- #1

- 245

- 1

## Main Question or Discussion Point

Affine spaces can be regarded as smooth manifolds if we take the natural topology and affine coordinate charts as atlas. So, if

[tex]\dot C(t_0): C^\infty(M) \rightarrow \mathbb{R}, f \mapsto \frac{d f(C(t))}{d t}\bigg|_{t=t_0} [/tex]

On the other hand, for

[tex]C'(t_0) = \lim_{t \to t_0} \frac {C(t) - C(t_0)}{t - t_0}[/tex]

[tex]C'(t_0)[/tex] and [tex]\dot C(t_0)[/tex] is related simply as

[tex]\dot C(t_0)(f) = \frac{d f(C(t))}{d t}\bigg|_{t=t_0} =f'(C(t_0))C'(t_0)[/tex]

where [tex]f'(C(t_0))[/tex] is the derivative of f at [tex]C(t_0)[/tex], that is a linear functional on

[tex]f(C(t))-f(C(t_0))= f'(C(t_0)) (C(t) - C(t_0)) + \mathcal{O}(\|{C(t) - C(t_0)\|^2})[/tex]

But this works only if a norm is also defined on

My question is: How can be related [tex]C'(t_0)[/tex] to[tex]\dot C(t_0)[/tex] without using norm or coordinates?

**is an***M**n*-dimensional affine space, then the tangent vector of a curve [tex] C: [0,1] \rightarrow M [/tex]in a point [tex]p = C(t_0)[/tex] can be defined as a derivation (as in any smooth manifold):[tex]\dot C(t_0): C^\infty(M) \rightarrow \mathbb{R}, f \mapsto \frac{d f(C(t))}{d t}\bigg|_{t=t_0} [/tex]

On the other hand, for

*is an affine space, the tangent vector of the curve C in the point [tex]p = C(t_0)[/tex] can be defined as an element of the underlying vectorspace***M***:***V**[tex]C'(t_0) = \lim_{t \to t_0} \frac {C(t) - C(t_0)}{t - t_0}[/tex]

[tex]C'(t_0)[/tex] and [tex]\dot C(t_0)[/tex] is related simply as

[tex]\dot C(t_0)(f) = \frac{d f(C(t))}{d t}\bigg|_{t=t_0} =f'(C(t_0))C'(t_0)[/tex]

where [tex]f'(C(t_0))[/tex] is the derivative of f at [tex]C(t_0)[/tex], that is a linear functional on

*which satisfy:***V**[tex]f(C(t))-f(C(t_0))= f'(C(t_0)) (C(t) - C(t_0)) + \mathcal{O}(\|{C(t) - C(t_0)\|^2})[/tex]

But this works only if a norm is also defined on

**. Evidently, the relation between [tex]C'(t_0)[/tex] and [tex]\dot C(t_0)[/tex] can also described using coordinates and then showing that the relation is independent from the coordinates chosen.***V*My question is: How can be related [tex]C'(t_0)[/tex] to[tex]\dot C(t_0)[/tex] without using norm or coordinates?

Last edited: