# Differential on a manifold

I got confused with the differential of a map on a manifold. I try to wrap up what I learned and what I am confused about. Let $X$ be a manifold and $c_1,c_2: (-\epsilon, \epsilon) \to X$ two curves satisfying $c_1(0) = x = c_2(0)$ for an $x \in X$. A tangent vector at $x$ is an equivalence class of curves satisfying the relation $c_1 \sim c_2 \Longleftrightarrow D(\varphi \circ c_1)(0) = D(\tilde{\varphi} \circ c_2)(0)$ where $(U, \varphi)$ and $(V,\tilde{\varphi})$ are charts around x and the tangent space of $X$ at $x$ is defined as the set of all these equivalence classes. If I have a map $f: X \to Y$ where $Y$ is another manifold, I can define the tangent map $T_xf: T_xX \to T_yY$ where $[c] \mapsto [f \circ c] \in T_yY$. But how can I define the derivative of $f$ at $x$ in this setup? I don't want to to this with derivations. Since the local representation of $f$ is $(\psi \circ f \circ \varphi^{-1}). (\varphi(x))$ cant I just differentiate this and obtain $Df$ in this way i.e. $Df(x) = D(\psi \circ f \circ \varphi^{-1})(\varphi(x))$? And is this related to the tangent map? If yes, what happened to $[f \circ c]$? I am sorry for my (maybe confusing) explanation but I can't get my head around this.

• See V. Arnol'd's "Ordinary Differential Equations." – avs Aug 16 '16 at 18:38
• I got the book from a library now. Thanks it also makes things clearer. – JDoe Aug 19 '16 at 8:06
• Good. The latest editions of the book are best and differ substantially from the early two. Also, keep an eye out for other books by the same author.;) – avs Aug 19 '16 at 16:51

Actually you can define the differential in 3 different way:

1. Through derivations, identifying (as you've done) functions that locally are alike since the differential is local notion and so you won't have anything to distinguish to functions which behave in the same manner locally but then change outside an open set;
2. Bringing everything back to ordinary differential of functions between $R^m$ and $R^n$ and then simply treat it with ordinary calculus instruments;
3. Thinking tangent vectors as velocities of curves and then calculating the velocity of all the possible curves in a point of the manifold.

Now, these 3 picture coincide on $R^n$ coincide so you might want to try them on something concrete as a half sphere or a paraboloid, but the most general picture (which can be extended in an easy way to all manifolds) happens to be first one with derivations. Once you get used you will understand that is indeed the most useful.

You just have to convice yourself that in $R^n$ there's an identification between directional derivatives, tangent vectors and possible velocities of a curve in a point. Once you got it you just have to convince yourself that what you're really looking for are directional derivatives and that a generalization for them are Derivations.

Sorry to be so vague, if you need some specific clarification tell me in the comments and I will edit the answer.

• Thanks for your detailed answer. To get things right: I thought how I introduced the tangent space was your third option (velocities of curves). – JDoe Aug 16 '16 at 19:42
• You can also define a tangent vector at $x\in X$ as a $\mathbb R$-linear map $D:C^\infty (X)\to \mathbb R$ satisfying Leibniz' rule: $D(fg)=f(x)D(g)+g(x)D(f)$. That definition is correct, elementary, crisp...and very bad! – Georges Elencwajg Aug 16 '16 at 23:52
• @GeorgesElencwajg Why is the definition via the Leibniz' rule "very bad"?I for myself found this definition the least applicable but what is the reason for your statement? But I still need to know if my definition is way number 3 in Dac0s' statement. – JDoe Aug 18 '16 at 9:41
• It is bad because it is very artificial to define a tangent vector as only able to act on global functions: you should be able to calculate the derivative of $\frac 1x$ at the point $x_0=3$ , even though the function $\frac 1x$ is not defined on all of $\mathbb R$! Moreover this definition completely breaks down in the algebraic or holomorphic case since the only global functions might be the constants . Good definitions should try to apply to the widest possible kinds of manifolds. – Georges Elencwajg Aug 18 '16 at 10:22
• And yes, your definition is Dac0's (a bit vague) number 3. The derivative of $f$ along $[c]$ at $x$ is the real number $(f\circ c)'(0)$. – Georges Elencwajg Aug 18 '16 at 10:30