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 it is invertible within a ball centered at that point. The instructor explains the conditions for a vector field to be locally invertible and the implications of its inverse being C¹. The lecture also covers the Inverse Function Theorem, stating that a vector field that is C¹ and has an invertible Jacobian matrix is locally invertible. Various examples are provided to illustrate the criteria for invertibility and non-invertibility of vector fields.