A point charge of $\rm+5.00 μC$ is located on the $x$-axis at $x=\rm5.00m$ , next to a spherical surface of radius $x=\rm4.00m$ centered at the origin.  According to Gauss's law, the net flux through the sphere is zero because it contains no charge. Yet the field due to the external charge is much stronger on the near side of the sphere (i.e., at $x=\rm4.00m$) than on the far side (at $x=\rm-4.00m$). How, then, can the flux into the sphere (on the near side) equal the flux out of it (on the far side)?  My Thoughts:  I know that the flux should be zero. I drew a picture, with the charge to the right of the sphere and drew lines radially from the point. However, I can't seem to wrap my head around why the net flux is zero. If I go through adding up all the $\rm\vec{E}.d\vec{A}$'s, does it have to do with the $\rm d\vec{A}$ being the opposite direction of the field on the close side to the particle, so flux is negative, but on the top, bottom, and left (front and back as well) sides of the sphere, the field is in a similar direction so flux would be positive?
If we have a point charge and outside of it we have a non-conducting Gaussian sphere, then Gauss's law says that the net flux should be zero. I agree that the total field lines coming in are equal to the number of field lines coming out, but isn't it true that the magnitude of the electric field at those segments of the sphere where the field is coming in is greater than those where it is coming out? So when we integrate, wouldn't these areas, having a greater electric field, contribute more to the net flux? I don't see why they would cancel each other out.
To show that $2$ | $x^{2n} - x^n$, $n \in \mathbb{Z^+}$ is the following adequate? Show that $2$ | $x^{2n} - x^n$, $n \in \mathbb{Z^+}$   Let $x^n = a$ then $\frac{a}{2}$ $\implies a = 2q + r$ such that $r = 0 \vee r = 1$.   So $x^{2n} = (x^n)^2 = a^2 = (2q + r)^2$, which means that $2|[x^{2n} - x^n ]= 2| [(2q + r)^2 - (2q + r)]$. If $r = 1$ we get $(4q^2 + 4q + 1) - 2q - 1 = 4q^2 + 2q = 2(2q^2 +1)$. Thus $2$|$x^{2n} - x^n$, $n \in \mathbb{Z^+}$. Is this reasoning correct? Is it adequate? FaceTime and iMessage - an error occurred during activation I couldn't log in to FaceTime so I did a factory reset of my MacBook. It worked for a brief period of time and then suddenly it just logged out.  I believe it might have been after iCloud synced with keychain or after I changed my local host name and computer name.  I have seen other users facing this issue but no solutions. Is there a troubleshooting guide or should I just contact Apple support since I can't use their service without activation to my Mac? Detect Chrome extension first run / update Compiling with g++ using multiple cores I play D&amp;D 5e and sometimes come here with questions. But sometimes I see things referenced here as sources that I don't have or aren't aware of....  Which are the "official rules" for D&amp;D 5e, and where do I find them? What am I doing wrong? Joint Density of sum of random variables! So, I have this question:    And, what I have tried so far is I let $z_1 = \frac{X + Y}{2}$ and let $z_2 = \frac{X - Y}{2}$ to use change of variables for this question. However, when I work this out and plug it into my joint distribution of $X$ and $Y$ above, I end up with the $z_2$ disappearing and only having $z_1$ left in my joint distribution of $z_1$ and $z_2$. So, then when I go to find the marginal for just $z_1$ (which is $\frac{X + Y}{2}$ &lt;- what I am looking for), there is no $z_2$ therefore, integrating wrt to $z_2$, I get $z_2$ just come up (integral of 1). This is an issue because the bounds are from 0 to $\inf$. Obviously, I can't have infinite in my marginal density so what am I doing wrong?   Also, for question 5, I work it out similarly except I let $z_2 = X$ but I just end up with the exact same density as $f_{x,y}$ as seen in question 4 - is that normal? Please help, I been doing this question for a long time trying various dummy transformations.  Also, for question 4, I found the marginals of $x$ and $y$ and realized that the independence consequence (of the product being equal to the joint) is satisfied but using convolution, I get the same issue of my integral coming out to infinite! Thank you for all the help!   EDIT: Am I doing my bounds wrong? the bound for z_2 is from 0 to inf or 0 to z_1? dpkg: error: parsing file '/var/lib/dpkg/updates/0045' near line 0: A gold reviewer badge? For your continued efforts in reviewing posts, even though the system sometimes messes up and doesn't let you actually review things and keeps displaying them over and over and over. So also, for your patience in putting up with the glitches in the review panel too! Haha  Anyways, a gold reviewer badge: 20000 reviews, over 4000 actioned  I just multiplied both numbers by 20. Think those numbers seem challenging? Is anyone up near those numbers or should it be multiplied by 10 instead? I prefer the higher amount because it would be gold and something harder to work for. :) But we also have to consider the longevity of when all these posts are eventually reviewed and it's only new posts being created.  