In this question I am looking for help in understanding the Algebraic Tangent vector and what the difference is between it and the "regular" Tangent vector.
A "differentiable function" near p is a pair $(f,U)$ where $U \subset$ is open with $p \in U$, and $f:U \rightarrow \mathbb{R}$ is a differentiable function. We say that two such pairs $(f,U), (g,W)$ are "equivalent" - the notation is $(f,U) \sim (g,W)$, if and only if there exists an open $V \subset U \cap W$ with $p \in V$ such that $f|_{V}$ and $g|_{V}$. So f and g coincide in a small neighborhood of p. $\sim$ is an equivalence relation.
Note: I am thinking that this is an equivalence relation (intuitively) because we are just looking at some neighborhood of point p - that which is $V \subset U \cap W$ and so other mapping $h: Y \rightarrow \mathbb{R}$ where $Z \subset U \cap W$ is also in the equivalence relation defined by:
$$E_{p} \equiv \left\{(f,U)|(f,U) \text{ is a differential function near } p \right\}/\sim$$
Now the equivalence class of a pair $(f,U)$ is denoted by $[f,U]$ and is called the "germ" of a differentiable function at $p$.
Note: From my understanding $[f,U]$, the germ, is showing that locally these functions $\in [f,U]$ are differentiable. I am guessing that is just scratching the surface - but furthermore other than knowing something has to be differentiable to find the tangent (at least in the sense of being $C^{1}$).
While I am trying to wrap my head around the above, it seems to get worse..
An element $v$ of the dual space of $E_{p}, v \in Hom(E_{p},\mathbb{R}) =\left\{v:E_{p} \rightarrow \mathbb{R} | \text{ is linear} \right\}$ is called a "derivation" if $\forall [f,U],[g,W] \in E_{p}$ the product rule $$v([f,U] \cdot [g,W]) = v([f,U]) \cdot [g,W](p) + [f,U] \cdot v([g,W])$$ holds
Defn: The Algebraic Tangent space of a space X at p is: $T_{p}^{alg}X \equiv \left\{v \in E_{p}^{*} | v \text{ a derivation}\right\}$
An element of $T_{p}^{alg}X$ is called an algebraic tangent vector of X at p.
So what is the point of the Algebraic Tangent vector and space?
Thanks much for any thoughts,
Brian
I don't think I can answer this coherently, so I'll break it up in different parts.
1. Germs.
What you wrote about germs seems right, as it is. Since looking at tangent spaces for non-differentiable functions doesn't make sense, differentiablitiy (at the point where you're looking at the tangent space) is a necessary condition. At places where I encountered the algebraic construction of tangent spaces, the (indeed!) equivalence relation $\sim$ was always defined on the set of differentiable maps.
I consider the next question to be a useful one wrt this subject: Show that the set of germs around 0 of functions from $\mathbb{R}^n$ to $\mathbb{R}$ has zero divisors.
2. Derivations.
The notation you use unfortunately makes things more complicated. The best way to think about derivations, I think, is as derivatives. I will assume you've written $f$ and $g$ instead of $f$ and $f'$ and reserve the prime for standard derivatives. If you have two maps $f,g$ and let $v$ be the standard derivative evaluated at $p$, then $v(\lambda f+g) = (\lambda f+g)'(p)= \lambda f'(p)+g'(p)$. This equailty still holds if you pass to $E_p$, justifying that you look at linear maps. The extra condition then corresponds to the product rule: $v(fg) = (fg)'(p) = (f'g)(p)+(fg')(p)$.
3. What's the point?
If it isn't clear why you should look at algebraic tangent vectors, this is a very good question that most certainly needs addressing. There are three different possible definitions of tangent spaces that can be used: the 'regular', or geometric, tangent space $T_pM$, the algebraic tangent space $T_p^{\text{alg}}M$ and the physicist's tangent space $T_p^{\text{ph}}M$. The main point in having these different definitions, is that the notions are equivalent and indeed, in differential geometry or differential topology, they are used interchangeably. Their main use is that in certain proofs one of the different notions is a lot easier to deal with than the others. For definitions of the notions as well as proofs of their being equivalent, I'd like to refer to the second chapter of Bröcker, Jänich, Introduction to differential topology.