﻿Tangent vectors to a manifold can be defined intrinsically or extrinsically. It turns out later that through some theorem, the two approaches are equal. We then choose to develop GR using the intrinsic approach, because it would be not economical to refer to a larger space that spacetime and its tangent space is embedded in, because then the question is why don’t you refer to this larger space as spacetime.


So we want to define tangent vectors intrinsically. How do we do it? It turns out that we have many options, so long as it confirms the behavior we expect of tangent vectors. The one we select is the directional derivative operator. A tangent vector literally is a directional derivative operator, because they have the same expected behavior. Specifically the behavior we care about is linearity and the Leibnitz rule, so that it essentially behaves like a derivation.


The main place we’ve seen derivations before is with c-infinity(R), or the smooth functions on R, where basically the derivation is the normal derivative we expect.


If we instead look at the derivation for c-infinity(M), we learn that it’s a sufficient condition for the directional derivative operator.


Anyways, what is a directional derivative operator anyways?


Basically, it’s an object waiting for a function that, once is given a function, turns into a real number. Or when you select a tangent vector at every point, it’s waiting for a function, and once given a function, turns into another function.


But, the at one point case is important to study in detail. Basically you have a curve that goes through a point p, and you have is so that c(p)=0, which you can easily do by reparameterization.


The directional derivative operator at a point waits for a function and returns the directional derivative at that point. What does the directional derivative mean? It’s the value of the slope of the tangent vector of that function at a point along a curve. The point is that it’s a number.


It’s defined sufficiently by the derivation as being (f*x)’(0)


Hmm I gotta review this, so that I can fully derive it on my own.


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The Frederic Schuller lectures goes like this:
* Logic
* Axioms
* Set theory
* Topology
* Manifolds
* Smooth manifolds
* Tensors
* Tangent spaces
* Fields
* Differential equations on fields & more complicated stuff / actual physics