Curl

The curl of a vector field is a vector-valued operation that measures the local rotation or "vorticity" of the field. It is applicable to vector fields in 3-dimensional Euclidean space and is often used in physics to describe phenomena such as fluid flow, electromagnetism, and more.

Given a vector field $\mathbf{F} = (F_x, F_y, F_z)$, where $F_x, F_y,$ and $F_z$ are its components along the x, y, and z directions, respectively, the curl of the vector field, denoted by $\nabla \times \mathbf{F}$ or $\text{curl} , \mathbf{F}$, is defined as:

$$\nabla \times \mathbf{F} = \begin{vmatrix} \mathbf{\hat{i}} & \mathbf{\hat{j}} & \mathbf{\hat{k}} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ F_x & F_y & F_z \end{vmatrix}$$

Expanding the determinant, we get:

$$\nabla \times \mathbf{F} = \left(\frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z}\right) \mathbf{\hat{i}} - \left(\frac{\partial F_z}{\partial x} - \frac{\partial F_x}{\partial z}\right) \mathbf{\hat{j}} + \left(\frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y}\right) \mathbf{\hat{k}}$$

The curl of a vector field gives a new vector field, where the magnitude of the curl at a point represents the local rotational intensity, and the direction of the curl vector indicates the axis around which the rotation occurs, following the right-hand rule. If the curl of a vector field is zero at every point, the field is considered irrotational.