I know about Wolfram, but in my opinion it isnt too helpful because all it really does is explain it in such a way that they assume you have a frickin PhD in math. Im starting to tie the pieces together now, but I must truthfully say that Vector Calculus is the most obscure stuff that ive ran into.
This is what I gather from what ive been doing so far. The Vector Field is pretty much an environment. It describes the environment that an object is in such as an explosion, a tornado, a stream, etc. It would be used for things such as air flow, hydrodynamics, magnetic fields, etc. Now, you have an object within this environment, and you want to see what the characteristics are when this object is in that environment. Its shows thing like overall force applied on the object, amount of work done to move it from a to b, and so fourth. Also, it shows how to use differential equations to find potential functions, and in turn these are used to simplify the line integrals. This is what ive got so far. If this sparks a memory, by all means say something because an input is helpfull.