Given a force field $F:A\subseteq\mathbb{R^3}\to\mathbb{R^3}$ where A is an open connected set and Given a regular curve $\phi:[a,b]\subseteq\mathbb{R}\to\mathbb{R^3}$ such that $\phi\left(\left[a,b\right]\right)\subseteq A$ my book defines the Work $W$ of $F$ to move a particle from $\phi(a)$ to $\phi(b)$ as: $$W=\int_\phi\langle F,T\rangle dS$$ Where $\langle F(x),T(x)\rangle$ is the scalar product between $F(x)$ and the tangent versor $T(x)$ to the curve at the point $x\in\phi([a,b])$. My problem with this definition is that we defined the tangent versor $T(t)$ to the curve at a point $t\in[a,b]$ as $T(t)=\frac{\phi^{'}(t)}{\Vert\phi^{'}(t)\Vert}$ so $T$ is a function that goes from $[a,b]\subseteq\mathbb{R}$ to $\mathbb{R^3}$
while to consider the line integral $$\int_\phi f dS$$ i should have that $f$ is a function defined in $\phi\left(\left[a,b\right]\right)\subseteq\mathbb{R^3}$ to $\mathbb{R}$
but if the curve is not injective i can't consider the function $\langle F,T\rangle:\phi\left(\left[a,b\right]\right)\subseteq\mathbb{R^3}$ to $\mathbb{R}$.
So i don't get if $\int_\phi\langle F,T\rangle dS$ is an abuse of notation to mean $\int_a^b\langle F(\phi(t)),T(t)\rangle\Vert\phi(t)\Vert dt$
Or if i need to suppose that $\phi$ is injective (or something similar like that $\phi$ is simple)
EDIT: the problem is that if $\phi$ is not injective i could have that for two different points $t_0$ and $t_1$ in [a,b] i have that $\phi(t_0)=\phi(t_1)$ and if $T(t_0)\neq T(t_1)$ i can't consider a function that for every $x\in \phi ([a,b])$ gives me the tangent versor to the curve $\phi$ at the point $x$. (because for the point $\phi(t_0)=\phi(t_1)$ i have two possible tangent versors to consider)
Your assertion
Is not correct. A line integral
$$\int_\phi f dS$$ is perfectly defined for $f : \mathbb R^3 \to \mathbb R$ by $$\int_a^b f(\phi(t)) \Vert \phi^\prime(t) \Vert dt.$$