I recently studied Stoke's Theorem for differential forms on manifolds in Loring Tu's, An Introduction to Manifolds. As an immediate corollary he shows how this trivialises the proof for the FTLI. I have a query concerning the proof of this corollary, because I believe that Tu states something, which to him is obvious but most probably isn't fully trivial, and as I am self studying differential geometry I would like to clarify my understanding.
Tu's Proof: Let $C$ be a smooth curve in $\mathbb R^3$ parameterized by $\textbf r(t)=(x(t),y(t),z(t))$, $a\leq t \leq b$, and $f:\mathbb R^3\to \mathbb R$. a smooth scalar field. The exact formulation of the Theorem Tu proves is: $$\int_C \nabla f\cdot d\textbf r=f(\textbf r(b))-f(\textbf r(a)).$$We take $C$ to be our manifold, and $f$ to be our $0$-form on $C$. From Stoke's Theorem $$\int_C df =\int_{\partial C}f,$$ and it is apparent that the right hand side equals $f(\textbf r(b))-f(\textbf r(a))$. Finally $$\int_C df=\int_C \frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy+\frac{\partial f}{\partial z}dz,$$ which gives the result.
My confusion stems from the final equality. To express the exterior derivative of $f$ on $C$ in local coordinates we first need to choose some chart on $C$, and Tu's proof suggests that he's chosen the standard chart for $\mathbb R^3$, however $C$ is clearly $1$-dimensional, so this cannot be the case. Now $C$ is actually an immersed submanifold in $\mathbb R ^3$, with inclusion map $\textbf r$ I believe (not sure though). This leads me to that believe what Tu means by the coordinates $x,y,z$ in the last equality is actually $\textbf r^*x,\textbf r^*y,\textbf r^*z$ with these $x,y,z$ the standard coordinates on $\mathbb R^3$ (Taking into account md2perpe's comment, maybe in fact Tu's co-ords should be $i^*x, i^*y, i^* z$, with $i$ the restricted identity function). This is all very fuzzy to me though, and I lack a feeling for what exactly is going on here, so any explanation about the exact nature of Tu's coordinate choice from someone more knowledgeable would be very appreciated.
References
Loring Tu. An Introduction to Manifolds. Springer, 2nd edition, 2011