Curl

Theorem: Curl in Cartesian Coordinates

Let $\mathbf{v}:\mathcal{D}\subseteq {\mathbb{R}}^3\to {\mathbb{R}}^3$ be a real vector field and let $\mathbf{p}\in {\mathbb{R}}^3$.

If $\mathbf{v}$ is differentiable at $\mathbf{p}$, then its curl there can be expressed via the partial derivatives of its component functions $v_1,\dots ,v_n$ with respect to Cartesian coordinates as follows:

$$\nabla \times \mathbf{v}(\mathbf{p})=\left[\begin{array}{c}\frac{\partial v_3}{\partial y}(\mathbf{p})-\frac{\partial v_2}{\partial z}(\mathbf{p})\\ \frac{\partial v_1}{\partial z}(\mathbf{p})-\frac{\partial v_3}{\partial x}(\mathbf{p})\\ \frac{\partial v_2}{\partial x}(\mathbf{p})-\frac{\partial v_1}{\partial y}(\mathbf{p})\end{array}\right]$$

PROOF

TODO