It often times just helps to write out exactly what the formula you are trying to get in the end looks like, and then interpreting that.

For example, you are trying to show that $F_\ast(\alpha X)=\alpha F_\ast(X)$. So, what does the left-hand side do to elements of $T_{F(p)}N$? Well, as you pointed out

$F_\ast(\alpha X)(f)=(\alpha X)(f\circ F)$

what does the right-hand side look like?

$(\alpha F_\ast(X))(f)=\alpha F_\ast(X)(f)=\alpha X(f\circ F)$

Make sense?

EDIT(to answer Jellybean's questions in the comment): I am not entirely sure what you mean here. The tangent space of $M$ at $p$ is thought of, or at least the book you're reading thinks of it, as the $\mathbb{R}$-derivations of germs of smooth functions $\mathcal{G}_{M,p}$ at $p$. A smooth map $F:M\to N$ allows one to take a derivation $X$ of $\mathcal{G}_{M,p}$ and "push it forward" to a derivation of $\mathcal{G}_{N,F(p)}$. The way it does this is basically the observation that if you start with an element of $\mathcal{G}_{N,F(p)}$, germs of smooth functions on $N$ at $F(p)$, there is a way to pull this back to an element of $\mathcal{G}_{M,p}$. Namely, if $f\in\mathcal{G}_{N,F(p)}$ then $f\circ F\in\mathcal{G}_{M,p}$. Thus, if you start with a derivation $X$ of $\mathcal{G}_{M,p}$ you want to use this to produce a derivation of $\mathcal{G}_{N,F(p)}$. 

But, a derivation of $\mathcal{G}_{N,F(p)}$ eats in, well, obviously, elements of $\mathcal{G}_{N,F(p)}$. Thus, $X$ itself doesn't act on $\mathcal{G}_{N,F(p)}$. But, we observed above that we have a method of taking an element of $\mathcal{G}_{N,F(p)}$ and pulling it back to an element of $\mathcal{G}_{M,p}$, which is the set of things that $X$ does eat. So, to create a derivation of $\mathcal{G}_{N,F(p)}$, which let's call $F_\ast(X)$, of $\mathcal{G}_{N,F(p)}$ we merely take an element of $\mathcal{G}_{N,F(p)}$, pull it back to $\mathcal{G}_{M,p}$ by the rule $f\mapsto f\circ F$, and now that we're back on $X$'s diningroom table we feed it to $X$ to get a real number out. Thus, the derivation $F_\ast(X)$ all put together acts as $f\mapsto f\circ F\mapsto X(f\circ F)$. You can check that this definition of $F_\ast(X)$ does, in fact, produce a a derivation of $\mathcal{G}_{N,F(p)}$. 

Thus, the association $X\mapsto F_\ast(X)$ produces a map $F_\ast:T_p M\to T_{F(p)}N$. This map, the map $F_\ast$, is the pushforward. It is a linear map between tangent spaces. 

I hope that helps.

Linearity of push forward $F_*$
Let me expand...

I have a client who is unhappy with their current hosting situation. They want me to move their website and domain across to my hosting platform. I can do this.

However I have realised that the company has:

1) A number of existing IMAP accounts

I understand that I can use a tool like imapsync in order to transfer the mailboxes and hopefully incur only a small amount of downtime with the emails during this process. Can anyone offer any advice regarding the transfer of IMAP mailboxes from one server to another.?

2) An SMTP Relay

Another company supplies my client with an SMTP relay. I've only really just found out what this is. I understand that emails are sometimes undelivered to the client and a mail relay routes incoming mails through another (more reliable?) server in order to keep trying to resend emails so none are missed.

Is this neccessary in this day and age? Does my Plesk 10 Panel Linux Server come with any internal mail relay? Maybe I could use another service like mailjet or elasticemail instead?


Am in in for a lot of hassle moving email accounts or is this a
fairly transparent process?
What should I look out for and how
can I make this hosting transfer seamless (or nearly seamless)?
When I move the domain to my server (and the IP changes) this will interrupt the current mail relay. Will my client's emails continue to be delivered?

