Curl (Real Vector Fields)#

Definition

TODO

Notation

\[\operatorname{curl} f(\boldsymbol{p}) \qquad \operatorname{rot} f(\boldsymbol{p}) \qquad \nabla \times f(\boldsymbol{p})\]

Theorem: Curl in Cartesian Coordinates

Let \(f: \mathcal{D} \subseteq \mathbb{R}^3 \to \mathbb{R}^3\) be a real vector field.

If \(f\) is totally differentiable at \(\boldsymbol{p} \in \operatorname{int} \mathcal{D}\), then its curl is given the partial derivatives of its component functions \(f_1,f_2, f_3\) as follows:

\[\operatorname{curl} f(\boldsymbol{p}) = \begin{bmatrix} \partial_2 f_3(\boldsymbol{p}) - \partial_3 f_2(\boldsymbol{p}) \\ \partial_3 f_1(\boldsymbol{p}) - \partial_1 f_3(\boldsymbol{p}) \\ \partial_1 f_2(\boldsymbol{p}) - \partial_2 f_1(\boldsymbol{p})\end{bmatrix}\]

Example: \(F(x,y,z) = \begin{bmatrix}xy & y^2 & xz\end{bmatrix}\)

Consider the real vector field \(F: \mathbb{R}^3 \to \mathbb{R}^3\) defined as follows:

\[F(x,y,z) = \begin{bmatrix}xy \\ y^2 \\ xz\end{bmatrix}\]

It is totally differentiable on \(\mathbb{R}^3\) and its curl is given by the partial derivatives of its component functions as follows:

\[\begin{aligned}\operatorname{curl} F(x,y,z) & = \begin{bmatrix}\partial_y (xz) - \partial_z(y^2) \\ \partial_z (xy) - \partial_x (xz) \\ \partial_x (y^2) - \partial_y (xy)\end{bmatrix} \\ & = \begin{bmatrix} 0 - 0 \\ 0 - z \\ 0 - x\end{bmatrix} \\ & = \begin{bmatrix}0 \\ -z \\ -x\end{bmatrix}\end{aligned}\]

Proof

TODO

Theorem: Divergence of Curl

Let \(f: \mathcal{D} \subseteq \mathbb{R}^3 \to \mathbb{R}^3\) be a real vector field.

If \(f\) is twice totally differentiable at an interior point \(\boldsymbol{p}\) of \(\mathcal{D}\), then the divergence of its curl there is zero:

\[\operatorname{div} \operatorname{curl} f(\boldsymbol{p}) = 0\]

Proof

TODO

Theorem: Stoke's Theorem

Let \(\phi: \mathcal{D}_{\phi} \subseteq \mathbb{R}^2 \to \mathbb{R}^n\) be a parametric surface which is regular except possibly on a null set of the Lebesgue measure such that the boundary of \(\phi(\mathcal{D}_{\phi})\) can be parameterized by a positively-oriented \(\partial \phi\)

\[\iint_{\phi} \operatorname{curl} f \cdot \mathrm{d}\boldsymbol{\phi} = \int_{\partial \phi} f \cdot \mathrm{d}\boldsymbol{\partial \phi}\]

Proof

TODO