From b35439f3b658907df36952abcbc3d1806818169b Mon Sep 17 00:00:00 2001
From: Luc <luc@bijl.us>
Date: Thu, 25 Jan 2024 16:32:29 +0100
Subject: [PATCH] Finished curl section in vector analysis.

---
 .../vector-analysis/curl.md                   | 72 +++++++++++++++++++
 .../vector-analysis/divergence.md             |  4 +-
 .../vector-analysis/vector-operators.md       |  2 +
 3 files changed, 76 insertions(+), 2 deletions(-)

diff --git a/docs/en/physics/mathematical-physics/vector-analysis/curl.md b/docs/en/physics/mathematical-physics/vector-analysis/curl.md
index e69de29..c63f7ce 100644
--- a/docs/en/physics/mathematical-physics/vector-analysis/curl.md
+++ b/docs/en/physics/mathematical-physics/vector-analysis/curl.md
@@ -0,0 +1,72 @@
+# 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](divergence.md#divergence-in-curvilinear-coordinates) 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 valid. 
+
+??? note "*Proof*:"
+
+    Will be added later.
\ No newline at end of file
diff --git a/docs/en/physics/mathematical-physics/vector-analysis/divergence.md b/docs/en/physics/mathematical-physics/vector-analysis/divergence.md
index 1528c18..dd36296 100644
--- a/docs/en/physics/mathematical-physics/vector-analysis/divergence.md
+++ b/docs/en/physics/mathematical-physics/vector-analysis/divergence.md
@@ -104,10 +104,10 @@ Assuming the system is stationary the time derivative of the particles inside th
 
 We may also give the divergence for ortho-curvilinear coordinate systems.
 
-> *Corollary*: the divergence of a flux density $\mathbf{\Gamma}: \mathbb{R}^3 \to \mathbb{R}^3$ for a ortho-curvilinear is given by
+> *Corollary*: the divergence of a flux density $\mathbf{\Gamma}: \mathbb{R}^3 \to \mathbb{R}^3$ for a ortho-curvilinear coordinate system is given by
 >
 > $$
->   \nabla \cdot \mathbf{\Gamma}(\mathbf{x}, t) = \frac{1}{h_1h_2h_3} \partial_i \Big(\Gamma^i(\mathbf{x},t)\frac{1}{h_i} h_1 h_2 h_3 \Big)
+>   \nabla \cdot \mathbf{\Gamma}(\mathbf{x}, t) = \frac{1}{h_1h_2h_3} \partial_i \Big(\Gamma^{(i)}(\mathbf{x},t)\frac{1}{h_i} h_1 h_2 h_3 \Big)
 > $$
 >
 > for all $\mathbf{x} \in \mathbb{R}^3$ and $i \in \{1, 2, 3\}$.
diff --git a/docs/en/physics/mathematical-physics/vector-analysis/vector-operators.md b/docs/en/physics/mathematical-physics/vector-analysis/vector-operators.md
index e69de29..51e806e 100644
--- a/docs/en/physics/mathematical-physics/vector-analysis/vector-operators.md
+++ b/docs/en/physics/mathematical-physics/vector-analysis/vector-operators.md
@@ -0,0 +1,2 @@
+# Vector operators
+