I can prove that $dx dy = r.dr d\theta$ by drawing a circle and calculating the area of a small square in the polar coordinates, but when I try proving it using the equations below, I fail to prove it. What is my mistake?  $x = r\cos(\theta) =&gt; dx = \cos(\theta).dr - \sin(\theta)r.d\theta$  $y = r\sin(\theta)=&gt;dy=\sin(\theta).dr + \cos(\theta)r.d\theta$  $=&gt; dxdy = r(\cos^2(\theta)-\sin^2(\theta))drd\theta=r\cos(2\theta).drd\theta$    I actually saw a similar question in this site:  My problem was that I didn't understand why $drd\theta = -d\theta dr$
In polar coordinate how we can get $dx\;dy=r\;dr\;d\theta$?  with these parameters:    $r=\sqrt{x^2+y^2}$ $x=r\cos\theta$ $y=r\sin\theta$  Tanks.
math fallacy problem: $-1= (-1)^3 = (-1)^{6/2} = \sqrt{(-1)^6}= 1$? I know there is something wrong with this but I don't know where. It's some kind of a math fallacy and it is driving me crazy. Here it is: $$-1= (-1)^3 = (-1)^{6/2} = \sqrt{(-1)^6}= 1?$$ What is the standard solution in JavaScript for handling big numbers (BigNum)? Find the Resistance between Two Terminals Let's say R = 49 ohms, and we want to find Req between a and d, how do we do that? Do we have to find Req for the whole circuit? Event binding on dynamically created elements? I would like to put an address in my document as follows:  |                             | |                     Name    | |                     Street #| |                     City    | |                     ZIP     | |                             |   where the | mark the left and right page borders. I would like the items of the address (Name, Street etc.) to be left-flushed, but the box right-flushed so that the longest item touches the right page border.  Is there a simple way to accomplish this? Why does sstatic.net serve cookies? According to :  This is a fast, cookieless domain intended for static content delivered to the   However, all responses from the server set a cookie anyway: $ http https://sstatic.net/ HTTP/1.1 200 OK CF-RAY: 162b88245bc406be-LHR Cache-Control: max-age=604800 Connection: keep-alive Content-Encoding: gzip Content-Type: text/html Date: Sun, 31 Aug 2014 19:11:04 GMT ETag: W/&quot;d5474fdfb5c3cf1:0&quot; Last-Modified: Fri, 29 Aug 2014 18:20:12 GMT Server: cloudflare-nginx Set-Cookie: __cfduid=db60cc6f3e7f6482e37f1f2b85c9dc5881409512264375; expires=Mon, 23-Dec-2019 23:50:00 GMT; path=/; domain=.sstatic.net; HttpOnly Transfer-Encoding: chunked Vary: Accept-Encoding [...]  Is this intentional, or just funny? Edited to add: I understand that the cookie is used to track visiting browsers so that cloudflare can distinguish them from misbehaving hosts using the same or a nearby IP address. However, if the policy is intentionally to serve cookies, then the self-documentation of sstatic.net as a &quot;cookieless domain&quot; is incorrect. Can anyone hint me to prove:  $\forall n\in \mathbb{N}: \exists$ Fibonacci numbers $ F_{i_1},\ldots,F_{i_k}$ such that: $$\sum F_{i_k}=n$$    Note: Every Fibonacci number can appear only once. Thanks Ragged Right linebreaking, such that the last line is fullest, and the first most empty Jewish Law prohibits the  However, butter, a milk product, is a kitchen staple for most cooked emulsions (eg. pan-sauces, sauces béchamels, or roux-based sauces). While I use appropriate cooking fats for making non-dairy versions for meat dishes, this frequently doesn't work.  What is a non-dairy fat which may be used to as an emulsifier or roux ingredient? I would like to add a new cable outlet to an outside wall and wanted to find out what kind of drill bit is needed to drill through and feed the coax cable as well as anything else to be mindful of. Comcast wanted $70 for a new outlet installation ($30 install, $40 tech visit), so I figured I could do it myself. The part that I'm most unfamiliar with is how to properly weather seal it. The outside just has vinyl siding and there's no electrical outlets nearby.  So far, I think I need the following   Coax Cable Coax Cable Wall Plate Long drill bit Add a 3-way Cable Splitter to cable box Weather sealer? What would you consider to be the best exception type to throw when an extension method is called on a null instance (where the extension method does not allow it)? Since extension methods are nothing but static methods you could argue that it should be ArgumentNullException, but on the other hand they're used like instance methods so it might be more natural to use the NullReferenceException. Let's take the following example:  public static string ToInvariantString(this IFormattable value, string format) {     return value.ToString(format, CultureInfo.InvariantCulture); }   This way a NullReferenceException will be thrown if the value parameter is null.   The other example would be:  public static string ToInvariantString(this IFormattable value, string format) {     if (value == null) throw new ArgumentNullException("value");     return value.ToString(format, CultureInfo.InvariantCulture); }   EDIT: In some of the answers you have pointed out that an extension methods can be called like a static method and in those cases a null reference exception would be wrong, which is a great point, and actually one of my concerns, not sure why I forgot to mention that in the question in the first place.   Someone also pointed out that it's wrong to throw a NullReferenceException, and yes, it is. That's why I don't throw it, I just let it happen (let the CLR throw it) by not guarding the method.  I think I favor the ArgumentNullException (that's what I've use so far) but I still think there is at least room to argue for an against the NullReferenceException since it seems more natural in most places where the method is going to be used. Fundamental Theorem of Calculus. Fundamental Theorem of Calculus says  Let $f$ be an integrable function on $[a,b].$ For $x$ in $[a,b],$  let $F(x)=\int_{a}^{x}f(t)dt.$ Then $F$ is continuous on $[a,b].$ If $f$ is continuous at $x_{0}$ in $(a,b),$ then $F$ is differentiable at $x_{0}$ and $F^{'}(x_{0})=f(x_{0}).$  Now my question is why we are saying that this is "Fundamental Theorem of whole Calculus".  