I know that physically, they describe relationships whereby, for instance a vector field, for each point in three dimensional space (a "vector"), we have a "vector" which has a direction or magnitude. Now I once asked what the difference between a vector field and a vector function is and the definition was given below: The problem is, I still can't picture it for a simple vector field: V(x,y) = u(x,y)i + v(x,y)j Someone later on did explain what the definition meant but I still can't put two and two together. So what is a tangent bundle and what is a vector (or topological) space in relation to a fluid flow. In terms of physics, it sounds to be like we have a global fixed reference frame which has its own x,y,z co-ordinate system and then we have a local co-ordinate frame which is, for this example, attached to a fluid element which then has a vector associated with it. But then why do we need a manifold and why do we need things to be tangent to it? If indeed the vector field maps from a manifold to some vector space, why do the vector spaces have to be attached to the manifold, because wouldn't that mean we can't place our global reference system anywhere in the flow field? Finally, why is the tangent bundle (presumably the collection of all the tangent or vector spaces) not a vector space in itself? So following on from that, can anyone give the precise definition for scalar fields and scalar functions and then tell me why they are different (mathematically rather than saying it doesn't have a direction etc). PS: I know there is some 2007 thread in these forums but I didn't really get it and its locked anyway.