mathematics-physics-wiki/docs/en/physics/mathematical-physics/vector-analysis/curl.md

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.

??? note "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.

??? note "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.

??? note "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.

??? note "Proof:"

Will be added later.