Manifolds
When does curved space look flat?
A manifold is a space that is complicated globally and indistinguishable from flat space locally. That one idea lets calculus be done on surfaces, on spacetime, and on spaces with no geometric picture at all — and the difficulty is entirely in making local agree with global.
Charts & atlases
How do you describe a space with no single map?
No flat map of the Earth is correct
Every world map distorts something, and the fix is to use several and agree on the overlaps. That fix is the definition of a manifold.
Smooth means the overlaps are smooth
Smoothness cannot be defined on the manifold directly, only through the charts — and then proved not to depend on which chart was used.
Tangent spaces
What is a direction on a space with no ambient room?
A vector with nowhere to point
On a surface in space a tangent vector is an arrow. With no surrounding space there is no arrow, so the definition has to be rebuilt from derivatives.
Vector fields and flows
A vector field is a direction chosen at every point; following it is an ordinary differential equation. The film shows the two pictures becoming one.
Differential forms
What is the right thing to integrate?
The object that was under the integral sign all along
The dx in an integral is not notation. Taking it seriously produces differential forms, and then Stokes' theorem absorbs four theorems of vector calculus into one line.
Riemannian metrics
What has to be added before you can measure anything?
Manifolds, as one volume
Every document in this subject is a chapter of the same book, compiled from one source with live cross-references and continuous numbering. Download the whole thing, or take chapters as you go — the page numbers and theorem references agree either way.
Download Volume XII ↓