 - 3*y*z", "-y*sin(x)". Combine these operations to create complex vector field expressions. For more advanced matrix operations, check out our matrix diagonalization calculator.

The right-hand rule helps visualize curl direction: curl your fingers in the direction of field rotation, and your thumb points in the direction of the curl vector. For a 2D field rotating counterclockwise, the curl points upward (positive z-direction). For clockwise rotation, it points downward (negative z-direction). The magnitude of the curl vector indicates the strength of rotation. This visual tool is essential for understanding vector field behavior.

Yes! Our site offers various mathematical calculators. While this tool specializes in curl calculations for vector fields, you can also use our diagonalization calculator for matrix eigenvalue problems and linear algebra operations. We're continually adding more advanced mathematics tools. For logarithmic calculations, explore our logarithm calculator.