Local Invertibility: Vector Fields

This lecture discusses the concept of local invertibility of vector fields, where a vector field is considered locally invertible at a point if there exists a ball around that point on which the vector field is invertible. The instructor explains that determining if a vector field is globally invertible is challenging due to the lack of simple criteria, but locally invertible vector fields simplify the analysis. The lecture also covers the Inverse Function Theorem, stating that a vector field that is continuously differentiable and has an invertible Jacobian matrix at a point is locally invertible at that point. The theorem also establishes that the inverse of such a vector field is also continuously differentiable locally.