Let $S=\{(x,y,z)|x^2+y^2\leq1,z=1+xy+x^2\}$. Now I want to calculate $\int_S<V,n>do$ for a given $V$. For this I need to calculate the normal unit vector. To get the direction I can take the gradient of $f=z-1-xy-x^2$. The proper way is to use the parameterisation $p=(x,y,1+xy+x^2)$ and just take the appropriate cross product. Now in this case it happens that the gradient of $f$ allready has the correct magnitude.
My question: Is there a (usable) parametrisation $P$ for a given surface (submanifold) $S$ for which the magnitude of $\nabla f$ is allready "correct" (i.e. if $f^{-1}(0)=S=P(A)$ then $\int_S<V,n>do=\int_A<V\circ P,\nabla f\circ P>d\mu$ for a given $V$)?
Maybe this is trivial or very badly written, if so let me know so I can edit!
A standard formula for a vector normal to the surface is:
$dS = $$<-\frac{\partial z}{\partial x},-\frac{\partial z}{\partial y}, 1> \:dx \:dy\\ <-y-2x,-x,1>\: dx \:dy$
And then since this is a surface bound by a cylinder, it is likely that you will what to covert to cylindrical.
$r<-r\sin t-2r \cos t,-r \cos t,1>\: dr \:dt$
Whether you actually need to unitize it depends on the problem. Usually you don't.