Cfrgtkky

Let $(M,g)$ be a smooth compact Riemannian manifold, and let $f in W^{1,p}(M)$ for $pge 1$. (I don't assume $p>dim M$).

I have seen in various sources that people refer to the weak derivative of $f$ as a linear functional $T_pM to mathbb{R}$, which is defined for almost every $p in M$. (an a.e.defined section of $T^*M$).

How exactly is this object defined? I couldn't find any precise details about this.

I define $W^{1,p}(M)$ to be the completion of the space of compactly supported smooth functions $C_c^{infty}(M)$ w.r.t the $|cdot|_{1,p}$ norm.

Optional: I suggest below $2$ approaches; I would like to know if they are compatible, i.e. if they both produce the same element in $(T_pM)^*$.

(Regarding the second approach, I am not even sure if it produces a well-defined functional).

Approach 1:

Given $f in W^{1,p}(M)$, there exist $f_n in C_c^{infty}(M)$, $f_n to f$ in $W^{1,p}$.

$df_n in Gamma(T^*M)$ is a Cauchy sequence in $L^p(M,T^*M)$, where $L^p(M,T^*M)$ is the completion of the space of smooth sections $Gamma(T^*M)$ w.r.t the natural $p$-norm. By completeness, $df_n$ converges to an element in $L^p(M,T^*M)$, which we can realize as a measurable section $T^*M$. We set $df=lim_{n to infty} df_n$.

Approach 2 ("Local picture"):

Let $phi:Usubseteq M to mathbb{R}^n$ be a surjective coordinate chart around $p in M$, and $phi(p)=0$. Set $f_{phi}=f|_U circ phi^{-1} :mathbb{R}^n to mathbb{R}$. Then $f_{phi} in W^{1,p}(mathbb{R}^n)$ (we might need to shrink $U$ to ensure nothing will explode). We define $df_p$ by the equation

$$ df_p circ d(phi^{-1})_0(e_i):= d(f_{phi})_0(e_i)=(partial_i f_{phi})(0). tag{1}$$

Does equation $(1)$ well-defines an element in $T_p^*M$ independently of the coordinate chart? Does it coincide with $lim_{n to infty} df_n$ from the previous approach?

You have a notion of weak derivative in this case: An almost everywhere section of $T^*M$, denoted $df$, is a weak derivative of $f$ if for every smooth, compactly supported 1-form $phi$ you have
$$
int_M g(df,phi) ,text{dVol} = int_M f,deltaphi ,text{dVol}.
$$
where the pairing is with respect to the metric $g$.
A function is in $W^{1,p}(M)$ if it is in $L^p$, and has a weak derivative in $L^p(T^*M)$.
Since this definition of weak derivative is consistent with taking limits in $L^p$, it is consistent with your approach 1.
I believe that taking $phi$ to be supported in a coordinate patch and writing the definition above in coordinates would just yield your approach 2.
The question why $W^{1,p}$ as defined here is actually the same as the completion should be very similar to the same question in Euclidean spaces (the classic $H=W$ question).