Contours of $\Psi$ on $(R, Z)$ plane are magnetic surfaces

Because $\Psi$ is constant along a magnetic field line and $\Psi$ is independent of $\phi$, it follows that the projection of a magnetic field line onto $(R, Z)$ plane is a contour of $\Psi$. On the other hand, it is ready to prove that the contour of $\Psi$ on $(R, Z)$ plane is also the projection of a magnetic field line onto the plane, i.e., a magnetic surface. [Proof. The contour of $\Psi$ on $(R, Z)$ plane is written

$$d \Psi = 0,$$ (13)

i.e.,

$$\frac{\partial \Psi}{\partial R} d R + \frac{\partial \Psi}{\partial Z} d Z = 0.$$ (14)

$$\Rightarrow \frac{1}{R} \frac{\partial \Psi}{\partial R} d R + \frac{1}{R} \frac{\partial \Psi}{\partial Z} d Z = 0.$$ (15)

Using Eqs. (4) and (5), the above equation is written

$$B_Z d R - B_R d Z = 0,$$ (16)

i.e.,

$$\frac{d Z}{d R} = \frac{B_Z}{B_R},$$ (17)

which is the equation of the projection of a magnetic field line on $(R, Z)$ plane. Thus, we prove that the contour of $\Psi$ is also the projection of a magnetic field line on $(R, Z)$ plane.]