The curl of a vector field
Definition: the Levi-Civita permutation symbol is defined as
\[ e_{ijk} = \begin{cases} 0 &\text{ if $i,j,k$ are identical}, \\ 1 &\text{ if the permutation $(i,j,k)$ is even}, \\ -1 &\text{ if the permutation $(i,j,k)$ is odd}.\end{cases} \]
The curl of a vector field may describe the circulation of a vector field and is defined below.
Definition: derivation and definition is missing for now.
Note that the "cross product " between the nabla operator and the vector field \(\mathbf{v}\) does not imply anything and is only there for notational sake. An alternative to this notation is using \(\text{rot } \mathbf{v}\) to denote the curl or rotation.
Theorem: the curl of a vector field \(\mathbf{v}: \mathbb{R}^3 \to \mathbb{R}^3\) for a curvilinear coordinate system is defined as
\[ \nabla \times \mathbf{v}(\mathbf{x}) = \frac{1}{\sqrt{g(\mathbf{x})}} e^{ijk} \partial_i \big(v_j(\mathbf{x}) \big) \mathbf{a}_k(\mathbf{x}), \]for all \(\mathbf{x} \in \mathbb{R}^3\).
Proof:
Will be added later.
The curl of a vector field for a ortho-curvilinear coordinate system may also be derived and can be found below.
Corollary: the curl of a vector field \(\mathbf{v}: \mathbb{R}^3 \to \mathbb{R}^3\) for a ortho-curvilinear coordinate system is defined as
\[ \nabla \times \mathbf{v}(\mathbf{x}) = \frac{1}{h_1 h_2 h_3} e^{ijk} \partial_i \big(h_j v_{(j)}(\mathbf{x}) \big) h_k \mathbf{e}_{(k)}, \]for all \(\mathbf{x} \in \mathbb{R}^3\).
Proof:
Will be added later.
Please note that the scaling factors may also depend on the position \(\mathbf{x} \in \mathbb{R}^3\) depending on the coordinate system.
Proposition: let \(\mathbf{v}: \mathbb{R}^3 \to \mathbb{R}^3\) be a vector field and \(f: \mathbb{R}^3 \to \mathbb{R}\) a scalar field then we have
\[ \begin{align*} \nabla \cdot \big(\nabla \times \mathbf{v}(\mathbf{x}) \big) &= 0, \\ \nabla \times \nabla f(\mathbf{x}) &= \mathbf{0}, \end{align*} \]for all \(\mathbf{x} \in \mathbb{R}^3\).
Proof:
Will be added later.
Similarly to the divergence theorem for the divergence, the curl is related to Kelvin-Stokes theorem given below.
Theorem: let \(\mathbf{v}: \mathbb{R}^3 \to \mathbb{R}^3\) be a smooth vector field and \(A \subset \mathbb{R}^3\) a closed surface with boundary curve \(C \subset \mathbb{R}^3\) piecewise smooth we have that
\[ \oint_C \big\langle \mathbf{v}(\mathbf{x}), d\mathbf{x} \big\rangle = \int_A \big\langle \nabla \times \mathbf{v}(\mathbf{x}), d\mathbf{A} \big\rangle, \]is true.
Proof:
Will be added later.