The curl flux integrand of a $C^2$ patch is a two-dimensional curl of the pulled-back field

Given: The open sets $O\subseteq {\mathbb{R}}^2$ and $U\subseteq {\mathbb{R}}^3$, the $C^2$ map $\phi :O\to {\mathbb{R}}^3$ with $\phi [O]\subseteq U$, and the $C^1$ field $F:U\to {\mathbb{R}}^3$.

In the present local setting, define the pulled-back functions directly by $P^{\ast }=⟨F\circ \phi ,{\phi }_u⟩$ and $Q^{\ast }=⟨F\circ \phi ,{\phi }_v⟩$ on $O$. For a regular patch over a finite elementary Green region these agree with the notation of The induced boundary chain and circulation of a $C^2$ patch over a finite elementary Green region.

For $x,y\in {\mathbb{R}}^m$, $⟨x,y⟩={\sum }_{i<m}{x}_i{y}_i$ (The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$).

For $u,v\in {\mathbb{R}}^3$, $u\times v=(u_y{v}_z-u_z{v}_y,u_z{v}_x-u_x{v}_z,u_x{v}_y-u_y{v}_x)$ (The cross product in ${\mathbb{R}}^3$), and the curl of a $C^1$ field is $\mathrm{curl}F=({\mathrm{\partial }}_y{F}_z-{\mathrm{\partial }}_z{F}_y,{\mathrm{\partial }}_z{F}_x-{\mathrm{\partial }}_x{F}_z,{\mathrm{\partial }}_x{F}_y-{\mathrm{\partial }}_y{F}_x)$ (Divergence and curl of a $C^1$ vector field).

A map is of class $C^k$ when each component is ($C^k$ Euclidean maps and diffeomorphisms), and a scalar is $C^k$ when every iterated derivative of length at most $k$ exists and is continuous ($C^k$ maps and multi-index derivative notation in Euclidean space).

If every partial derivative ${\mathrm{\partial }}_j{f}_i(a)$ exists, the Jacobian matrix is $Jf(a)=({\mathrm{\partial }}_j{f}_i(a){)}_{i<n,j<m}$ (The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case).

For a $C^1$ field $F$ on an open $U\subseteq {\mathbb{R}}^3$, a point $p\in U$ and $u,v\in {\mathbb{R}}^3$, $⟨DF(p)u,v⟩-⟨DF(p)v,u⟩=⟨\mathrm{curl}F(p),u\times v⟩$ (The curl measures the antisymmetric part of the total derivative).

If $f$ is $C^2$ on an open subset of ${\mathbb{R}}^m$, then ${\mathrm{\partial }}_i{\mathrm{\partial }}_{j}f={\mathrm{\partial }}_j{\mathrm{\partial }}_{i}f$ for every pair of coordinate indices (Clairaut--Schwarz theorem for continuous second partial derivatives).

If $f$ is totally differentiable at $a$ and $g$ at $f(a)$, then $D(g\circ f)(a)=Dg(f(a))\circ Df(a)$ (The chain rule for total derivatives: $D(g\circ f)(a)=Dg(f(a))\circ Df(a)$).

If $f$ is totally differentiable at $a$ then $D_{w}f(a)=Df(a)w$ for every $w$, and the matrix of $Df(a)$ is $Jf(a)$ (A total derivative computes every directional derivative, and its matrix is the Jacobian).

If every partial derivative of $f$ exists on a neighbourhood of $a$ and is continuous at $a$, then $f$ is totally differentiable at $a$ with $Df(a)$ the linear map of matrix $Jf(a)$ (If all partial derivatives exist on a neighbourhood and are continuous at a point, then the map is totally differentiable there with Jacobian derivative).

For real functions of one real variable differentiable at a point, $f+g$ is differentiable there with $(f+g{)}^{\mathrm{\prime }}=f^{\mathrm{\prime }}+g^{\mathrm{\prime }}$ and $fg$ is differentiable there with $(fg{)}^{\mathrm{\prime }}=f^{\mathrm{\prime }}g+f{g}^{\mathrm{\prime }}$ (Sums, scalar multiples, products and quotients: $(f+g{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)+g^{\mathrm{\prime }}(c)$, $(\alpha f{)}^{\mathrm{\prime }}(c)=\alpha f^{\mathrm{\prime }}(c)$, $(fg{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)g(c)+f(c)g^{\mathrm{\prime }}(c)$, and $(f\mathrm{/}g{)}^{\mathrm{\prime }}(c)=(f^{\mathrm{\prime }}(c)g(c)-f(c)g^{\mathrm{\prime }}(c))\mathrm{/}g(c{)}^2$ when $g(c)\mathrm{\ne }0$).