What is the algorithm for finding the constraint force using the method of Lagrange multipliers?
How are constraint forces represented in Lagrangian mechanics?
What is the physical significance of the off-diagonal moment of inertia matrix elements? The tensor of moment of inertia contains six off-diagonal matrix elements, which vanish if we choose a reference frame aligned with the principal axes of the rotating rigid body; the angular momentum vector is then parallel to the angular velocity. But while considering the general case, what are the off-diagonal moment of inertia matrix elements? That is, do they have any physical significance as [say] the components of a vector? Or is it merely a mathematical construction with no definite physical meaning (which seems rather wrong to me)?  A similar thread exists  but they are more interested in the principal axes of the body. It also says:     The physical significance of the off-diagonal components is that   you're using a coordinate system not aligned with the principal   directions of the object. They tell us nothing interesting about the   object itself.   Is that all or is there more to it, perhaps related to properties of tensors in general? What is the " := " operator in (VB).NET or what is it good for? If $[G' : G'']\leq p^2$, then $G'$ is abelian. If $a^{n}-1$ is prime then $a=2$ and $n$ is prime? I was doing some basic Number Theory problems and came across this problem :     Show that if $a$ and $n$ are positive integers with $n\gt 1$ and $a^{n} - 1$ is prime, then $a = 2$ and $n$ is prime   My Solution : (Sloppy)        $a^{n}-1$ = $(a-1)$ . $(a^{n-1} + a^{n-2} + ... + a + 1)$   This means that $(a-1)$ | $(a^{n}-1)$    But $(a^{n}-1)$ is prime   So , $(a-1)$ = 1 $\Rightarrow$ $a = 2$   Now , let $n$ be composite   $n = kl$ , where $1 \lt k \lt n$ and $1 \lt l \lt n$   Now , $a^{kl}-1$ = $(a^{k}-1)$ . $(a^{k.{(l-1)}} + a^{k.{(l-2)}} + ... + a^k + 1)$   This means that $2^{n} -1$ is composite    Hence , we have achieved a contradiction      My Question : Am I correct ? This is a question about American English usage of the word "already". As a UK resident I don't completely understand when I hear Americans give commands like "Stop it already!" In the UK the word already is not normally used in the imperative mood and the sentence I've just quoted would leave an English person thinking "If you're saying I've already stopped it why are you asking me to stop it again?" How do I copy to the clipboard in JavaScript? What is the best way to copy text to the clipboard (multi-browser)? I have tried: function copyToClipboard(text) {     if (window.clipboardData) { // Internet Explorer         window.clipboardData.setData(&quot;Text&quot;, text);     } else {         unsafeWindow.netscape.security.PrivilegeManager.enablePrivilege(&quot;UniversalXPConnect&quot;);         const clipboardHelper = Components.classes[&quot;@mozilla.org/widget/clipboardhelper;1&quot;].getService(Components.interfaces.nsIClipboardHelper);         clipboardHelper.copyString(text);     } }  But in Internet Explorer it gives a syntax error. In Firefox, it says unsafeWindow is not defined. A nice trick without using : I've taken multivariate calculus and am wondering if I can see a specific function where the order of taking the partial derivative matters. I've been told that there are some exceptions where $ \dfrac{\partial ^2 f}{\partial x \partial y} \ne \dfrac{\partial ^2 f}{\partial y \partial x} $, so I'm curious to see what this looks like.  EDIT: And why would this true? Windows Phone ReverseGeocoding to get Address from Lat and Long You have 6 red balls, 6 blue, and 6 white. You randomly select a sample of 3 balls without replacement. The balls, save for the color, are indistinguishable from each other. What are the odds that your sample contains at least a ball from each color?  