Surface integral on an inclined ellipse/Stokes theorem I have encountered a problem related to Stoke´s Theorem. We are given the intersection between a circular base cylinder that is parallel to the z axis (R3), and a plane that cuts it obliquely, and therefore we end up with an ellipse, that from the top looks like a circle (the contour of the cylinder). If we apply the Theorem on the intersection, we can instead of computing the line integral of a vector field we are given, compute the flux of the curl of the vector field through the surface it encloses, that in cylindrical coordinates ranges between (0,2pi) and (0,r), a fixed r, which is the same as the area of a circle. Does this imply that every cross-section of the cylinder has the same area, or where am I losing it?

You forgot the $z$ values. The intersection cannot be represented only using $x,y$, hence besides $\theta,r$, you have another $z$ component in the cylindrical system, which can be represented by the function of the plane.

What you have so far is only the projection of the surface onto $xy$-plane.