Curl
High school
Consider a flow, like water moving in a pond, and imagine placing a tiny paddle wheel at a point in it. The curl of the flow at that point measures how much the flow swirls around the point: how fast the paddle wheel would spin.
Undergraduate
Consider a vector field $F = (F_1, F_2, F_3) : \reals^3 \to \reals^3$. The curl of $F$ is the vector field
\[\nabla \times F = \left( \frac{\partial F_3}{\partial y} - \frac{\partial F_2}{\partial z}, \frac{\partial F_1}{\partial z} - \frac{\partial F_3}{\partial x}, \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y} \right).\]It measures the rotation of $F$ around each point.
Distinctions involving this term
- Types of derivatives in vector calculus: The Jacobian of a vector field contains all of its first partial derivatives. The divergence (a scalar) and curl (a vector) summarize parts of it. The Laplacian is a second-order operator on scalar functions.