Magnetic field due to a coil of N turns and a solenoid
Why is the B field of a solenoid equal to $\mu_0 i n$ while that of a loop is $\frac{\mu_0 i R^2}{2(R^2+z^2)^{3/2}}$?
Coordinate-Free Definition of Trace. $\DeclareMathOperator{\tr}{trace}$ I am reading the  on the trace operator. The section titled Coordinate-Free Definition defines the trace as follows.  Let $V$ be a finite dimensional vector space over a field $F$ and define a bilinear map $f:V\times V^*\to F$ as $f(v, \omega)=\omega(v)$ for all $(v, \omega)\in V\times V^*$. This maps induces a unique linear map $\tr:V\otimes V^*\to F$. Since $\text{End}(V)$ has a canonical isomorphism with $V\otimes V^*$, we have now a notion of trace of a linear operator on $V$.  The Question: The second paragraph of the section in the article says that  This also clarifies why $\tr(AB)=\tr(BA)$.  I can't see how $\tr(AB)=\tr(BA)$ follows from this definition at all. Can somebody give me a hint? I am studying elementary Algebraic Topology recently. I have seen that a topological space is identified with a group. We are telling the group as Fundamental Group. So every topological space $X$ and for any $x \epsilon X$ there is a fundamental group $\pi_1 (X,x)$. I have a question about the converse. If we consider any arbitrary group $G$, then does there exist a topological space $X$ and an element $x\epsilon X$ such that $\pi_1 (X,x)=G$? I can type:  dirname ~/home/blah/file.zip   for instance, and this works fine alone, but when I use this syntax as the parameter for some command it always reads 'dirname' as the desired directory. i.e.:  cd dirname ~/home/blah/file.zip bash: cd: dirname: No such file or directory   Basically, how do I get 'dirname file' to be read as one entity? I have a generic method with this (dummy) code (yes I'm aware IList has predicates, but my code is not using IList but some other collection, anyway this is irrelevant for the question...)  static T FindThing&lt;T&gt;(IList collection, int id) where T : IThing, new() {     foreach T thing in collecion     {         if (thing.Id == id)             return thing;     }     return null;  // ERROR: Cannot convert null to type parameter 'T' because it could be a value type. Consider using 'default(T)' instead. }   This gives me a build error      "Cannot convert null to type parameter   'T' because it could be a value type.   Consider using 'default(T)' instead."   Can I avoid this error? File: \boot\bcd Status: 0xc00000e9 windows 8 I am having this error for quite a few months. When the problem occurs, my windows freezes at boot time with black screen. After few reset attempts, sometimes it shows the entitled error. after few attempts, the windows starts but everything become slow. If I again restart, it works well. But at a point windows never starts and I have to reinstall it. I have already reinstalled 3/4 times in the past 2/3 months. I checked for any HDD errors with EASEUS Partition Master Home Edition with surface test, chkdisk at boot time etc for my C drive and F drive but no errors found. Startup repair also fails to solve the problem.  Windows Boot Manager   Windows failed to start. A recent hardware or software change might be the cause.   To fix the problem:  &nbsp;&nbsp;1. Insert your Windows installation disc and restart your computer.  &nbsp;&nbsp;2. Choose your language settings, and then click "Next." &nbsp;&nbsp;3. Click "repair your computer."  If you don't have this disc, contact your system administrator or computer manufacturer for assistance.   &nbsp;&nbsp;File: \Boot\BCD  &nbsp;&nbsp;Status: 0xc00000e9  My laptop's info:  Model: Dell n5110 corei5 OS: Windows 8 PRO 64 bit Anti-virus:  Avast Home edition  Additional info: My Program Files, Program Files (x86) and Users directory is located at F drive and linked with symbolic link (mklink). Also I have changed these basic locations from regedit. I also use eclipse android intelx86 emulator, virtualbox and Genymotion emulator most often. OS X updater claims installation is successful, but new OS is not installed Adding Layer Files (.lyr, .lyrx) to QGIS At college we use ArcGIS Desktop 10, but that does not function on my Mac so I have QGIS there instead.  I want to do my exercises with QGIS, but in ArcGIS Desktop we use .lyr files with ArcMap and .lyrx files with ArcGIS Pro, and they are not supported by QGIS.  Does anyone know, how I can connect the *.lyr files to my *.shp files to use them on my Mac with QGIS? I've been working with Corel Draw previously and now i use Illustrator more.Still getting used to some of their features though. I want to insert an image into a circle I've drawn but i don't know how to go about it.It is called power clipping in Corel. Any answers would be appreciated. lists Ubuntu 4.10 (Warty Warthog) as the first Ubuntu released by Canonical Ltd.  