Divergence of Electric Field Due to a Point Charge I am trying to formally learn electrodynamics on my own (I only took an introductory course). I have come across the differential form of Gauss's Law.   $$ \nabla \cdot \mathbf E = \frac {\rho}{\epsilon_0}.$$  That's fine and all, but I run into what I believe to be a conceptual misunderstanding when evaluating this for a point charge.   I know the math looks better in spherical coordinates, but I will be using Cartesian.   So when I calculate the divergence I obtain:  $$ \nabla \cdot \mathbf E = \nabla \cdot kQ\langle\frac{x}{(x^2+y^2+z^2)^{\frac{3}{2}}},\frac{y}{(x^2+y^2+z^2)^{\frac{3}{2}}},\frac{z}{(x^2+y^2+z^2)^{\frac{3}{2}}}\rangle = \frac{-3(x^2+y^2+z^2)}{(x^2+y^2+z^2)^{\frac{5}{2}}}+\frac{3}{(x^2+y^2+z^2)^{\frac{3}{2}}}.$$  This can further be simplified:  $$\frac{-3(x^2+y^2+z^2)}{(x^2+y^2+z^2)^{\frac{5}{2}}}+\frac{3}{(x^2+y^2+z^2)^{\frac{3}{2}}} = \frac{3}{(x^2+y^2+z^2)^{\frac{3}{2}}}-\frac{3}{(x^2+y^2+z^2)^{\frac{3}{2}}} = \frac{3-3}{(x^2+y^2+z^2)^{\frac{3}{2}}}.$$   Now instinctively I would say that 3-3 is zero and then the while thing is zero everywhere. I am confused as to why (purely mathematically) this expression is not equal to zero at the origin. I completely understand why it physically has to be that way. And I also understand that it is modeled with the delta dirac function. But what (again, mathematically) is stopping me from saying that equation is just zero even at the origin?
Divergence of $\frac{ \hat {\bf r}}{r^2} \equiv \frac{{\bf r}}{r^3}$, what is the 'paradox'? 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?
Combinatorics identity question I encountered this particular combinatorial identity :$$\begin{align}&amp;1\dbinom{n-1}{r-1}+2\dbinom{n-2}{r-1}+\cdots+(n+1-r)\dbinom{r-1}{r-1}\\=&amp;\dbinom{n}{r}+\dbinom{n-1}{r}+\cdots+\dbinom{r}{r}\\=&amp; \dbinom{n+1}{r+1}\end{align}$$How does one figure it out? Err 80 appears when I turn on my Canon EOS 5D Mark II Recently when on holiday in Italy my 5DII stopped powering up. Turn the switch on and - nada. Pulling the battery and popping it back in brings the camera to life, however as soon as the camera goes in to sleep mode, or I turn the power off, the problem returns. Sometimes the camera shows Error 80 at this point:     Err 80      Shooting is not possible due to an error.    Turn the camera off and on again or re-install the battery.   I've seen various posts on the interwebs about pulling the clock battery and leaving the camera for a while - have tried this and also replaced the clock battery to no avail.   Has anyone else had this problem - did the body have to go back to Canon for repair or did you manage to fix it? How can I make a tool drop blocks with CanPlaceOn tag In Adventure mode, I'm trying to let a player get blocks that he can place later, is there a way to do it directly or do I have to change them in his inventory?  Situation: The player gets a tool that will allow him to clear a boulder blocking the road. Further down the path, he encounters a broken bridge, and I want him to use the blocks he gathered before to rebuild it.  At the moment, if I build the boulder with blocks having the CanPlaceOn tag, the tag disappear when the player destroys them and they go in its inventory. I am trying to copy the contents of a folder to another folder in a different directory using terminal.  Would somebody be able to provide me an example of the command line syntax required to achieve this? I'm using Apache JMeter 2.3, which now supports "attempt HTTPS spoofing" under the Proxy Server element.   I've tried this on several different servers, and have had no success.  Has anyone been able to successfully record from an HTTPS source with this setting?  Or barring successfully recording, can anyone share a work-around? When available, I simply have HTTPS turned off at the server level, but this is not always feasible. Thoughts? MacOS partition showing as "FFFFFFFF-FFFF-FFFF-FFFF-FFFFFFFF" bashrc lazy substitution I've been using SSH Tunnel manager to easily set up and stop port tunneling over SSH to various servers at my company. That saved me the pain of opening up a new terminal window which will hang out there just for the sake of having a tunnel open.  This worked great (well , sort of) but SSH Tunnel Manager is a PowerPC application. With Lion, it is not supported any more.  