I'm studying an introduction to Sobolev spaces and I found this remark which I can't prove:
If $u\in\mathcal{C}^1(\Omega)$, then $u$ admits weak gradient $Du$ and $Du=\nabla u$ a.e. in $\Omega$.
So I have two prove two things:
If $u\in\mathcal{C}^1(\Omega)$, then $u$ admits weak gradient: indeed if $u\in\mathcal{C}^1(\Omega)\subset\mathcal{C}^0(\Omega)$, which is dense in $L^2(\Omega)$, then $u\in L^2(\Omega)$ (but at this point I'd like to prove $u\in H^1_0(\Omega)$ -by this I denote the Sobolev space wrt the Dirichlet homogeneous problem-, and by definition it follows $u$ admits weak gradient, but I don't know how to conlcude);
$Du=\nabla u$ a.e. in $\Omega$: I have no idea. I know I have to prove $$\int_{\Omega} w_i\phi dx=- \int_{\Omega}u\frac{\partial \phi}{\partial x_i} dx, $$ where $w_i$ is the $i^{th}$- component of $Du$, and $\phi\in\mathcal{C}_c^{\infty}(\Omega)$, but apart from this I don't weven know what to write or which result I should use.
I know I should post my attempts, but I'm quite new to the subject and I havr many doubts. Any hint or advice would be much appreciate, thanks in advance-