Understanding the definition and meaning of cotangent space

Question

I would like to understand definition and also meaning of cotangent space. Let's first of all define tangent space: generally we know that equation of tangent line of function $f(x)$ at point $x_0$ is equal to

$$y=f(x_0)+f'(x_0)*(x-x_0)$$

Is tangent space somehow related to tangent lines? For example, family of tangent lines? According to Wikipedia, cotangent space is a dual of tangent space; as I know duality for example in vectors means that take a vector and produce scalar, general definition of tangent space is given on the following mathematics site

http://mathworld.wolfram.com/TangentSpace.html

According to this, I understood that if we have point $x$ in compact manifold $M$, and if we attach at $x$ a copy of $n$-dimensional real space as a tangent to $M$, then resulting space is tangent space, so does it means that at given point $x$, we are attaching $\mathbb{R}^n$ space as a tangent to $M$? If so, then what does cotangent do? Does it take vector from tangent space and produce scalar?

Answer 1

One possible way of defining tangent space to a manifold is to say that it is the possible values of tangencies that a path $\gamma(t)$ in the space can have. i.e. it is the vector space of all possible derivatives of maps from the interval into the manifold.

If you consider a surface embedded in $\mathbb{R}^3$ (picture a sphere or something simple... like a soccer ball), and you imagine an ant who is condemned to live on the surface of the ball. Consider all velocity vectors the ant is allowed to have when he passes through a specific spot on the ball. This gives you the tangent space at that point.

The cotangent bundle is indeed the dual space to this, but I often find it hard to imagine dual spaces. Instead, I would think of the cotangent bundle as being where differentials of functions live. Picture a function defined on the sphere -- maybe the temperature at that location. The derivative of this function at a given point is something that eats a direction and spits out a number -- this is the infinitesimal change of temperature as you consider displacing the point an infinitesimal amount in that direction.

In other words, the tangent bundle consists of values of derivatives of maps $\gamma \colon \mathbb{R} \to M$ and the cotangent bundle tells you about differentials of maps $f \colon M \to \mathbb{R}$.