Ueber Riemann's Theorie der Algebraischen Functionen

In this work the author begins by showing how the mathematics of complex functions can be visualized as steady‑state fluid flows in a plane. By treating the real part of a function as a velocity potential, the accompanying imaginary part naturally yields the orthogonal streamlines, turning abstract equations into a vivid picture of liquids sliding between parallel sheets or spreading as a thin membrane. The text walks the listener through the basic differential relations, explains the invariance under scaling and rotation, and introduces the notion of conjugate flows that mirror each other’s behavior.

The early chapters focus on the special points where the flow becomes singular, describing how these “intersection points” arise and how their local structure can be uncovered using power‑series expansions and polar coordinates. Through clear examples and careful diagrams, the author lays the groundwork for Riemann’s broader theory of algebraic functions, inviting listeners to see the deep link between physics and pure mathematics before the more intricate geometry of later sections unfolds.