I am working through some questions and I am stuck on the working of this one:  Where the working to the answer is here:  Can someone explain the highlighted section as I can't see where it comes from?
I just started in Griffith's Introduction to electrodynamics and I stumbled upon the divergence of $\frac{ \hat r}{r^2} \equiv \frac{{\bf r}}{r^3}$, now from the book, Griffiths says:  Now what is the paradox, exactly? Ignoring any physical intuition behind this (point charge at the origin) how are we supposed to believe that the source of $\vec v$ is concentrated at the origin mathematically? Or are we forced to believe that because there was a contradiction with the divergence theorem? Also how would the situation differ if $\vec v$ was the same vector function but not for a point charge? Or is it impossible?
Overheard on an elevator today,     I didn't realize it was him.   Corrected by the know-it-all,     He.  "I didn't realize it was he."   The know-it-all then went on a rant about how everybody is a dolt for not knowing that.  I swear I have never heard this before in my life.  Is the know-it-all correct?    If so, why is "I didn't realize it was he," grammatically correct?  What rule makes "him" incorrect here? Here is some information we got : We know the value of $n$, with size $1043$. We know the value of $p$ (size $20$) and $q$ (size $1023$) as the factors. $e = 65537.$ $\varphi(n)$ = $(q-1)(p-1)$ When I calculated $\gcd$ and $\text{modinv}$, I got : $\gcd(e,\varphi(n)) = 65537$ $modinv(e,\varphi(n)) = 1 $ So we can tell that they are not relatively prime. So, how to compute the d, and get the value of m? I'm not that good with math, so I cant understanding well the theory. so can anyone please make an example implementation or write a clear formula? Why can't I change my desktop background? How do I search for questions with specific tags?       For more information about searching, see &quot;&quot; in the . How do I start applications automatically on login? Finite groups with exactly one maximal subgroup line 56 syntax error: unexpected end of file I have typed out the code for this mychoice.sh bash example and I keep getting the syntax error: unexpected end of file.  Am I missing something?  mychoice.sh  #!/bin/bash  E='echo -e';e='echo -en';trap ";exit" 2 ESC=$( $e "\e") TPUT(){ $e "\e[${1};${2}H";} CLEAR(){ $e "\ec";} CIVIS(){ $e "\e[25l"} DRAW(){ $e "\e%\e(0";} WRITE(){ $e "\e(B";} MARK(){ $e "\e[7m";} UNMARK(){ $e "\e[27m";} R(){ CLEAR ;stty sane;$e "\ec\e[37;44m[J";}; HEAD(){ DRAW     for each in $(seq 1 13);do     $E "   x                    x"     done     WRITE;MARK;TPUT; 1 5     $E "BASH SELECTION MENU         ";UNMARK;}     i=0; CLEAR; CIVIS;NULL=/dev/null FOOT(){ MARK;TPUT 13 5     printf "ENTER - SELECT,NEXT         ";UNMARK;} ARROW(){ read -s -n3 key 2&gt;/dev/null &gt;&amp;2      if [[$key = $ESC[A ]];then echo up;fi      if [[$key = $ESC[B ]];then echo dn;fi;} M0(){ TPUT  4 20; $e "Login info";} M1(){ TPUT  5 20; $e "Network";} M2(){ TPUT  6 20; $e "Disk";} M3(){ TPUT  7 20; $e "Routing";} M4(){ TPUT  8 20; $e "Time";} M5(){ TPUT  9 20; $e "About  ";} M6(){ TPUT 10 20; $e "Exit   ";} LM=6 MENU(){ for each in $(seq 0 $LM);do M${each};done;} POS(){ if [[ $cur == up ]];then ((i--));fi     if [[ $cur == dn ]];then ((i++));fi     if [[ $i -lt 0 ]];then i=$LM;fi     if [[ $i -gt $LM ]];then i=0;fi;} REFRESH(){ after=$((i+1)); before=$((i-!))     if [[ $before -lt 0 ]];then before=$LM;fi     if [[ $after -gt 0 ]];then after=0;fi     if [[ $j -lt $i ]];then UNMARK;M$before;else UNMARK;M$after;fi     if [[ $after -eq 0 ]] || [$before -eq $LM ];then          UNMARK; M$before; M$after;fi;j=$i;UNMARK;M$before;M$after;} INIT(){ R;HEAD;FOOT;MENU;} SC(){ REFRESH;MARK;$S;$b;cur='ARROW';} ES(){ MARK;$e "ENTER = main menu ";$b;read;INIT;};INIT while [[ "$0" != " " ]]; do case $i in     0) S=M2;SC;if [[ $cur == "" ]];then R;$e "\nS(w     )\n";ES;fi;;     1) S=M2;SC;if [[ $cur == "" ]];then R;$e "\nS(ifconfig  )\n";ES;fi;;     2) S=M2;SC;if [[ $cur == "" ]];then R;$e "\nS(df -h )\n";ES;fi;;     3) S=M2;SC;if [[ $cur == "" ]];then R;$e "\nS(route -n  )\n";ES;fi;;     4) S=M2;SC;if [[ $cur == "" ]];then R;$e "\nS(date  )\n";ES;fi;;     5) S=M2;SC;if [[ $cur == "" ]];then R;$e "\nS($e by oTo )\n";ES;fi;;     6) S=M2;SC;if [[ $cur == "" ]];then R;exit 0;fi;;  esac;POS;done; Definite article before name of languages Should I use the definite article before name of languages?   (the?) Persian language (the?) German language (the?) Arabic language (the?) Russian language (the?) English language   Some language name may also refer to other things, so you need language word after their name. subequations and array in braces I need something like this:  \left\{ first line  (1a) second line (1b) third line  (1c) \right.   all lines should be aligned to left limit of Holder norms: $\sup\limits_{x\in [a,b]} f(x) = \lim\limits_{n\rightarrow\infty} \left(\int_a^b (f(x))^n \;dx\right)^{\frac{1}{n}}$ Show that $$\sup_{x\in [a,b]} f(x) = \lim_{n\rightarrow\infty} \left(\int_a^b (f(x))^n \;dx\right)^{\frac{1}{n}}$$ for f continuous and positive on [a,b].  