Equivalent definition of Tangent Spaces

Question

There are about 4 definitions of tangent spaces 1) using velocities of curve 2) via derivations 3)via cotangent spaces 4) as directional derivatives. I am not getting the intuition about what tangent space is and why are they equivalent.Most importantly I am having problems connecting the viwepoints.Please help me. Here is link to tangent space definition

Answer 1

The "intuition" behind tangent spaces is purely multi-variable calculus. The only new aspect introduced by differential geometry is the desire to apply the methods of multi-variable calculus in more general contexts: e.g. to take our knowledge of calculus on the Euclidean plane and apply it to calculus done on the unit sphere or a torus or some other two-dimensional manifold.

All of those things are familiar constructions from multi-variable calculus.

1. If $f$ is a curve, then $f'(0)$ is a column-vector.
2. If $v$ is a column-vector, then $\nabla_v$ is the directional derivative in its direction
3. If $w$ is a row-vector, then $wv$ is a scalar.
4. Directional derivatives have the properties of derivations

The tangent space to any point of $\mathbb{R}^n$ is just $\mathbb{R}^n$ again, viewed as the set of column vectors. (the tangent bundle is $\mathbb{R}^n \times \mathbb{R}^n$, the space that would contain the graph of any vector field)

All four of the technical definitions you cite are simply trying to pick out what a vector should be based on its properties. e.g. if you know the value $\omega v$ for every covector $\omega$, that's enough to figure out what $v$ is; and every point in the dual space to the cotangent space corresponds to a vector.