Anyways, my stats for those who are interested (I wouldn't get the badge to start because I only have ~2000 actioned):  Upvoted            189 Downvoted          11 Edited             926 Commented          127 Voted to Close     44 Flagged            447 Reviewed           21609   As far as the name of the badge, I really like "The Owl is Watching" but not sure if that would be too long. Sounds snazzy though. Ideas?    Personally, I was on a quest to try and clear out all the posts in a single tab before it started showing me the same things over and over (and over) and then it got frustrating because I wasn't even making progress. Quest denied. :(    Lately I've noticed a massive build-up of unreviewed posts. When the review panel was first implemented, there would only ever be maybe 50 unreviewed posts at a time (in a given tab) for my monthly view. Now I log in and there's a thousand or two thousand pending. I think a lot of people quit reviewing after the first badge, so having a gold badge with a high threshold would really improve activity in the review panel and keep people going to improve the community. the structure "like for somebody to do something" Is there a problem with the ClearBits data dump? I would like to convert a Mp4 file in HD into a Mp4 file of lower resolution. Is there any software that can be used for this? How do I prove equality of x and y? If $0\leq x,y\leq\frac{\pi}{2}$ and $\cos x +\cos y -\cos(x+y)=\frac{3}{2}$, then how can I prove that $x=y=\frac{\pi}{3}$? Your help is appreciated.I tried various formulas but nothing is working. Forcing series convergence Find all natural numbers $n$ such that ${(n+1)}^{1/2}-{(n-1)}^{1/2}$ is rational Page title with substitution on views page I have a view page that requires a dynamic page title. It is a simple substitution of the label field.  If I use {{ label }} it works fine for the title displayed on the page, but it doesn't substitute the label for the actual page title (in meta)  I was able to fix this by using hook_views_post_render using the following code  // Set the route title. $route = \Drupal::routeMatch()-&gt;getCurrentRouteMatch()-&gt;getRouteObject(); $route-&gt;setDefault('_title', $title);  $view-&gt;setTitle($title);   However, the title only works if the page is not cached. For the cached pages, the page title is missing. This feels like a bug in drupal, but I am hoping that there is a simple way to get this done. Command for argmin or argmax? How can you get the argmin or argmax in Latex? Two solutions I found are:  \underset{x}{\operatorname{argmax}}   \DeclareMathOperator*{\argmin}{arg\,min}   Any other ideas?  [1]   [2] Badge Notification Repeats Not sure if someone else has already raised this - I've got eleven notifications that I've received a Caucus badge on English Language Learners. Perhaps one notification for each community that I'm a member of?  I am a member of the community, and did visit the election. Also, the badge hasn't been awarded more that once (so I think it's just the notification that's at fault).  Note that this seems to happen for other badges, it has been reported for: Caucus, Nice Answer, Enthusiast. Show that $K[x,xy,xy^2,\dots]$ is not Noetherian Here is the problem I am stuck on: Fix a field $K$ and consider the subring $A \leq K[x,y]$ generated by $K \cup \{x,xy,\dots,\}$. Show that $A$ is not Noetherian.  I figure that taking ideals $I_n = (x,xy,\dots,xy^n)$ should give an infinite strictly ascending chain, which would establish that $A$ is not Noetherian, but I cannot figure out how to show that each $I_n$ is a proper subset of $I_{n+1}$. Any help here? Is it possible to restore a MySql database only using files from Data directory? I have a Lenovo Ideapad with pre-installed Windows 8.1 In order to install Ubuntu I disabled secure boot and changed boot mode from UEFI to Legacy. This allowed me to install Ubuntu and the grub loader is also showing an option to either boot into Windows or Ubuntu, but whenever I try booting to Windows it says Cannot load.  This happens because the boot order is set to Legacy mode and Windows 8 will boot only with UEFI. I am unable to access my BIOS from where I can change the boot order. I press F1/F2 on booting but it doesn't load the BIOS, and directly goes to the grub loader screen.  How can I make my Windows 8 run along with Ubuntu?