Can this be solved using divergence theorem? $\iint_{S^+} dydz + dzdx +dxdy$

Question

$$ \iint_{S^+} \mathrm{d}y \, \mathrm{d}z + \mathrm{d}z \, \mathrm{d}x + \mathrm{d}x \, \mathrm{d}y $$ Bounded by $x^2+y^2=1$, $z=0$. If I use the divergence theorem (Gauss-Ostrogradsky) I get zero. But ifI solve this using the regular method I get $\pi$. The area is closed, it is a circle. Why can’t I use the divergence theorem here?