What's is the use of field lines of vector field? I have this equation F( p(t)) = p'(t), where F is a functions of a vector field and p is a curve. My teacher said this equation gives us a curve that coincides with the vector field. My question is once I found this curve, what can I do with it to take advantage of this property? and what would a curve other than p '(t) be useful for?
 A: Field lines are a peculiarity of smooth vector fields. They divide the space into non-intersecting curves in a way that is called a "foliation". These arise as solutions to differential equations.
Field lines/flow curves allow a more general treatment of derivatives over manifolds, by defining "a flow along which to derive". The core problem of multivariable calculus being the generalization of the derivative: when you have one dimension, there is no ambiguity, only one direction (with a given orientation) along which to derive; when you have multiple dimensions, you need to build a system that can represent "derivation along any oriented 1D line in an nD space".
Using flow curves is a way of cleanly defining "a specific direction in which to derive at every point $p$ on a manifold". "Cleanly" as in: it'll have some nice properties that allow a better understanding of the vector field as a whole on the manifold. This scheme is the core of using Lie algebras of vector fields, in the domain of differential geometry and for differential equations.
If you want to gain some intuition into why this is the case, I suggest you check this post, with an answer of mine: (gradient: field of tangent vectors vs. normal to surface at a point).
It'll give you some insight into how one goes from multivariable calculus to differential geometry. Though you'll probably need a more formal treatment of the subject to really understand the algebraic depth of the matter. Check out eigenchris on YouTube, he's a good resource as well if this interests you and you're just starting out.
