I'm trying to find a mapping $f$ from the 2D real projective plane to $\mathbb{R}^3$ which

1. is smooth

2. has non-singular directional derivative. That is,

   $\forall x, v, \quad v \ne 0 \implies D_v f(x) \ne 0$.

This is different from an embedding, which is impossible because the surface would have to intersect itself. Here, I am allowing this intersection.

1 and 2 are possible for at least one other non-orientable manifold—the Klein bottle. Is there some simple way of bending the real projective plane such that it is smooth and has non-singular derivative in $\mathbb{R}^3$, like the Klein bottle?