We develop a covariant formulation of classical fluid dynamics rooted in the Lagrangian perspective, using the language of exterior calculus and differential geometry. In this framework, the observer is not limited to rigid frames. In particular, we can consider observers (Lagrangian observers) whose frame of reference comoves with the fluid. For such an observer the geometry of space evolves according to the dynamics. We introduce the concepts of frame circulation and frame vorticity, which generalize inertial effects and allow for a natural classification of observers. For Lagrangian observers, the flow appears motionless while the metric evolves, leading to a set of coupled equations for the metric and frame circulation. Several exact nonlinear solutions are presented, including analogues of trochoidal and internal waves. Finally, we derive a generalized Orr–Sommerfeld equation in quasi-Lagrangian coordinates and apply it to prove the stability of Couette flow at all Reynolds numbers. This framework opens new avenues for the geometric analysis of turbulence, stability, and mixing from a fluid-following viewpoint.

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• Flow instability
• Eulerian flow field
• Lagrangian flow field