Show that the level-set $S_F(0)$ $$F: \Bbb R^3 \to \Bbb R, F(x,y,z) = 3xz+e^{y}+ye^z$$ is a smooth $2$-dimensional surface.
Computing the gradient I get that $\nabla F(x,y,z)=(3z,e^y+e^z, 3x+ye^z)$. However, I cannot seem to find the definition to show that this would be smooth. Should I consider the implicit function theorem here to compute the Jacobian of the parametrization and look at the rank of the matrix?