Skip to main content

Mathematics for Physics

308 Curl

A curl is a vector calculated from the derivative of a mapping of vectors that is used to calculate the change in a vector value within a neighborhood.

definition [d] (Curl = curl = rot = \(\nabla \times \mathbf{F}\)) From Kreyszig: let \(\mathbf{F}(x,y,z) = [F_{x}, F_{y}, F_{z}] = F_{x}\,\mathbf{i} + F_{y}\,\mathbf{j} + F_{z}\,\mathbf{k}\) be a differentiable vector function of the Cartesian coordinates \(x\), \(y\), \(z\). Then the curl of the vector function \(\mathbf{F}\), or of the vector field given by \(\mathbf{F}\), is defined by the symbolic determinant

  • \(\displaystyle \operatorname{curl} \mathbf{F} = \nabla \times \mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \dfrac{\partial}{\partial x} & \dfrac{\partial}{\partial y} & \dfrac{\partial}{\partial z} \\ F_{x} & F_{y} & F_{z} \end{vmatrix}\)

which equals

  • \(\displaystyle \left( \dfrac{\partial F_{z}}{\partial y} - \dfrac{\partial F_{y}}{\partial z} \right)\mathbf{i} + \left( \dfrac{\partial F_{x}}{\partial z} - \dfrac{\partial F_{z}}{\partial x} \right)\mathbf{j} + \left( \dfrac{\partial F_{y}}{\partial x} - \dfrac{\partial F_{x}}{\partial y} \right)\mathbf{k}\) .

Instead of \(\operatorname{curl} \mathbf{F}\) one also uses the notation \(\operatorname{rot} \mathbf{F}\).

where

  • \(\mathbf{F}\) is a vector field.
  • \(F_{x}\), \(F_{y}\), \(F_{z}\) are the Cartesian components of \(\mathbf{F}\).
  • \(x\), \(y\), \(z\) are Cartesian coordinates.
  • \(\operatorname{curl} \mathbf{F}\) and \(\nabla \times \mathbf{F}\) denote the curl of \(\mathbf{F}\).