What is a good replacement? Debugging JavaScript in IE7 I need to debug JavaScript in Internet Explorer 7. Unfortunately, its default debugger doesn't provide me with much information. It tells me the page that the error showed up on (not the specific script) and gives me a line number.  I don't know if that is related to my problem. It'd be nice if it could narrow down the error to a line number on a specific script (like Firebug can). Is there an addon to debug JavaScript in IE7 like Firebug does in Firefox? Thank you! See also: TestMethod do not support Web service callouts, test skipped I have written a test class for a trigger on the lead object.  When i run the test class i get the following error:  Methods defined as TestMethod do not support Web service callouts, test skipped  There is an other trigger on lead that has a call to a webservice so i get this error.  I have wrapped the call with:  if (!Test.isRunningTest()) {    // do callout }   But i’m still getting the same error.  Any other way to run correctly the test class?  Thank you in advantage. Let $f(x)$ be a monotone non-increasing function such that $\int_{0}^{\infty}f(x) dx$ converges. Prove: $\lim\limits_{x \to \infty} xf(x)=0$. My question is, why can't I simply contradict any other possibility by using the Integral Limit Comparison Test with $1\over x$? I am, after all, to show that $f(x)=o(x)$ as $x\to \infty$. I really don't understand why monotony is crucial here. I could use some help. Google Drive storage mismatch I have a paid 100GB Google Drive plan. It is reporting that I am at 99% capacity.  According to the the storage both within the desktop and web application, I have used 90.53GB within Drive.    My Drive is mapped to my OS X filesystem, and when fully synched it reports only 61.28GB used.    So what is going on here? I need to do a cleanup but cannot account for the 30GB difference. I am writing a journal which has two column. I am using {figure} as well as {figure*} to place the figures in one column as well as stretch across two columns.   My problem is the order of the figures are not correct. e.g. their order of appearance is:                  Figure 1.                 Figure 2.                 Figure 5.                 Figure 3.                 Figure 4.   What should I do to make them appear in order? Can all real polynomials be factored into quadratic and linear factors? How to remove ":" (colon) when displaying field name? I created a custom content type with some text fields. In the view settings of that content type I defined both to appear and as well their titles (names of the fields) to be displayed above the content of the fields respectively.  But how can I remove the trailing colon of the titles? Is there a way of modifying the way field names are display just for certain content types? I do not want this altered site wide (e.g. for all fields of all content types).  PS: I'm already using the "views" module to define a special layout for this custom content type. Probably there is a way to carry out these changes with the help of the "views" module and a custom layout template?  /edit/ I did not create a view (as with the "views" module) to display that content but used Drupal's own content type view configuration (maybe called "display"). I'm working with a localized Drupal and retranslating the different words.  Remark: Using Drupal 7.10 with Views 7.x-3.0-rc3 How do you give a C# auto-property an initial value?  I either use the constructor, or revert to the old syntax.    Using the Constructor:  class Person  {     public Person()     {         Name = "Initial Name";     }     public string Name { get; set; } }   Using normal property syntax  (with an initial value)  private string name = "Initial Name"; public string Name  {     get      {         return name;     }     set     {         name = value;     } }   Is there a better way? How to find the list of all available packages along with the lastest version number available in the repositories? I want to list all the packages(installed or not installed) along with the latest version number available in the repositories.  For example:  The output of apt-cache policy chromium-browser shows like below  $ apt-cache policy chromium-browser chromium-browser:   Installed: (none)   Candidate: 34.0.1847.116-0ubuntu2   Version table:      34.0.1847.116-0ubuntu2 0         500 http://ftp.cuhk.edu.hk/pub/Linux/ubuntu/ trusty/universe amd64 Packages   So the latest version of chromium-browser package available in the repository is 34.0.1847.116-0ubuntu2. Like that, i want to list the version number along with the package names that are available in repositories. And all i want to do this through command-line.  