Given two form $\omega$, find $\mu$ such that $d\mu=\omega$?
Does there exist a $1$-form $\alpha$ with $d\alpha = \omega$?
Here's the problem: if $S = \mathbb{Q}[x]/(x^2)$, compute the group  of automorphisms $\mbox{Aut}(S)$.   First off, I have the solution and I know it's isomorphic to the multiplicative group $\mathbb{Q}\backslash\{0\}$, so I'm trying to understand it. It uses what is called the substitution principle and the fact that the restriction of any endomorphism $\phi: \mathbb{Q}[x]\to \mathbb{Q}[x]$ to $\mathbb{Q}$ is the identity. It says that these two facts implies that the endomorphisms of $\mathbb{Q}[x]$ are in 1-1 correspondence with $h \in \mathbb{Q}[x]$, where the endomorphism corresponding to $h$ is $\phi_h : \mathbb{Q}[x] \to \mathbb{Q}[x]$ given by $\phi_h(f) = f(h)$ for $f \in \mathbb{Q}[x]$. I guess I just don't understand how it applied the substitution principle? It's probably obvious, but for some reason I am not getting it.  An explanation would be greatly appreciated! Or if you have a different solution, that'd be cool too. Thanks!! Does a 2100W angle grinder need an earth pin? I have recently purchased a Makita 2100W angle grinder. It has just the live (L) and neutral (N) connectors unlike the one I previously used, which had a third protected earth (PE) connection.   Would it be better if I changed the connections to a 3 pin connector? CSS3's attr() doesn't work in major browsers Leading zero in umask 0022 Given $f(x) = x + |x|$ for what values of $x$ is $f$ differentiable Problem : Given $f(x) = x + |x|$ for what values of $x$ is $f$ differentiable?   For the sake of generality, let's assume that it is unknown to us that $|x|$ is not differentiable at $x = 0$  Attempted Solution :  Using the definition of differentiability, a function is differentiable over an interval $I$ $\text{iff}$  $$f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}, \ \ \forall x\in I$$  Now implicitly this definition of differentiability requires $$\lim_{h\to 0^+} \frac{f(x+h)-f(x)}{h} = \lim_{h\to 0^-} \frac{f(x+h)-f(x)}{h}$$  Therefore $f$ will be differentiable only when  $$\lim_{h\to 0^+} \frac{(x + h + |x+h|)-(x + |x|)}{h} = \lim_{h\to 0^-} \frac{(x - h + |x-h|)-(x + |x|)}{h}$$  But it is unclear what to do next as $x$ is just arbitrary $$ \begin{equation} \begin{aligned} |x+h| &amp;= \begin{cases}x+h &amp; \text{if} &amp; x \geq -h\\ -x-h &amp; \text{if} &amp; x &lt; -h \end{cases} \\ &amp;= \begin{cases}x+h &amp; \text{if} &amp; x \geq 0\\ -x-h &amp; \text{if} &amp; x &lt; 0 \end{cases} \end{aligned} \end{equation}$$    A Wrong Solution :  We could take the derivative of $f$, (and evaluate the domain of the derivative $f'(x)$ to values for which it is defined), using the rules of differentiation, but we would get a wrong answer.  $$f'(x) = \frac{|x| + x}{|x|}$$ $$\implies f'(x) = \begin{cases} 2 &amp; \text{if} &amp; x \geq 0 \\ 0 &amp; \text{if} &amp; x &lt; 0 \end{cases}$$  This implies the derivative $f'(0)$ exists, when it does not as per the definition of differentiability. Why is that so?    Questions:   How can a solution be found using the definition of differentiablity? Why does taking the derivative of $f$, $f'(x)$ and evaluating it's domain, not give the correct values of $x$ for which $f$ is differentiable? How do I use the different shutter speeds my camera offers? What is the difference between these two:     Suzuki was nearly close to go out of business just after the World War period.   Vs.     Suzuki was nearly close to going out of business just after the World War period.(from a video)   When talking about the past, do these two sentences different from each other? Like, if you use verbing(going) with infinitive(to) in the past, does it mean that there is another event happening? Or none. What is a "loop device" when mounting? Draft mode for pgfplots My current desktop reso is 1366x768    In cs 1.6 I have selected display option as Normal and Resolution as 1024x768.     In this settings I don't get those vertical black side bars which make the playing screen square.     Atm it is rectangular but I wanted square.    How do I get the vertical black bars while playing cs 1.6? This is actually a two part question:   Which methods are there to calculate average slope by unit (say km²) for a given area? Are there any special requirements on the data - like additional information besides height measurements? Which open source GIS implement methods usable to calculate average slope on a grid (e.g. TIF file)? Ratio test and the Root test Prove the definitions of $e$ to be equivalent How to prove that the following definitions of $e^x$ are equivalent, with as simple tools as possible and without any knowledge of $e$ or logarithms ?  $$\sum_{n=0}^{\infty} \frac{x^n}{n!}=\lim_{n\to\infty}\left(1+\frac{x}{n}\right)^n$$  Also preferably, prove that this is $a^x$ for some real number $a&gt;0$. The browser's "Back" button cause problems with the review buttons PDF video Embedding Problem I have embed an mp4 video in the PDF file as given below:  \documentclass{beamer} \usetheme{Singapore} \usepackage{media9} \usepackage{hyperref} \begin{document}  \begin{frame}\frametitle{Title}  \includemedia[ width=\paperwidth,height=0.7\linewidth, activate=pageopen, flashvars={aaa.mp4 &amp;autoPlay=true} ]{}{C:/Users/name/Desktop/aaa.mp4}  \end{frame}  \end{document}    There is no compile time error in the code but after running it this massage comes along a  blank pd f file....  "To view the Flash @R Technology content in this pd f file, please install the required version of Flash Player"  I would like to add here that I already have installed the latest version of Flash Player(Adobe Flash Player 12 ActiveX). So,Can anyone tell what version is required to be installed to obtain Video in pd f file. How to evaluate these two integrals. How to determine a Python variable's type? How do I see the type of a variable whether it is unsigned 32 bit, signed 16 bit, etc.?  How do I view it? I have a text file abc.txt and its contents are:     /lag/cnn/org/one.txt    /lag/cnn/org/two.txt    /lag/cnn/org/three.txt   If I use:  $ tar -cvf allfiles.tar -T abc.txt   I'm getting the tar of files in the list. Similarly is it possible to copy those files in abc.txt to a folder?  I tried this:              $ cp --files-from test1.txt ./Folder   But it is not working. Why does the co-efficent of kinetic friction of a block on an inclined plane not change when you increase the angle of the plane? I'm trying to understand Python's approach to variable scope. In this example, why is f() able to alter the value of x, as perceived within main(), but not the value of n?  def f(n, x):     n = 2     x.append(4)     print('In f():', n, x)  def main():     n = 1     x = [0,1,2,3]     print('Before:', n, x)     f(n, x)     print('After: ', n, x)  main()   Output:  Before: 1 [0, 1, 2, 3] In f(): 2 [0, 1, 2, 3, 4] After:  1 [0, 1, 2, 3, 4]