I thought about finding the chance that the sample will not contain at least one color and subtracting that from one:  $$P(\text{sample is missing at least one color})=P(\text{sample is missing one color})+P(\text{sample is missing two colors}) $$  Essentially the second probability statement on the right hand side is the probability that all the balls in the sample are the same color. The case where the sample does not contain three colors doesn't make sense, so I didn't include it. $$=\binom{3}{1} \left[\left(\frac{12}{18}\right) \cdot \left(\frac{11}{17}\right) \cdot\left(\frac{10}{16}\right)\right]+\binom{3}{2}\left[\left(\frac{6}{18}\right) \cdot \left(\frac{5}{17}\right) \cdot\left(\frac{4}{16}\right)\right] \approx 0.882353$$  When all of this is done, we simply subtract the probability from one to get $0.117647$.  What am I doing wrong? Generic htaccess redirect www to non-www I would like to redirect www.example.com to example.com. The following htaccess code makes this happen:  RewriteCond %{HTTP_HOST} ^www\.example\.com [NC] RewriteRule ^(.*)$ http://example.com/$1 [L,R=301]   But, is there a way to do this in a generic fashion without hardcoding the domain name? If I can generally reach web pages on the Internet but can't reach a specific one, how do I troubleshoot what the cause is as an end user?       This question was a .   Read the  for more details or  yourself My homework has this problem:     Prove that there exist irrational numbers $x$ and $y$ such that $x +  y$ is rational.   There is an easy solution that I found on mathbitsnotebook.com:     \begin{align} \left(2+6\sqrt{7}\right) + \left(-6\sqrt{7}\right) = 2.\end{align}   Can we prove the same without using subtraction? Card game probability 3 players I don’t know how to handle this 3 players case, may anyone give a hand? Thanks!     When playing card game, there are 3 players playing 32 cards, each   receive 10 cards, 2 remind on the table. Overall are 4 aces and 4   kings in the game. How likely is P (E | D)? D = "The table remains 2   aces" E = "I (at least) get one king".   I guess I should have steps as follows.   P(E) = P(1King)+P(2K)+P(3K)+P(4K) P(D) = (4/32) * (4/32) = 1/256 P (E | D) = P(E) + P(D)   Equations may be used:  Counting combinations      The number of combinations of n objects taken r at a time is     nCr = n(n - 1)(n - 2) . . . (n - r + 1)/r! = n! / r!(n - r)!                 Number of favorable outcomes P(I) =     Total number of possible outcome Automate taking a screenshot of specified area Setting: Let $\sum a_n z^n$ have radius of convergence $R$.  We have that  $$ R = \frac{1}{\underset{n \rightarrow \infty}{\limsup} \left|a_n \right|^{1/n}} $$  via Hadamard's formula for the radius of convergence.  Question: What are the radii of convergence for (i) $\sum a_n z^{2n}$ and (ii) $\sum a_n^2 z^n$?  Attempt for $\sum a_n^2 z^n$:   For ease of notation, let $R_1$ and $R_2$ denote the radii of convergence for power series $\sum a_n z^{2n}$ and $\sum a_n^2 z^n$ respectively. From Hadamard's formula, we then have  $$ R_2 = \frac{1}{\underset{n \rightarrow \infty}{\limsup} \left|a_n^2 \right|^{1/n}} = \left( \frac{1}{\underset{n \rightarrow \infty}{\limsup} \left|a_n \right|^{1/n}} \right)^2 = R^2 $$   But what about $R_1$? Suppose I wanted to travel to one of the recently discovered potentially Earth-like planets such as Kepler 186f that is 490 light years away. Assuming I had a powerful rocket and enough fuel, how long would it take me? How to insert text before the first line of a file? Can anyone suggest a detailed installation process for Ubuntu on Windows 8.2? Why is setting a variable before a command legal in bash? I've just encountered several answers such as to  that use the construct:  while IFS=, read xx yy zz;do     echo $xx $yy $zz done &lt; input_file   where the IFS variable is set before the read command.  I've been reading through the  but can't figure out why this is legal.  I tried  $ x="once upon" y="a time" echo $x $y   from the bash command prompt but got nothing echoed. Can someone point me to where that syntax is defined in the reference that allows the IFS variable to be set in that way? Is it a special case or can I do something similar w/ other variables? How can I make things link url references, my index, table of contents, and cross references clickable links in documents produced by LaTeX? I generally use pdflatex to produce my documents.