I want the output to be like this,  chromium-browser 34.0.1847.116-0ubuntu2 xxxxxxxxxxxxxx   yyyyyyyyyyy ............     ............ Is there a way to log all Microsoft SQL Server queries executed on a certain server, with their respective durations?  I'd like to be able to analyze all queries will a tool like what  does for PostgreSQL. While we're at it: do you know of any such tools? Limit involving binomial coefficients: $\lim_{n\to\infty}\left(\binom{n}{0}\binom{n}{1}\dots\binom{n}{n}\right)^{\frac{1}{n(n+1)}}$ I am facing difficulty with the following limit. $$ \lim_{n\to\infty}\left(\binom{n}{0}\binom{n}{1}\dots\binom{n}{n}\right)^{\frac{1}{n(n+1)}} $$ I tried to take log both sides but I could not simplify the resulting expression. Please help in this regard. Thanks. Best way to return multiple values from a function? The canonical way to return multiple values in languages that support it is often . Option: Using a tuple Consider this trivial example: def f(x):   y0 = x + 1   y1 = x * 3   y2 = y0 ** y3   return (y0, y1, y2)  However, this quickly gets problematic as the number of values returned increases. What if you want to return four or five values? Sure, you could keep tupling them, but it gets easy to forget which value is where. It's also rather ugly to unpack them wherever you want to receive them. Option: Using a dictionary The next logical step seems to be to introduce some sort of 'record notation'. In Python, the obvious way to do this is by means of a dict. Consider the following: def g(x):   y0 = x + 1   y1 = x * 3   y2 = y0 ** y3   return {'y0': y0, 'y1': y1 ,'y2': y2}  (Just to be clear, y0, y1, and y2 are just meant as abstract identifiers. As pointed out, in practice you'd use meaningful identifiers.) Now, we have a mechanism whereby we can project out a particular member of the returned object. For example, result['y0']  Option: Using a class However, there is another option. We could instead return a specialized structure. I've framed this in the context of Python, but I'm sure it applies to other languages as well. Indeed, if you were working in C this might very well be your only option. Here goes: class ReturnValue:   def __init__(self, y0, y1, y2):      self.y0 = y0      self.y1 = y1      self.y2 = y2  def g(x):   y0 = x + 1   y1 = x * 3   y2 = y0 ** y3   return ReturnValue(y0, y1, y2)  In Python the previous two are perhaps very similar in terms of plumbing - after all { y0, y1, y2 } just end up being entries in the internal __dict__ of the ReturnValue. There is one additional feature provided by Python though for tiny objects, the __slots__ attribute. The class could be expressed as: class ReturnValue(object):   __slots__ = [&quot;y0&quot;, &quot;y1&quot;, &quot;y2&quot;]   def __init__(self, y0, y1, y2):      self.y0 = y0      self.y1 = y1      self.y2 = y2  From the :  The __slots__ declaration takes a sequence of instance variables and reserves just enough space in each instance to hold a value for each variable. Space is saved because __dict__ is not created for each instance.  Option: Using a  (Python 3.7+) Using Python 3.7's new dataclasses, return a class with automatically added special methods, typing and other useful tools: @dataclass class Returnvalue:     y0: int     y1: float     y3: int  def total_cost(x):     y0 = x + 1     y1 = x * 3     y2 = y0 ** y3     return ReturnValue(y0, y1, y2)  Option: Using a list Another suggestion which I'd overlooked comes from Bill the Lizard: def h(x):   result = [x + 1]   result.append(x * 3)   result.append(y0 ** y3)   return result  This is my least favorite method though. I suppose I'm tainted by exposure to Haskell, but the idea of mixed-type lists has always felt uncomfortable to me. In this particular example the list is -not- mixed type, but it conceivably could be. A list used in this way really doesn't gain anything with respect to the tuple as far as I can tell. The only real difference between lists and tuples in Python is that lists are , whereas tuples are not. I personally tend to carry over the conventions from functional programming: use lists for any number of elements of the same type, and tuples for a fixed number of elements of predetermined types. Question After the lengthy preamble, comes the inevitable question. Which method (do you think) is best?