What Are Differential Forms and How Do They Apply in Differential Geometry?

[fisico30]

Thanks mathwonk!

I am used to the idea of differential as an infinitesimal delta of something: dx is the infinitesimal distance, dq the infinitesimal amount of charge...
Infinitesimal intended as something infinitesimally small, but always bigger than zero.

Q1: In differential geometry, the differential changes meaning:
is is true that it represents a unit tangent vector?
is it a functional (operates on a function and outputs a number)?
why its symbol is dx? how is it related to the infinitesimal, differential in calculus?

Q2:p-forms. I am not sure I understand what they are. Could you provide a baisc explanation/example of 1-form and 2 form?

Q3: Manifold: is it correct to say that a manifold is a space, flat or curved?Lethe explains that a surface (a sphere) is a curved 2-manifold. how do we figure out if a space is curved or flat? Is there a curvature function related to the metric distance?
But, what is a space, first of all? For instance, could the electric field, which is a vector field be called a manifold? If so, why?
It seems that a manifold is a general, abstract concept applicable to many things which satisfy a certain criterion of membership But what is that criterion?
(like anything can be a vector, as long as it satisfies those 10 rules that make it be a vector...)

thanks