# Geometric Tangent Vector

1. Mar 3, 2013

### BrainHurts

So let $ℝ^{n}_{a}$={(a,v) : a $\in$ $ℝ^{n}$, v $\in$ $ℝ^{n}$}

so any geometric tangent vector, which is an element of $ℝ^{n}_{a}$ yields a map

Dv|af = Dvf(a) = $\frac{d}{dt}|_{t=0}$f(a+tv)

this operation is linear over ℝ and satisfies the product rule

Dv|a(fg) = f(a)Dvg + g(a)Dvf

if v|a = $\sum_{i=1}^n$ viei|a, then by the chain rule
Dv|af can be written as:

Dv|af $\sum_{i=1}^n$ vi $\frac{∂f}{∂x_{i}}(a)$

not seeing how the chain rule applies and how the result as such.

2. Mar 3, 2013

### dx

xi(t) = ai + tvi

(d/dt) f(a + tv) = (d/dt) f(xi(t))

= Σ (∂f/∂xi) (dxi/dt)

= Σ vi(∂f/∂xi)