`0`

. A Curve is a
differentiable map $C: (a, b) \to M$ where $a < 0 < b$.
`f(m): M -> R`

with respect to a curve `c(t): I -> M`

, denoted as `c[f]`

. `g(t) = (f . c)(t) :: I -c-> M -f-> R = I -> R`

.
This this is the value `dg/dt(t0) = (d (f . c) / dt) (0)`

.
`p`

: `m`

dimensional manifold `M`

, a tangent vector at a point `p`

is an
equivalence class of curves that have `c(0) = p`

, such that `c1(t) ~ c2(t)`

iff
:
- For a (all) charts
`(O, ch)`

such that`c1(0) ∈ O`

,`d/dt (ch . c1: R -> R^m) = d/dt (ch . c2: R -> R^m)`

.

`TpM`

): `p`

forms a vector space `TpM`

.
We prove this by creating a bijection from every curve to a vector `R^n`

.
Let `(U, ch: U -> R)`

be a chart around the point `p`

, where `p ∈ U ⊆ M`

. Now,
the bijection is defined as:
```
forward: (I -> M) -> R^n
forward(c) = d/dt (c . ch)
reverse: R^n -> (I -> M)
reverse(v)(t) = ch^-1 (tv)
```

`TpM*`

): dual space of the tangent space / Space of all linear functions from `TpM`

to `R`

. - Associated to every function
`f`

, there is a cotangent vector, colorfully called`df`

. The definition is`df: TpM -> R`

,`df(c: I -> M) = c[f]`

. That is, given a curve`c`

, we take the directional derivative of the function`f`

along the curve`c`

. We need to prove that this is constant for all vectors in the equivalence class and blah.

`push(f): TpM -> TpN`

`c: I -> M`

, the pushforward
is the curve `f . c : I -> N`

. This extends to the equivalence classes
and provides us a way to move curves in `M`

to curves in `N`

, and thus
gives us a mapping from the tangent spaces.
This satisfies the identity:
```
push(f)(v)[g] === v[g . f]
```

`pull(f): TpN* -> TpM*`

`wn : TpN -> R`

, the pullback is defined as
` wn . push(f) : TpM -> R`

.
This satisfies the identity:
```
(pull wn)(v) === wn (push v)
(pull (wn : TpN->R): TpM->R) (v : TpM) : R = (wn: TpN->R) (push (v: TpM): TpN) : R
```