I´m looking for a book with the proof of the result that says the associated Lie algebra of a Lie group $G$, let be $G_{Lie}$, is isomorphic to the tangent space $T_e(G)$, where $e \in G$ is the neutral element.  Let me said the chapter or theorem, not only the book, please.  Thank you all
Define the Lie algebra $\mathfrak{g}$ of a Lie group G to be the set of all left-invariant vector fields of $G$. I want to prove that   $$          f: \mathfrak{g} \rightarrow T_eG \\              X \mapsto X(e) $$ is a linear isomorphism.  The only thing that I have left to prove is surjectivity.Letting $v\in T_eG$, we set $X\in \mathcal{X}(G)$ such that $\forall x\in G,\,X(x)=L_{x_{*,e}}(v)$ (I still have to show that such an $X$ is indeed a vector field on $G$). We get that  $$ f(X)=X(e)=L_{e_{*,e}}(v)=v $$  where the last equality holds because  $$       L_{e_{*,e}}(v)f = v(f \circ L_e) = v(f \circ id_G) = v(f) $$ for any $f\in C^\infty(G)$. (I am thinking of the tangent vectors as derivations).  Now all we have left to do is prove that such an $X$ is left-invariant, and it is therefore an element of $\mathfrak{g}$, but I do not know how. I tried to prove that  $$            \forall f\in C^\infty(G),\, L_{{x}_*,y}(X(y))f=X(L_x(y))f $$ but with no success.
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Windows 8 Pro clean install - everything slow; hard drive at constantly at 100% Yesterday, I have downloaded and installed Windows 8 Professional, after downloading it (legally) and burning it to a DVD, on my 3-year-old  (Intel Core Duo 2x2.4&nbsp;GHz, 4&nbsp;GB RAM,  hard disk drive).   The installation failed two or three times, each time differently (error codes, for example 0xc000021a, with an infinite boot loop), but finally I completely formatted the hard drive and installed. Seemingly, everything was OK.  However, after booting successfully, the OS is extremely slow. When I open the task manager, it shows that the HDD is at 100% - however, no application or background process or Windows process can be made responsible. I searched the Internet and found that Windows might try to "index" the hard drive, but I did't find a process that isn't also on my second computer, which still uses the Release Preview (restarting every two hours). Also, the startup is empty, and HD Tune and HDD Health show that the drive health is OK, even though I don't know how reliable these two programs are.  I looked for diagnose software from Toshiba, but they only offer tools for "Fujitsu branded Toshiba devices"; I tried it out, and it did not work on my hard drive.  Does this sound like a hard drive failure? The laptop is 3.5-years-old, has been carried around sometimes roughly and was used (and on) most of the time. However, it would have failed exactly the night before installing Windows&nbsp;8 Profesional, which seems - unlikely.  If this is a software error - then what did I do wrong if this occurs from the moment of installation (even the personalization at the end of installation took half an hour)? How can I fix this? I tried using integration by parts twice, the same way we do for  $\int \sin {(\sqrt{x})}$ but in the second integral, I'm not getting an expression that is equal to  $\int x\sin {(\sqrt{x})}$.  I let $\sqrt x = t$ thus, $$\int t^2 \cdot \sin({t})\cdot 2t dt =  2\int t^3\sin(t)dt = 2[(-\cos(t)\cdot t^3 + \int 3t^2\cos(t))] = 2[-\cos(t)\cdot t^3+(\sin(t)\cdot 3t^3 - \int 6t \cdot \sin(t))]]$$  which I can't find useful. Extruding surface while keeping the shape here is the picture of what I would like to get(example nr. 2). I need the edge frame to be same in width and length. Also the extrusion is on surface (without depth) I tried extruding region, but it did not give me the result I wanted.   I know I could just adjust the scale and do it manually, but that would be eyeballing without achievign accuracy. I would like to know if it is possible to extrude or inset while keeping the shape (even edge space) Escape a string for a sed replace pattern Prove that $\sqrt{n} \le \sum_{k=1}^n \frac{1}{\sqrt{k}} \le 2 \sqrt{n} - 1$ is true for $n \in \mathbb{N}^{\ge 1}$ I'm trying to solve  proposed by the department of mathematics of Oxford University. I don't know how to give a valid proof for the third one which says the following:     Prove that for $n \in \mathbb{N}^{\ge 1}$: $$ \sqrt{n} \le  \sum_{k=1}^n \frac{1}{\sqrt{k}} \le 2 \sqrt{n} - 1 $$   My first approach has been to convert the summation into a function in terms of $n$ so that I could find the maximum and minimum values in the interval. However, there doesn't seem to be an easy way to express the summation as a function. Thanks in advance! How to add USB 3.0 port to your laptop How do I set up a folder so that anything created in it inherits permissions? What happens to the Windows 7 key when upgrading to Windows 8? Microsoft is offering "Microsoft Windows 8 Professional Upgrade" at a discounted price compared to the OEM version. However it states:     If you currently have a personal computer running Windows 7, Windows XP or Windows Vista then you can upgrade to Windows 8 Pro (Professional).   My question is what happens to the Windows 7 key that I use to upgrade? Will it become the Windows 8 key or will it generate a new Windows 8 key?  Basically if I upgrade using my Windows 7 key from my laptop for my desktop will I be able to continue using the laptop on Windows 7 while concurrently using Windows 8 upgrade on my Desktop? Cannot deserialize instance of time from VALUE_STRING value please I tried to find counterexamples to see that $l^p$ is not norm with $p&lt;1$ in the triangle inequality but I have problems with convergence when I choose some successions. Thanks. Ubuntu 20.04 still supporting Python2 $\operatorname{Supp}(M)=V(\operatorname{Ann}M)$ if $M$ is finitely generated I've always wondered this - why can't you declare variables after a case label in a switch statement?  In C++ you can declare variables pretty much anywhere (and declaring them close to first use is obviously a good thing) but the following still won't work:  switch (val)   {   case VAL:     // This won't work   int newVal = 42;     break; case ANOTHER_VAL:     ...   break; }     The above gives me the following error (MSC):     initialization of 'newVal' is skipped by 'case' label   This seems to be a limitation in other languages too.  Why is this such a problem?