so let $\vec{v}: \Omega \subset \mathbb R^n \to \mathbb R^n$ be a vector-field.  Now we know:  (I) $\vec{v}$ conservative $\quad\Rightarrow\quad \nabla\times\vec{v}=0$  If further, $\Omega$ is simply-connected i.e. path-connected and null-homotopic, we can also state:  (II) $\nabla\times\vec{v}=0 \quad \Rightarrow \quad \vec{v}$ is conservative.  Question: Why do we need null-homotopy for (II)?
A field that is conservative must have a curl of zero everywhere. However, I was wondering whether the opposite holds for functions continuous everywhere: if the curl is zero, is the field conservative? Can someone please give me an intuitive explanation and insight into this, and if it is true, why? Also, please try to not be too rigorous (only in grade 9).
Show that $(2,0,4) , (4,1,-1) , (6,7,7)$ form a right triangle What I tried:  Let $A(2,0,4)$, $B(4,1,-1)$, $C(6,7,7)$ then  $$\vec{AB}=(2,1,-5), \vec{AC}=(4,7,3), \vec{BC}=(2,6,8)$$  Then I calculated the angle between vectors:  $$\begin{aligned} \alpha_1 &amp;= \cos^{-1}\left(\frac{(2,1,-5)(4,7,3)}{\sqrt{2^2+1^2+(-5)^2}\sqrt{4^2+7^2+3^2}}\right) \\ &amp;= \cos^{-1}(0)=90° \\ \alpha_2 &amp;= \cos^{-1}\left(\frac{(4,7,3)(2,6,8)}{\sqrt{4^2+7^2+3^2}\sqrt{2^2+6^2+8^2}}\right) \\ &amp;= \cos^{-1}\left(\frac{74}{\sqrt{74}\sqrt{104}}\right)=32.49\\ \alpha_3 &amp;= \cos^{-1}\left(\frac{(2,6,8)(2,1,-5)}{\sqrt{2^2+6^2+8^2}\sqrt{2^2+1^2+(-5)^2}}\right) \\ &amp;= \cos^{-1}\left(\frac{-30}{\sqrt{104}\sqrt{30}}\right)=122.5° \end {aligned}$$  As you can see, these angles don't even form a triangle, what am I doing wrong, any thoughts? I'm confused about the use of symbols $\vdash$ and  $ \models$. Reading the answers to  and  I see that:  the turnstile symbol $ \vdash $ denotes syntactic implication. Then $ S \vdash \psi$ means that $\psi$ can be derived from the formulae in $S$  the double turnstile $ \models $, denotes semantic implication. Then the notation $A \models B$ means simply that B is true in every model in which A is true.  So I understand that, e.g., $(ZFC+ A) \vdash B$ means that $A \Rightarrow B$ is a theorem in ZFC and, in this sense $\vdash$ is a mathematical symbol for a theorem. While $(ZFC+ A) \models B$ means that $ B$ is true in every model of ZFC in which $A$ is true, and, in this sense $\models$ is not a mathematical but a metamathematical symbol.  I understand correctly? And those are the only uses of the two symbols? Is it acceptable to say "fed up OF" - surely it should be "fed up WITH" or "bored WITH something" rather than "bored OF something"? I hear people using of when I have always thought it was with in the circumstances explained above.  It seems to be getting more common. Is a contradiction enough to prove a set equality to $\varnothing$? How do I reset my main keychain and completely delete the old one? I changed my password and its been coming up with all of these popups asking for my new password when I first login. If anyone can help with updating my keychain I would appreciate it or you can just help me completely delete and reset it. How to type directly with keyboard the U+200C character? How to type directly with keyboard the U+200C character in Notepad or other places... in Windows 8.1? (without depending on a certain font)  I don't want to copy and paste ways. I don't want any complicated ways. I just want something like Alt+### for typing U+200C Directly at any text area in windows 8.1 without depending to a certain font. Just like ALT+129 for U+200B.  (I do not need ALT+129, That's for U+200B. I need U+200C)  the U+200C character name is: ZERO WIDTH NON-JOINER (ZWNJ)  Here is the U200C character info: Universal extractors With an  of archive/compression file formats, is there a single free/open-source command line tool to rule them all? Perhaps something with a consistent / unified set of flags? (note my  friendly )  I once run into a set of aliases meant to largely simplify the task of compressing/de-compressing files with bindings to tar and other utils, but I can't find this anymore.  Update: How can I configure something like atool to not use unzip to extract zip files (which apparently can't handle files larger than 4GB) and to use gunzip instead?  $ aunpack large_file.zip error:  Zip file too big (greater than 4294959102 bytes) Archive:  large_file.zip warning [large_file.zip]:  1491344848 extra bytes at beginning or within zipfile   (attempting to process anyway) error [large_file.zip]:  start of central directory not found;   zipfile corrupt.   (please check that you have transferred or created the zipfile in the   appropriate BINARY mode and that you have compiled UnZip properly) aunpack: unzip ...: non-zero return-code I am failing to add a heading for the appendix to the ToC with the correct page number:  \documentclass{scrbook} \begin{document} \tableofcontents \chapter{test} \appendix \addcontentsline{toc}{part}{\appendixname} \chapter{First Appendix} \end{document}     I also tried to use the \addappheadtotoc command from the  package, but that changes nothing (expect that it writes 'Appendices' instead).  I assume that the appendix package provides the solution, but I could not figure out how to use it. How do I get an iball 3.5G modem working on Ubuntu? I purchased iball 3.5G  connect wireless modem 1 month before. I am trying to connect it with my laptop which is having ubuntu Operating System. After plug-in it to the ubuntu system it is not connecting with net.   This modem is plug and play but after plugging it nothing is displaying to the system.  I am expecting step-by-step method to connect this modem with my ubuntu 12.04 LTS sysetm from your expert's advice. I used to work as a consultant and have been spoiled with nice looking slides. For instance, being able to create slides that have 2, 3, or 4 graphs (perfectly positioned and with nice headers!) at the click of a mouse button via macros.  Similarly, there are many other kinds of word slides that had multiple columns, etc. that would be nice to recreate as well as objects like "main point" boxes at the bottom that summarize a main point and ways to add a transparent highlight to any part of the screen.  Is this kind of functionality far beyond anything that LaTeX presentations can do or is there an archive somewhere of code for making more intricate presentations? Conditional \endinput I want to prevent processing rest part of a package in case if it is loaded not from lualatex. For some reason I cannot load ifluatex package, so I borrowed definition of \ifluatex from that package and invented the following chunk of code:  \expandafter\ifx\csname luatexversion\endcsname\relax % = not \inluatex   %\typeout{^^J********************** not LuaTeX^^J}   \PackageError{babel}{     ^^J\space\space * Option russianu is intended for LuaTeX only.              ^^J\space\space * Option russian  is used instead.                     }   \relax   \input{russianb.ldf}     \let\@next@action\endinput \else   %\typein{^^J********************** LuaTeX^^J}   \let\@next@action\relax \fi \@next@action   It seemed to work but Is there a better solution. In particular, it is desirable to avoid introduction of \@next@action because it is used the only time.  Just to explain: my goal is to hack babel with option russian to make it working painlessly with LuaLaTeX without any additional hacks (as described, eg, in  and ) Does "no larger than" imply shapability? When I receive a message on an SE post, I get a notification like this:    If I double tap this notification, the SE application opens, but my feed is empty approximately half the time. It looks like this:    If I refresh the feed (slide from the top), the feed does refresh and shows the new notification.   The notification, above, leads to  comment.  I haven't figured out the pattern yet, but this occurs roughly half the time I open a notification. It's occurred for chat messages, comments and answers - but not consistently. I have received notifications for the same post (for example, the two comments after the one linked above) that do not show an empty feed. But, a chat message barely 10 minutes later will show the empty feed again.  I am utilizing the beta app - version 1.0.72 on Android 5.0.1. 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Button for playing audio from document that uses this trick. What is the difference between N and N-1 in calculating population variance? I did not get the why there are N and N-1 while calculating population variance. When we use N and when we use N-1?      It says that when population is very big there is no difference between N and N-1 but it does not tell why is there N-1 at the beginning.  Edit: Please don't confuse with n and n-1 which are used in estimating.  Edit2: I'm not talking about population estimation. After an AJAX request, sometimes my application may return an empty object, like:  var a = {};   How can I check whether that's the case?