I know this has been covered before in , but I still don't fully understand why cancelling differentials —not just inside an integral, but in general— is not considered okay or valid among mathematicians.  I'm currently studying physics at college, and my teachers —those who are physicists and not mathematicians, at least— do it all the time without it causing any trouble. In fact, the first (and less rigorous) method we were taught for solving differential equations relies heavily on treating differentials like numbers and derivatives like fractions. I don't know if it has a proper name, but I mean something like:  $$ \frac{\text{d}I}{\text{d}t}\cos(\omega t) = I\omega \sin(\omega t)  \implies  \frac{\text{d}I}{I}=\omega \tan(\omega t)\space\text{d}t $$  Now, I know that differentials aren't 'really' numbers. I mean, they can be understood both as the "linear part" of a variation or as a linear form, and the former looks like it should be okay to treat them like numbers.  Even the chain rule and the derivative of inverse functions seem to tell us that treating them like so should be consistent, when you write them using Leibniz's notation.  Is there really a situation where treating differentials like numerators and denominators causes any trouble?
Background  I am a first year calculus student so I would prefer if answers remained in Layman's terms.  It is common knowledge and seems to me a mantra that I keep hearing over and over again to "not treat differentials/derivatives as fractions".  I am of course, in particular, referring to Leibniz notation.  However, aside from a quick response such as "oh, it's because its not a fraction but rather a type of operator", I never really got a full answer as to why we can't treat it as such. It just kind of sits at the edge of taboo in my mind where it sometimes gets used and sometimes doesn't.  Confusion is further compounded when a lot of things seem to just work out if we treat them just as fractions (e.g. u-substitution / related-rates)    Example     Air is being pumped into a balloon at a rate of $100cm^3/s$. We want the rate of change of radius when the radius is at $25cm$.   $$\text{we are given}\ \frac{dv}{dt}=100cm^3/s$$ $$\text{we want}\ \frac{dr}{dt}\ \text{when}\ r=25cm$$ Thus we will solve this by using the relation $v=\frac{4}{3}\pi r^3$ $$\frac{dv}{dt}=\frac{dv}{dr}\frac{dr}{dt}$$ $$\frac{dv}{dt}\frac{dr}{dv}=\frac{dr}{dt}$$ $$100\frac{1}{4\pi r^2}=\frac{1}{25\pi}$$ So the answer is $\frac{dr}{dt}=\frac{1}{25\pi}$ when $r=25cm$  *Note the manipulation of derivatives just as if they were common fractions using algebra.    Question  When exactly can I treat differentials/derivatives as fractions and when can I not?   Please keep in mind that at the end of the day, I am a first year college student. An answer that is easy to understand is preferred over one that is more mathematically rigorous but less friendly to a beginner such as me.
Can I embed a custom font in an iPhone application? I would like to have an app include a custom font for rendering text, load it, and then use it with standard UIKit elements like UILabel. Is this possible? What is the word for the emotion I feel when I see someone being humiliated? When I see someone else being embarrassed / humiliated, for example a singer forgetting their words in a concert, I would say something like "I feel embarrassed for them". But is there a better word to describe this feeling?  I would say it's a mix of:   Empathetic discomfort for someone else's humiliation Tension; hope that spectators will be forgiving Frustration that I cannot help them   Other scenarios I would feel the same emotion:   Seeing someone make a presentation, the slides are all out of order, and their face gets red with embarrassment while there's a tense silence in the room. Seeing someone make an embarrassing social blunder in front of unforgiving people. Seeing a speaker who keeps making a distracting and annoying gesture, and they don't realize that the whole audience is annoyed. Seeing someone be taken advantage of, like seeing someone naively and happily accept a terrible offer on a car. Seeing a person of strong image (for example the CEO of your company) do something like accidentally leave their webcam on after a conference call, and employees see him/her let out a big burp and pick their nose or something.   In these cases it's important that the audience is not forgiving, even if they don't confront the person about it. The person may or may not know they are being humiliated.  "Pity" as proposed in an answer seems very close, but I think it lacks a few components. I think "pity" has a chronic connotation - that you are referring to the general state of things rather than a specific event.  I think "pity" also depends on the subject suffering, but in my example they are often oblivious.  It could very well be that there is no word for this at all.  NOTE: Some people do not feel the same way I do in this scenario, so you may not think this question makes any sense. Persist sort/filter selections Add field "tome" to biblatex entries I need to add a new field in the @book or @InProceedings entries: this field is "tome", for I have for instance the following title:  Francesco Bertolini, Società di trasmissione orale: mito e folclore, in Lo spazio letterario della Grecia antica, a cura di G. Cambiano, L. Canfora e D. Lanza, vol. I, tomo I, Salerno, Roma 1992, pp. 47--75.   So the following code is not enough:  @InProceedings{ bertolini:1992,   author                   = "Francesco Bertolini",   title                    = "Società di trasmissione orale: mito e folclore",   booktitle                = "Lo spazio letterario della Grecia antica",   editor                   = "G. Cambiano and L. Canfora and D. Lanza",   publisher                = "Salerno",   location                 = "Roma",   year                     = "1992",   volume                   = "1,   pages                    = "47--75", }   Using othes codes, I was able only to produce the "tome" field, but with a general trouble about the other fields:  \DeclareSourcemap{   \maps[datatype=bibtex]{     \map{       \step[fieldsource=tome,fieldtarget=usera]     }   } } \DeclareFieldFormat{usera}{tomo\space{#1}}   \DeclareBibliographyDriver{book}{%   \usebibmacro{author/editor}%   \setunit{\labelnamepunct}\newblock   \usebibmacro{title}%   \newunit\newblock   \usebibmacro{journal+issuetitle}%   \setunit{\bibpagespunct}%   \printfield{pages}%   \newunit\newblock   \printfield{usera}%   \usebibmacro{finentry}} \renewcommand*{\finentrypunct}{}   (useful for articles, but not for my case). By real, I'm not able to understand  which bibmascros I have to invocate...  