198 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.
Note: Also written \(\nabla \times \mathbf{v}\). Also written \(\operatorname{rot} \mathbf{v}\).
definition [d] (Curl = curl = rot = \(\nabla \times \mathbf{v}\)) From Kreyszig: let \(\mathbf{v}(x,y,z) = [v_{1}, v_{2}, v_{3}] = v_{1}\,\mathbf{i} + v_{2}\,\mathbf{j} + v_{3}\,\mathbf{k}\) be a differentiable vector function of the Cartesian coordinates \(x\), \(y\), \(z\). Then the curl of the vector function \(\mathbf{v}\), or of the vector field given by \(\mathbf{v}\), is defined by the symbolic determinant
- \(\displaystyle \operatorname{curl} \mathbf{v} = \nabla \times \mathbf{v} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \dfrac{\partial}{\partial x} & \dfrac{\partial}{\partial y} & \dfrac{\partial}{\partial z} \\ v_{1} & v_{2} & v_{3} \end{vmatrix}\)
which equals
- \(\displaystyle \left( \dfrac{\partial v_{3}}{\partial y} - \dfrac{\partial v_{2}}{\partial z} \right)\mathbf{i} + \left( \dfrac{\partial v_{1}}{\partial z} - \dfrac{\partial v_{3}}{\partial x} \right)\mathbf{j} + \left( \dfrac{\partial v_{2}}{\partial x} - \dfrac{\partial v_{1}}{\partial y} \right)\mathbf{k}\) .
Instead of \(\operatorname{curl} \mathbf{v}\) one also uses the notation \(\operatorname{rot} \mathbf{v}\).
where
- \(\mathbf{v}\) is a differentiable vector function.
- \(v_{1}\), \(v_{2}\), \(v_{3}\) are the Cartesian components of \(\mathbf{v}\).
- \(x\), \(y\), \(z\) are Cartesian coordinates.
- \(\operatorname{curl} \mathbf{v}\) and \(\nabla \times \mathbf{v}\) denote the curl of \(\mathbf{v}\).