In the book I'm reading, the Stokes' Theorem is stated as follows:
Let $\Omega\subset\mathbb{R}^2$ be open, bounded and of class $C^1$. Let $V\subset\mathbb{R}^3$ be an open set. Let $f:V\rightarrow\mathbb{R}^3, f=(f_1,f_2,f_3)$ be a vector field of class $C^1$ over $V$. Let $\gamma:[0,1]\rightarrow\mathbb{R}^2$ be a closed Jordan curve of class $C^1$. Let $F:U\rightarrow\mathbb{R}^3$ be an immersion** of class $C^2$, where $U\subset\mathbb{R}^2$ open such that $\overline{\Omega}\subset U$. If $\Gamma=F\circ\gamma$, $S=F(\overline{\Omega})$ and $\partial S=F(\partial\Omega)$, then $$\int_{S}\langle rotf(\sigma):\upsilon(\sigma)\rangle d\sigma=\int_{\partial S}f\cdot d\Gamma$$
Here, $\upsilon(\sigma)$ is the unitary vector normal to $S$ (I think), and $rot f$ is the curl of $f$.
**I'm not sure "immersion" is the correct translation, but the properties I want $F$ to have is be injective, differentiable in $U$, $\forall u\in U$$F'(u)$ is injective and $F$ is a function of class $C^2$.
What I want to know is if this theorem written correctly. More specifically, I want to know if we need to assume anything else about $F$ and $\gamma$.
By the definition of $\Gamma$, it seems we must choose $\gamma$ such that $\gamma([0,1])\subset U$, even though it is not clearly stated. Do we need to choose $\gamma$ this way for the theorem to make sense? And if so, did the author need to state it clearly for it to be correct or is it correct the way it is written?
Also, the first step of the proof is $\int_{\partial S}f\cdot d\Gamma=\int_0^1\langle f(\Gamma(t)):\Gamma'(t)\rangle dt$. Do we also need to choose $F$ such that $F(\gamma([0,1]))\subset V$, and if so, did the author need to state it more clearly? Or am I seeing problems where there are none?