Curl Calculator

The Curl Calculator finds the curl of a 3D vector field F(x, y, z) = (P, Q, R) at a point — a vector that measures the field's local rotation, or "spin," at that point. It shows the simplified symbolic curl (still a function of x, y, z) alongside the numeric curl vector at your chosen point, with a full step-by-step derivation of every partial derivative involved.

It's built for multivariable calculus and vector calculus students checking a homework answer, and for anyone who needs curl F without differentiating six partial derivatives by hand.

How to use it

1. Enter the three components of your vector field: P(x, y, z), Q(x, y, z), and R(x, y, z) — standard math notation works: ^ for powers, * for multiplication, plus sqrt(), sin(), cos(), log(), and the other functions supported by mathjs's expression parser.
2. Enter the point (x, y, z) to evaluate the curl at.
3. Read curl F at (x, y, z) for the numeric curl vector, or the individual i, j, k components separately.
4. Step-by-Step Derivation shows all six partial derivatives, the simplified symbolic curl, and the final numeric evaluation.

Results update automatically as you type. For example, the rotation field F = (-y, x, 0) at (1, 2, 3) returns a curl of (0, 0, 2) — a constant spin around the z-axis, independent of where you evaluate it.

What is curl?

For a vector field F(x, y, z) = (P, Q, R), the curl is defined as:

curl F = ∇ × F = (∂R/∂y − ∂Q/∂z, ∂P/∂z − ∂R/∂x, ∂Q/∂x − ∂P/∂y)

Physically, curl measures how much a vector field "circulates" or rotates around a point — if you dropped a tiny paddle wheel into the field at that point, curl F points along its spin axis (by the right-hand rule) and its magnitude is proportional to how fast the wheel would spin. A curl of zero everywhere means the field has no local rotation anywhere — it's called irrotational, and (on a simply-connected domain) it's also a conservative field, meaning it's the gradient of some scalar potential function.

Why does the calculator show the symbolic curl too?

Curl is itself a vector field — a function of (x, y, z) — not just a number at one point. Seeing the simplified symbolic expression for each component (before plugging in numbers) makes it possible to check whether curl F is zero everywhere (conservative), constant (like a rigid rotation), or varies from point to point, which the single evaluated vector alone can't show.

What functions can I enter?

Anything mathjs's expression evaluator understands: polynomials (x^2*y - 2*z), trigonometric functions (sin(x*z), cos(y)), roots and logarithms (sqrt(x), log(y)), and combinations of these, using x, y, and z as the three variables. Angles in trig functions are in radians by default — write pi for π.

To evaluate arbitrary math expressions directly, see the Math Evaluator, or to find the average rate of change of a single-variable function, see the Average Rate of Change Calculator.

Privacy

This calculator runs entirely in your browser. Your vector field and point are never uploaded, logged, or stored — the computation happens locally on your device.