External unit normal field of cylinder

External unit normal field of cylinder

Hello, I consider $$ A:=\left\{(x,y,z)\in\mathbb{R}^3:x^2+y^2\leq 1,0\leq
z\leq 1\right\} $$ and $$ v(x,y,z):=(x^3,x^2y,zx^2). $$ and the task is to
calculate the surface of $A$ by using the integral theorem of Gauß.
I guess it is meant to use $$ \int\limits_{\partial A}\langle v,w\rangle\,
dS=\int\limits_A\mbox{div } v\, d^3x, $$ where $w$ is the external unit
normal field of $\partial A$.
Am I right that the external normal field of $\partial A$ is given by
$w(x)=e_3$, when $x$ is in the "cover" of the cylinder $A$, $w(x)=-e_3$,
when $x$ is in the "ground" of the cylinder "A" and $w(x)=(x,y,0)$ is $x$
is the rest of the cylinder $A$?