Riemann Mapping Theorem

Boeing uses conformal maps for wing profiles: the Joukowski transform underlies the aerodynamics of the 737 MAX, designed in the 1960s without CFD.

• **Aerodynamics:** Joukowski transform gives analytic flow solution around any airfoil via the cylinder solution.
• **Electrostatics:** conformal maps solve Laplace problems for complex geometries (capacitors, electrodes).
• **Quantum conductance (NIST):** Hall resistance on graphene computed via conformal mapping (van der Pauw method).

Conformal Maps

Boeing uses conformal maps to design airfoil profiles: the Joukowski transform converts the analytically solvable flow around a cylinder into flow around a wing. The 737 MAX profile was developed using this method as far back as the 1960s.

Riemann Mapping Theorem

In 2D quantum conductance measurements of graphene (NIST 2019), the Hall resistance is computed via conformal mapping of an arbitrary sample plate onto a standard square. The Riemann mapping theorem guarantees the existence of such a map.

Key ideas

• **Conformality:** analytic f with f'(z0)!=0 preserves angles. Scale |f'(z)|, rotation arg f'(z).
• **Joukowski transform:** w=z+R^2/z maps circles to airfoil profiles. Mobius: circles map to circles.
• **Riemann mapping theorem:** any simply connected U (strictly in C) is conformally equivalent to the disk. Normalization gives uniqueness.