I can show that LHS is greater than or equal to RHS but I can't show the other direction. Sequences or 'chains' of adjoint functors Suppose we have (some categories and some functors such that) $F_1$ is left adjoint to $G_1$, $G_1$ left adjoint to $F_2$, $F_2$ left adjoint to $G_2$. Will $F_1$ then be equal to $F_2$ (and $G_1$ to $G_2$)?  Or can there exist arbitrarily long 'chains' of adjoint functors that do not trivialize? Double Negation in C++ I just came onto a project with a pretty huge code base.   I'm mostly dealing with C++ and a lot of the code they write uses double negation for their boolean logic.    if (!!variable &amp;&amp; (!!api.lookup("some-string"))) {        do_some_stuff();  }                                      I know these guys are intelligent programmers, it's obvious they aren't doing this by accident.   I'm no seasoned C++ expert, my only guess at why they are doing this is that they want to make absolutely positive that the value being evaluated is the actual boolean representation. So they negate it, then negate that again to get it back to its actual boolean value.   Is this correct, or am I missing something? I have been wondering why anyone ever thought that we could find an expansion for eigenstates of some perturbed Hamiltonian in terms fo those for the basis Hamiltonian. My lecturer insisted that this was because the eigenstates of the Hamiltonian are complete', however from my (limited) understanding of the mathematical basis of QM:   When you are given a Hamiltonian, it comes with the Hilbert space of functions of which it acts (subject to some conditions we impose, such as the Lebesgue square-integrability of functions) When you are given another Hamiltonian, there is no guarantee that it will act on the same Hilbert space Therefore, that the zeroth order Hamiltonian has eigenstates forming a complete basis for the space on which ti acts does not guarantee that these same eignestates form a complete basis for the Hilbert space on which the purturbed Hamiltonian acts.   As a simple example. I can imagine adding a small perturbing magnetic field to a Hamiltonian in which there initially is none, such that we lose the degeneracy in the eigenstates accounting for different spins of a particle. One could say that we have the same Hilbert space before and after but we lose degeneracy; or I would have been tempted to say that our initial Hilbert space did not have these extra dimensions corresponding to the spin states. So rather than the eigenstates of the basis Hamiltonian also forming a complete basis for the purtubed Hamiltonian, but having eigenvalue degeneracy which is lifted on the purtubration, I would have said that our Hilbert space has changed. After all, if we take the former view (i.e. a basis accouting for spin but with degeneracy) then it seems like for any problem we would need to know and account for EVERY possible system property that can be discerned. This doesn't seem reasonable to me....  NOTE:  1) My question was asked here previously:   although there is only a discussion and no conclusive answer given. I see that the distinction is important in relativistic QM, and the case of adding a potential onto a free particle Hamiltonian seems to be a special one. Although I am still not sure what the answer is for general purturbations.  2) Similar question here:   but it seems to completely avoid answering the question. Getting polygon shapefile node coordinates and point order Can a multiclass character use either of its spellcasting modifiers for spells? Link2SD not working completely Is there consensus in the field of statistics that one book is the absolute best source and completely covering every aspect of GLM - detailing everything from estimation to inference? What can I use to find duplicate photos, including photos that have been resized? Simple & easy way to jail users I need a simple and easy way to jail users in their home directories in Oneiric. Do you have a simple configuration for jailing users, with full help, or some good web links?  I would be offering an online free public server with 10 to 20 GB free space. I don't know how many users. I want to give them SSH and SFTP so that they can connect through FileZilla. Why do we use Hermitian operators in QM? Position, momentum, energy and other observables yield real-valued measurements. The Hilbert-space formalism accounts for this physical fact by associating observables with Hermitian ('self-adjoint') operators. The eigenvalues of the operator are the allowed values of the observable. Since Hermitian operators have a real spectrum, all is well.  However, there are non-Hermitian operators with real eigenvalues, too. Consider the real triangular matrix:  $$ \left( \begin{array}{ccc} 1 &amp; 0 &amp; 0 \\ 8 &amp; 4 &amp; 0 \\ 5 &amp; 9 &amp; 3 \end{array} \right)  $$  Obviously this matrix isn't Hermitian, but it does have real eigenvalues, as can be easily verified.  Why can't this matrix represent an observable in QM? What other properties do Hermitian matrices have, which (for example) triangular matrices lack, that makes them desirable for this purpose?