But I wanted to know whether there ever was an Ubuntu 1.x, as in Windows or Mac OS. If yes, is it available somewhere? Did the Council of Elrond know that destroying the Ring would destroy Sauron? Did they decide to destroy it simply because they were Wise and knew it was dangerous to wield? Did they create the Fellowship simply because they were heroic; and decided to face Sauron after the Ring was destroyed?  Or did they know that destroying the ring would defeat Sauron, and they wouldn't actually have to do anything once that is taken care of? What is the purpose of .bashrc and how does it work? How can I make Chrome open bookmarked pages in new tab by default? Current situation:  I am currently browsing a page (say, www.google.com). I want to open a bookmarked page (say, www.bing.com), but I want to keep the current page (Google) opened. I need to click on new tab, then click on the bookmark for Bing. Alternatively, I can right click and open Bing in new tab.  But I want none of those because why click twice when one is enough?    What I want:  I am currently on Google. I want to open Bing, so I click on the bookmark and Bing opens in a new tab. Google remains open.    I am aware of a duplicate question linked . However it didn't really provide a satisfactory answer. Changing every link to javascript breaks the icon and is a pain to do; middle clicking doesn't really work because I'm on a laptop and the touchpad does not support middle clicking. I hope there is an extension for Chrome that can redefine this behavior to the way I want. A hidden setting in Chrome would be even better but I doubt Chrome offers that. Value of Vandermonde type determinant using the \lstlistoflistings always named as Contents in table of contents.  I did look into   and  but the solutions there did not work.  Example: (my complete preamble)  \documentclass[11pt, oneside]{Thesis}  \usepackage{tikz} \usepackage{pgfplots} \usepackage{pgfplots,filecontents} \usepackage{caption} \usepackage{subcaption} \usepackage{booktabs,colortbl} \usepackage{graphicx} \usepackage{rotating} \usepackage{listings} \usepackage{color} \usepackage{comment} \usepackage[official]{eurosym} \usepackage{colortbl} \usepackage{algorithm}  \usepackage{algpseudocode} \usepackage{indentfirst} \usepackage{latexsym} \usepackage{titlesec} \usepackage[square, numbers, comma, sort&amp;compress]{natbib}   \hypersetup{urlcolor=blue, colorlinks=true}  \renewcommand{\lstlistlistingname}{List of Code}   \begin{document}  \tableofcontents \lstlistoflistings \lstinputlisting[language=python,firstline=1,lastline=68,caption=some caption here., label=code:getbb]{Code/getBoundingBox.py} \end{document}   The renew command changes the header in the page but not in the table of contents. Any ideas?  EDIT x2:  Managed to get the naming correct by editing this in the Thesis.cls   }  \renewcommand\lstlistoflistings{  \btypeout{List of Listings}  \addtotoc{List of Listings}  \begin{spacing}{1}{ \setlength{\parskip}{1pt} \if@twocolumn   \@restonecoltrue\onecolumn \else   \@restonecolfalse \fi \chapter*{\lstlistlistingname   \@mkboth{       \MakeUppercase\lstlistlistingname}{\MakeUppercase\lstlistlistingname}} \@starttoc{lot} \if@restonecol\twocolumn\fi \cleardoublepage  }\end{spacing}   Now the name appears correctly in the TOC but the Listings do no show in the Listings page, its just empty..  A MWE can be found here: Making a (simple) directory tree Prove that for all non-negative integers $m,n$, $\frac{(2m)!(2n)!}{m!n!(m + n)!}$ is an integer. Prove that for all non-negative integers $m,n$, $\frac{(2m)!(2n)!}{m!n!(m + n)!}$ is an integer.   I'm not familiar to factorial and I don't have much idea, can someone show me how to prove this? Thank you. Material setings in properties window is missing How to limit execution time of a function call? There is a socket related function call in my code, that function is from another module thus out of my control, the problem is that it blocks for hours occasionally, which is totally unacceptable, How can I limit the function execution time from my code? I guess the solution must utilize another thread. How to cook fall-off-the-bone beef spare ribs in oven or on stove top? All,  I would like to cook spare ribs to a fall-off-the-bone consistency.  I don't have access to a grill.  I know how to do that with pork shoulders (dry rubbed, then uncovered in the oven at low heat for 6-8hrs), but spare ribs are a different type of animal.  They are individually smaller pieces (my ribs are sawed in chunks), and they are fairly fatty, with a big bone.  What should I do?  Should I pan-sear them first, do I need some liquid in the dish?  Should I cover them?  Thanks,  JDelage Suppose you have a physical . It is true that as amplitude increases, the period increases. Can we demonstrate this fact without explicitly finding the period (which is pretty involved and pretty messy) in:    an intuitive fashion,  rigorously?