In this session, he talks about differential geometry starting from coordinate charts: to have this clear picture starting from a punched plane is very helpful in later variable coordinate etc. transformation.

then to understand the concept of pull back and push forward. pull back means pull back of functions, or change of variables.

To understand pull back of functions vividly via the same picture:

So for example

conceptually, we redefine notion of identity to notion of equivalent functions, let’s use below diagram to illustrate

Next, concepts of tangent vector and cotangent vectors, there are prerequisite to understand differential geometry!

So now we take an entire new perspective in understanding coordinate etc. from view of tangent vector, it’s just taking the derivative at that point or operator operating on that point to function f:

A tangent vector is a vector that lies in the tangent space of a point on a smooth manifold. it is an operator operates on a function by taking derivatives or differentiates.!

“tangent vector v acting or operating on a function f” is how we carry out computation and why learn differential geometry to handle complexity. here is a concrete and simple example to illustrate:

But what’s the intuitive meaning of alpha and beta here? it’s flexible to “measure” any direction in this case 1 and -1 direction at point (2, 3)

What if alpha and beta are functions? here is another example:

Moving on next to another important concept: Push forward, which basically is just chain rule on variables!

from the visual, one can see the pull back is on function sai, while push forward is on variable r, theta to new variable sai start in subscription position.

Now what’s the fundamental relationship between pull back and push forward?

It’s equivalent to have f push forward then act on f and have v act on or operate on or take derivatives or differentiate the pull back of f: