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Curl

math vector calculus

The curl measures the rotation or circulation of the vector field F \small \mathbf{F} at any given point in space.

Cartesian Coordinates #

In cartesian coordinates, the curl of a vector field F \small \mathbf{F} is given by the determinant of the matrix:

∇×F=∣ijk∂∂x∂∂y∂∂zFxFyFz∣ \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,
  • i \small \mathbf{i} , j \small \mathbf{j} , and k \small \mathbf{k} are the unit vectors in the x, y, and z directions, respectively, and
  • Fx \small F_x , Fy \small F_y , and Fz \small F_z are the components of the vector field F \small \mathbf{F} .

Written out, this is:

∇×F=(∂Fz∂y−∂Fy∂z)i+(∂Fx∂z−∂Fz∂x)j+(∂Fy∂x−∂Fx∂y)k \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 F \small \mathbf{F} is given by:

∇×F=(1ρ∂Fz∂φ−∂Fφ∂z)ρ^+(∂Fρ∂z−∂Fz∂ρ)φ^+1ρ(∂∂ρ(ρFφ)−∂Fρ∂φ)z^ \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:

  • ρ \small \rho is the distance from the z-axis, φ \varphi is the angle from the x-axis, and z \small z is the height above the xy-plane, and
  • ρ^ \small \hat{\rho} , φ^ \small \hat{\varphi} , and z^ \small \hat{z} are the unit vectors in the ρ \small \rho , φ \small \varphi , and z \small z directions, respectively.

Spherical Coordinates #

In spherical coordinates, the curl of a vector field F \small \mathbf{F} is given by:

∇×F=1rsin⁡θ(∂(sin⁡θFφ)∂θ−∂Fθ∂φ)r^+1r(1sin⁡θ∂Fr∂φ−∂(rFφ)∂r)θ^+1r(∂(rFθ)∂r−∂Fr∂θ)φ^ \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:

  • r \small r is the radial coordinate, θ \small \theta is the polar coordinate, and φ \small \varphi is the azimuthal coordinate, and
  • r^,θ^,φ^ \hat{r}, \hat{\theta}, \hat{\varphi} are the unit vectors in the r,θ,φ \small r, \theta, \varphi directions respectively.

Sources #

See also #

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