the total variation of a function $u\in L^1(\Omega)$, $\Omega\subset \mathbb{R}^n$, can be defined as
$$ \sup \{ \int_\Omega u \; \mathrm{div} g \; dx:\; g \in C_c^1(\Omega,\mathbb{R}^n), \; \lvert g(x) \rvert \leq 1,\; x \in \Omega \} $$
for (weakly) differentiable functions $u$, this supremum equals the $L^1$ norm of the (weak) gradient $\int_\Omega \lvert \nabla u \rvert\; dx$. however, i can't seem to find the rigorous argument to show this. can someone help me?