Vector Fields and Their Flows

Vector fields are one of the most intuitive geometric objects we can study on manifolds. They assign a tangent vector to each point of the manifold, giving us a way to describe motion and change across the entire space.

The Power of Vector Fields

Vector fields appear naturally in many contexts:

• As the velocity field of a fluid
• As the gradient of a function
• As generators of symmetries
• As solutions to differential equations

Topics We’ll Cover:

1. Definition and examples of vector fields
2. Local and global flows
3. The Lie bracket
4. Complete vector fields
5. Applications to physics

Vector fields form the foundation for understanding dynamics on manifolds, making them essential tools in both pure mathematics and applications.