In these lecture notes https://www3.nd.edu/~lnicolae/Lectures.pdf I read how one can define sobolev spaces between a riemannian manifold and a vector bundle over it. First I don't understand the definition that he makes of weak derivative. It goes like this
Let $u\in L^1_{loc}(E)$ and $v\in L^1_{loc}(T^*M^{\otimes m}\otimes E)$. We say that $\nabla^m u=v$ weakly if $\int_{M}\langle v,\phi \rangle dV_g=\int_M \langle u, (\nabla ^m)^* \phi\rangle dV_g $ for any $u\in C_0^{\infty}(T^*M^{\otimes m}\otimes E)$.
Where is the $\phi $ coming from ? Shouldn't it be the one that is arbitrary ? Also what is the notation $(\nabla ^m)^*$ ?
Now I am trying to work out a concrete example and I am having some trouble checking what really is the weak derivative.
Suppose we have a manifold $M$, and $x:[0,1]\rightarrow T^*M$ be a smooth map. We can define the sobolev space $W^{1,1}([0,1], x^*(T^*M))$. Now I am interested in seeing what is the weak derivative of $x$, will it be $\dot x$? I know that if we are working with sobolev spaces in $\mathbb{R}^n$ the weak derivative coincides with the usual derivative , if the latter exists, but I am not sure the same happens in this case? Will the weak derivative be $\dot x$?
Any help is appreciated , thanks in advance.