In vector calculus, the curl, also known as rotor, is a vector operator that describes the infinitesimal circulation of a vector field in three-dimensional Euclidean space. The curl at a point in the field is …

In this section, we examine two important operations on a vector field: divergence and curl. They are important to the field of calculus for several reasons, including the use of curl and divergence to …

Nov 16, 2022 · In this section we will introduce the concepts of the curl and the divergence of a vector field. We will also give two vector forms of Green’s Theorem and show how the curl can be used to …

Jan 14, 2026 · Curl of a vector field F in three-dimensional space is denoted by ∇ × F ∇ × F. Geometrically, it measures the tendency of the vector field to rotate about a point.

What Is the Curl of a Vector? The curl of a vector field, ∇ × 𝐅, at any given point, is simply the limiting value of the closed line integral projected in a plane that is perpendicular to ̂ 𝐧. Mathematically, we can …

It is worth making explicit a fact that we have used implicitly throughout this section: the curl of a vector field is itself a vector field! That is, evaluating curl (F) at a point gives a vector.

The magnitude of the curl vector at P measures how quickly the particles rotate around this axis. In other words, the curl at a point is a measure of the vector field’s “spin” at that point.

If a fluid flows in three-dimensional space along a vector field, the rotation of that fluid around each point, represented as a vector, is given by the curl of the original vector field evaluated at that point.

Apr 8, 2026 · Curl Calculator Calculate the curl ∇×F of any 2D or 3D vector field with step-by-step cross product determinant expansion. Enter component functions P, Q (and R for 3D), get symbolic curl, …

The net result is that the wheel rotates very little. Let's now look at the curl of this vector field. Shown below is the same animation, but with the curl drawn as a surface over the vector field. Additionally, …