Introduction to differentiable manifold – formal definition of differentiable manifold

See Manifold

A differentiable structure is what we need to add to a set in order to do differential Calculus on it. In particular differentiation (derivatives) of functions defined on the set. It is what it's needed to define differentiability of functions in a consistent way. It also allows to do Integration (what do you integrate? Differential forms)

Lecture notes

Coordinate chart

Diffeomorphism

Tangent space

Cotangent space

Some common manifolds. The differentiable structure induces a topology (Topology)

Relation between differentiable and topological structures

Some invariants

De Rham cohomology

Level set manifold

For $F: M \to N$ a Smooth function, then $F^{-1}(c)$ is a Smooth manifold.

In this case, the Tangent space is $T_a \cong Ker(DF_a)$, that is the kernel of the Differential of the map. That is, all directions along which $F$ doesn't vary, which corresponds to the directions "inside" the level set, defined as all points with a fixed value of $F$.