Differential Geometry 06 : vectors - part 1

In this video we investigate objects that are arguably the most important things in differential geometry - vectors! We begin with a slightly informal overview of the three different definitions we have for vectors - for want of better names we have referred to them as the geometer's, the algebraist's, and the physicist's definition. We describe how our high school notions of displacement and velocity vectors in 3D can be extended to these ideas that work in more general manifolds. After this introduction, we focus on the algebraists definition of a tangent vector as a linear functional tied to a point. We begin by defining the linear functional tied to the tangent of a curve and then go on to the general definition of a tangent vector. We show that you can define addition and multiplication by a scalar on the collection of all tangent vectors at a point - leading to the linear vector space called the tangent plane. We also show why vectors at different points can not be added or subtracted - and thus noted that differentiation of vector fields on a general manifold requires additional structure.

[harshit]

Hlo sir,
Actually I am a master student and in my college a course on General Theory of Relativity is going on and Ihave opted the course and I am facing a lot of difficulty in that subject due to the advance mathematics part so could u plz tell whether first I should go through the lectures on Topology or I should start direct Differntial Geometry and your course on Tensor Analysis is sufficient or not?
Plz Sir help me and plz reply the msg.