Differentiating Vector Fields: How Not to Do It

This lecture discusses the differentiation of vector fields on embedded submanifolds, focusing on the challenges and incorrect approaches. The instructor explains the concept of smooth vector fields, extensions, and the computation of gradients. Through examples on Riemannian submanifolds, the lecture demonstrates the computation of gradients and the importance of choosing the right differentiation method. The discussion extends to the projection of vectors and the implications for different types of manifolds.