It is clear that it connects intergal and differential calculus. But i am not getting the exact concept, why this is so important as its name is fundamental theorem of calculus. Why we says that it is fundamental theorem of calculus?  Please suggest me. Thanks in advance. When and why should I use % !TEX TS-program and % !TEX encoding? Comma — a punctuation error? Consider:     This book provides a solid psychological understanding of these experiments, and adds a few expansions and conclusions of its own. It also provides …   My question regards the comma. I have been told that having the comma there is an error.  According to  this seems to be the case according to “some style guides”. The reason is that the stuff after the comma is not independent; it’s dependent on what comes before.  What do you think? It would seem to me the comma helps a lot, because those two “and”s without any comma make the sentence rather cluttered. That’s why I put the comma there. So I’m not sure. How do I make sure that values from MySQL keep their type in PHP? I have encountered a statement in Lang's Algebra (Revised Third Edition, page 53) concerning morphisms of objects that seems strange to me: "In practice, in this book we shall see that most of our morphisms are actually mappings, or closely related to mappings."  I have always been under the impression that the terms 'mapping' and 'morphism' are synonymous in the context of categories. Perhaps it is just the case that Lang defines the two in a way that they disagree, but I can't find such an instance. Are they in fact different? If so, what is an example of a morphism that is not a mapping? I see the following error while trying run the command shown below. I read somewhere that my /boot partition is low on disk space. How can I increase the size of the /boot partition so I can install more software? I have a 500GB hard disk, so there is enough space to play with.  sudo apt-get install libdvdread4  gzip: stdout: No space left on device                                                                                                                                                                                  E: mkinitramfs failure cpio 141 gzip 1                                                                                                                                                                                 update-initramfs: failed for /boot/initrd.img-3.2.0-33-generic with 1.                                                                                                                                                 run-parts: /etc/kernel/postinst.d/initramfs-tools exited with return code 1                                                                                                                                            Failed to process /etc/kernel/postinst.d at /var/lib/dpkg/info/linux-image-3.2.0-33-generic.postinst line 1010.                                                                                                        dpkg: error processing linux-image-3.2.0-33-generic (--configure):                                                                                                                                                      subprocess installed post-installation script returned error exit status 2                                                                                                                                            dpkg: dependency problems prevent configuration of linux-image-server:                                                                                                                                                 linux-image-server depends on linux-image-3.2.0-33-generic; however:                                                                                                                                                    Package linux-image-3.2.0-33-generic is not configured yet.    dpkg: error processing linux-image-server (--configure):     dependency problems - leaving unconfigured    dpkg: dependency problems prevent configuration of linux-server:     linux-server depends on linux-image-server (= 3.2.0.33.36); however:      Package linux-image-server is not configured yet.    dpkg: error processing linux-server (--configure):     dependency problems - leaving unconfigured    No apport report written because the error message indicates its a followup error from a   previous failure.                                                                                                           No apport report written because the error message indicates its a followup error from a previous failure.  Errors were encountered while processing:  linux-image-3.2.0-33-generic  linux-image-server  linux-server N: Ignoring file 'michael-gruz-canon-precise.list.1' in directory '/etc/apt/sources.list.d/' as it has an invalid filename extension N: Ignoring file 'michael-gruz-canon-precise.list.1' in directory '/etc/apt/sources.list.d/' as it has an invalid filename extension   Listed below is the output of du  Filesystem              1K-blocks      Used Available Use% Mounted on /dev/mapper/ubuntu-root 712660664 104095912 572363692  16% / udev                      3964792         4   3964788   1% /dev tmpfs                     1591012      1064   1589948   1% /run none                         5120         0      5120   0% /run/lock none                      3977528       684   3976844   1% /run/shm /dev/sda1                  233191    219821       929 100% /boot Capturing console output from a .NET application (C#) How do I invoke a console application from my .NET application and capture all the output generated in the console?   (Remember, I don't want to save the information first in a file and then relist as I would love to receive it as live.) Too many of your edits were rejected, try again in 7 days How to find out who deleted some data SQL Server My boss had a query from a customer yesterday asking how they could find out who deleted some data in their SQL Server database (it is the express edition if that matters).  I thought this could be found from the transaction log (providing it hadn't been truncated) - is this correct? And if so how do you actually go about finding this information out?