Conformal maps

Möbius transformations, the Riemann mapping theorem, and the Joukowski airfoil.

Riemann’s mapping theorem: the upper half-plane mapped conformally onto the disk

Riemann's mapping theorem: the upper half-plane mapped conformally onto the disk

The Joukowski map: turning circles into airfoils

The Joukowski map: turning circles into airfoils

Möbius transformations map circles to circles

Möbius transformations map circles to circles

Schwarz’s lemma: self-maps of the disk fixing 0 cannot expand

Schwarz's lemma: self-maps of the disk fixing 0 cannot expand