Curl
The curl measures the rotation or circulation of the vector field \( \small \mathbf{F} \) at any given point in space.
Cartesian Coordinates #
In cartesian coordinates, the curl of a vector field \( \small \mathbf{F} \) is given by the determinant of the matrix:
$$ \nabla \times \mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ F_x & F_y & F_z \end{vmatrix} $$
Where:
- \( \small \nabla \) represents the del operator,
- \(\partial \) denotes the partial derivative,
- \( \small \mathbf{i} \), \( \small \mathbf{j} \), and \( \small \mathbf{k} \) are the unit vectors in the x, y, and z directions, respectively, and
- \( \small F_x \), \( \small F_y \), and \( \small F_z \) are the components of the vector field \( \small \mathbf{F} \).
Written out, this is:
$$ \nabla \times \mathbf{F} = \left( \frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z} \right) \mathbf{i} + \left( \frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x} \right) \mathbf{j} + \left( \frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y} \right) \mathbf{k} $$
Cylindrical Coordinates #
In cylindrical coordinates, the curl of a vector field \( \small \mathbf{F} \) is given by:
$$ \nabla \times \mathbf{F} = \left( \frac{1}{\rho} \frac{\partial F_z}{\partial \varphi} - \frac{\partial F_\varphi}{\partial z} \right) \hat{\rho} + \left( \frac{\partial F_\rho}{\partial z} - \frac{\partial F_z}{\partial \rho} \right) \hat{\varphi} + \frac{1}{\rho} \left( \frac{\partial}{\partial \rho} \left( \rho F_\varphi \right) - \frac{\partial F_\rho}{\partial \varphi} \right) \hat{z} $$
Where:
- \( \small \rho \) is the distance from the z-axis, \( \varphi \) is the angle from the x-axis, and \( \small z \) is the height above the xy-plane, and
- \( \small \hat{\rho} \), \( \small \hat{\varphi} \), and \( \small \hat{z} \) are the unit vectors in the \( \small \rho \), \( \small \varphi \), and \( \small z \) directions, respectively.
Spherical Coordinates #
In spherical coordinates, the curl of a vector field \( \small \mathbf{F} \) is given by:
$$ \nabla \times \mathbf{F} = \frac{1}{r \sin \theta} \left( \frac{\partial (\sin \theta F_\varphi)}{\partial \theta} - \frac{\partial F_\theta}{\partial \varphi} \right) \hat{r} + \frac{1}{r} \left( \frac{1}{\sin \theta} \frac{\partial F_r}{\partial \varphi} - \frac{\partial (r F_\varphi)}{\partial r} \right) \hat{\theta} + \frac{1}{r} \left( \frac{\partial (r F_\theta)}{\partial r} - \frac{\partial F_r}{\partial \theta} \right) \hat{\varphi} $$
Where:
- \( \small r \) is the radial coordinate, \( \small \theta \) is the polar coordinate, and \( \small \varphi \) is the azimuthal coordinate, and
- \( \hat{r}, \hat{\theta}, \hat{\varphi} \) are the unit vectors in the \( \small r, \theta, \varphi \) directions respectively.