I saw from "Advanced Engineering Mathematics, 10th Edition" by Kreyszig, p. 400, that the solution $V$ of the Laplace's equation,  $$\nabla^2 V = \frac{\partial^2V}{\partial x^2}+\frac{\partial^2V}{\partial y^2}+\frac{\partial^2V}{\partial z^2} = 0,$$  is a potential function.  However I thought I can manipulate the expression as $\nabla^2V=0 \ \rightarrow \ \nabla \cdot (\nabla V)=0$. Because $\vec{F}=-\nabla V$, I can write the expression as $\nabla \cdot \vec{F} = 0$.  I did an example calculation with the gravitational force:  $$\vec{F} = - \frac{GMm}{r^2} \hat{r} = - \frac{GMm}{(x^2+y^2+z^2)^{3/2}} \ (x, y, z)$$  and I got $\text{div} \ \vec{F} = \vec{0}$.  But why is this not introduced in most mathematics / physics books?  Or is there any exception?
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?
Should we close 'fix my program' questions Recently, I've seen many questions with this structure:   a wall of code a request for the community to debug it by inspection   You can find a nice selection .  I offer, as a thesis, that these questions are not consistent with the stated goals of SO. They do not, usually, create a resource of use to other people.   As a community, we could politely but firmly tell these askers to do enough of their own work to come up with a concrete, specific, programming question, other than 'can someone tell me what's wrong with this?'  Don't get me wrong. I'm sympathetic to people with problems. I have not been downvoting or voting to close these, because the community seems to be inclined in general to be helpful. I'm asking, are we really doing the right thing by encouraging the site to populate with these?  If the consensus is 'yes' then I'll just slink back to my cave. I'm not interested in starting a campaign here. Can the rogue repeatedly hide in combat to sneak attack the same enemy? One of my players is a rogue and has been, on his turn, using Cunning Action to hide as a bonus action and immediately afterwards attacking. I have been ruling that this does work to grant him advantage as an unseen attacker as I believe that is the RAW, but it seems a little strange. Is he really, in the span of 6 seconds, ducking behind a corner and popping back out and this completely disorientates his target? I guess I just wanted to make sure I was ruling correctly. In a semiconductor memory chip, each bit of binary data is stored in a tiny circuit called a memory cell consisting of one to several transistors. Volatile type.   Suppose an application stored its data in a particular segment on the computers RAM.  How would the CPU know what data to extract and how would it sort through the other data to get to it. If everything in its core level is 0 or 1 its hard to distinguish what purpose 'x' memory has. How to perform a true GIS clip of polygons layer using a polygon layer in R? Suppose $f$ is entire and $|f(z)| \leq A + B |z|^{3/2}$. Show that $f$ is a linear polynomial.   My attempt: Since $f$ is entire, we know that it has a Taylor series expansion on the circle $|z| = R$, that is $f(z) = \sum_{n = 0}^\infty a_n z^n$. By using Cauchy's estimate, we can write   $|f^n(0)| \leq \frac{n! M}{R^n}$. Since this holds for any $R&gt;0$, it follows that for $n\geq 2$, $|f^n(0)| \leq 0$. This implies that $f^n(0) = 0$ for $n\geq 2$. Hence $f$ is a polynomial of degree at most $1$.  Edit 1 : On the circle $|z| = R$, $|f(z)|\leq A + B R^{3/2}$.  Thank you very much.  Kindly suggest me if I am wrong? Is there any other apporoach to solve such kind of problems? In Star Trek, does the original die in teleportation? I am just curious.  If I am teleported is the original me dead and the teleporter just makes another copy of me there[at destination]? "No root file system defined" error while installing ubuntu I am new to linux/ ubuntu and just downloaded the v11 of ubuntu to learn linux. I faced this "No root file system is defined" error while trying to install it. This is how I was proceeding -    Downloaded ubuntu v11 from the ubuntu.com website, wrote the iso into a bootable cd, and then got into the linux installation process on restarting; everything fine till here. In one of the screen during installation, there are three choices - install alongside windows, something else etc.. Since I have windows 7 in the C drive, I selected the first option - "Install alongside another OS".      Let me interrupt here to give my existing system details - HardDisk   size: 640GB.  Number of existing partitions - 5 C drive - label   windows - 160GB - has windows 7 installed D drive - label linux -   160GB - does not have any data E drive - label backup - 100GB - has   some data F drive - label entertainment - 100GB  - has some data G   drive - label work - 90GB  - has some data All of the above are in   NTFS format.    In the screen that follows, an image is shown with 2 parts and each part is shown to have size equal to 80GB. There is a horizontal slider in the middle of these two images, with which I can resize the partitions. The right side partition is labeled Linux. There is also an option to do 'advanced partitions'. On clicking advanced partitions on the above screen, a new screen is shown in which the partitions existing in my computer are shown - but the labels are different. I see that there are two partitions with size = 160GB each (as expected), and so I assumed that the first 160GB would be my existing C drive and the second on would be the partition which I had labeled as linux (drive D). So I selected the linux partition and clicked on proceed/ install now, and then this error was shown - "No root file system defined".. and I cannot proceed after this.   Please let me know two things here:   During the screen in point number 3 above, the image with slider was shown. Which partition (drive c: or d: or anyother) is being referred to here? Which one am I actually resizing - is it the existing c: drive which has windows 7 installed already, that is getting shrunk and then ubuntu getting installed in the remaining space? What does ""No root file system defined" error mean? How do I select the proper partition and continue with the installation?    P.S.:  I had already partitioned my drives as mentioned above, and the D drive with 160GB was specifically left free for installing ubuntu. I would like to Select D drive during setup and install ubuntu here, while still keeping the already installed windows7 in C drive.  Thanks! How do I efficiently iterate over each entry in a Java Map? I'm going on a vacation from Europe to the US and was planning of bringing along some Xbox360 or PS3 games, but now I'm curious if they will be compatible with my European Xbox.  In that past there would be a clear difference between NTSC and PAL games, but is this still the case with the current gen of consoles? using a:hover to change the color of an image Is "has or will read" grammatical? Assigning strings to arrays of characters I am a little surprised by the following.  Example 1:  char s[100] = "abcd"; // declare and initialize - WORKS   Example 2:  char s[100]; // declare s = "hello"; // initalize - DOESN'T WORK ('lvalue required' error)   I'm wondering why the second approach doesn't work. It seems natural that it should (it works with other data types)? Could someone explain me the logic behind this? The new notice for protected questions looks like this:     Earn 10 reputation in order to answer this question. The reputation requirement helps protect this question from spam and non-answer activity.  The phrasing of this notice leaves out a lot of info, and is incorrect in many cases:  In the case where the user has just created their account on the site (whether or not they have the association bonus of +100), the default 1 for signing up does count toward the required 10. The message implies that the user has to earn 10 more reputation for a total of 11 (or 111), while the actual requirement is 10 (or 110).  If the user just registered and started off with the association bonus, then later lost part of it (e.g. through bounties or making a downvoted post), earning 10 reputation is not enough to answer protected questions. According to the literal wording of the notice, earning 10 reputation is enough regardless of previous rep loss, but this is not the case.   The easiest fix that can be made, to at least fix part of the problem, would be to explicitly indicate that the association bonus does not count. This was implemented in the prior protected question notice in response to , but was omitted from the new notice. This omission  , and adding this back would remove that. Can we please fix this notice? Change \headheight for a \pagestyle Open CSV File with Different Delimiter Why does paper become translucent when smeared with oil but not (so much) with water? When I smear oil onto a scrap of paper and rub it in, the paper becomes quite translucent; but when I attempt the same with water it doesn't as much. Why? Find $\lim_{n\to\infty}2^n\underbrace{\sqrt{2-\sqrt{2+\sqrt{2+\dots+\sqrt2}}}}_{n \textrm{ square roots}}$. Find $\displaystyle \lim_{n\to\infty}2^n\underbrace{\sqrt{2-\sqrt{2+\sqrt{2+\dots+\sqrt2}}}}_{n \textrm{ square roots}}$.   By geometry method, I know that this is $\pi$. But is there algebraic method to find this ?   Thank you. Estimating Uniform distribution endpoints using data with errors Suppose I have a random variable $X$ ~ $Unif(0,\theta)$ where I want to estimate $\theta$. I draw a sample $X_1,...,X_n$.One way is to get a point estimate using e.g.  maximum likelihood estimation via the sufficient statistic $X_{(n)}$.  However, suppose I want a 95% confidence interval of sorts on the value of $\theta$, how should I construct one?  The reason why I am interested in confidence interval is because of the following issue, which is actually my main question (the above prelude on confidence interval is just a suggestion on possible approach). So actually my sample came with some noise, in a sense that I actually observe $X_1+E_1,...,X_n+E_n$ where $E_i$~$N(0,\sigma_i^2)$ independently from each other and from the $X$'s. How should I make some form of statistical estimation about $\theta$? I would prefer some form of confidence estimate. This question is regarding the minecraft Beta 1.8 prerelease -- I have a beautiful house in single player survival which I'd like to record while flying.  I want to be able to change the game mode of my survival map to creative to be able to fly through it without installing a mod. I know it is possible if you run the map on a multiplayer server via the command /gamemode &lt;user&gt; 0|1 but I want to achieve this in single player :)  I think this would be possible through editing your level.dat file as I think that's where the game mode for the map is stored. Possible Duplicate:             I faced this issue today when I was replying to a comment posted which used @Transaction annotation.     @Trevor: @Transaction and ... @Transaction   When typing this comment, I got a warning.