I tried to write a biblatex-tome.dbx:  \ProvidesFile{biblatex-tome.dbx}   \DeclareDatamodelFields[type=field,datatype=literal,skipout=false]{tome}   \DeclareDatamodelEntryfields{tome}   Inside the main .tex file I put just this code:  \DeclareFieldFormat{tome}{\mkbibacro{tome}\addcolon\space #1} \newcommand*{\patchmoretome}[1]{   \xpatchbibdriver{#1}%     {\printfield{tome}}     {\printfield{tome}%           \newunit\newblock      \printfield{tome}}%     {\typeout{patching #1 to include more tome succeded}}%     {\typeout{patching #1 to include more tome failed}}% } % patch all drivers \patchmoretome{article} \patchmoretome{book} \patchmoretome{collection} \patchmoretome{inbook} \patchmoretome{incollection} \patchmoretome{inproceedings} \patchmoretome{manual} \patchmoretome{periodical} \patchmoretome{proceedings} \patchmoretome{report} \patchmoretome{thesis}   but it doesn't print the field in bibliographic entry. So I'm very confused about a (re)writing of driver Using InDesign CS6, how can I simultaneously edit A4 and A5 versions of a poster? I'm on Adobe Creative Cloud, so I have the latest InDesign, and I'm attempting to design a poster for my choir's next concert. Traditionally, I've done this in Photoshop, in two separate files, however I've recently started using InDesign after converting to it to create wedding albums through Blurb.  I've created an A4 document, and created an alternate-layout for it in A5. I've then lined them up side-by-side, and I've added a black frame to the A4 page, to act as the black background for my page (by the way, feel free to advise on better ways to do this).  My assumption is that I've failed to understand how this works, as I was hoping that I'd magically get a black frame on the A5 version, and that I could use liquid layout to get this to automatically scale (along with all other elements), in order to create the two versions simultaneously.  All help/advice would be greatly appreciated! I have seen many examples of how to render sprites from a spritesheet but I havent grasped why it is the most common way of dealing with sprites in 2d games.  I have started out with 2d sprite rendering in the few demo applications I've made by dealing with each animation frame for any given sprite type as its own texture - and this collection of textures is stored in a dictionary. This seems to work for me, and suits my workflow pretty well, as I tend to make my animations as gif/mng files and then extract the frames to individual pngs.  Is there a noticeable performance advantage to rendering from a single sheet rather than from individual textures? With modern hardware that is capable of drawing millions of polygons to the screen a hundred times a second, does it even matter for my 2d games which just deal with a few dozen 50x100px rectangles?  The implementation details of loading a texture into graphics memory and displaying it in XNA seems pretty abstracted. All I know is that textures are bound to the graphics device when they are loaded, then during the game loop, the textures get rendered in batches. So it's not clear to me whether my choice affects performance.  I suspect that there are some very good reasons most 2d game developers seem to be using them, I just don't understand why. What should I do when I see a question that is a duplicate of another one?   Should I answer it? Should I downvote it? Should I comment? Should I edit the question to indicate it's a duplicate? If I can, should I (vote to) close the question? What happens when I vote to close as duplicate? Or: Why did a comment with my name on it just appear? Should I flag it for moderator attention? What about similar or related questions? I would like to know how to unload an assembly that is loaded into the main AppDomain.  I have the following code:  var assembly = Assembly.LoadFrom( FilePathHere );   I need/want to be able to unload this assembly when I am done.  Thanks for your help. Why don't Jedi (or Sith) use the Force to snap their foes' necks? I understand that the Jedi use the Force for self defence, but still, couldn't they just use the Force to snap their foes' necks and be done with it? Like when Jaina Solo was hunting down Darth Caedus. Or even when Caedus was duelling Grand Master Skywalker. If he just used the Force to snap Skywalker's neck (like he did to poor Lieutenant Tebut), it will all be over very fast.  Is there a canon reason why Jedi and Sith don't use their powers to directly harm their opponents? Deleting my own question I am writing a WP plugin. I have created a button for the user to click in thw WP dashboard. I need to run a function when a user clicks the button, shown with a red arrow. The code snippet for the button is:&lt;p&gt;&lt;button class="button button-primary"&gt;Update Media Titles and ALT Text&lt;/button&gt;&lt;/p&gt;. I have already created the function in my class like so:  public function kh_update_media_seo() {      //update media files title and alt tags here }   I can handle the code that goes in the function alone, I only need help making the button in the WP dashboard fire off this specific function when clicked.  Pardon me if this sounds dump or straight forward. It's my first time doing this.  My plugin is a one file plugin if that helps. My specific question is with the ASUS M3N78-VM motherboard that my home server uses. I currently have 4x2GB installed for a total of 8GB but I was hoping to upgrade to say 16GB (4x4GB). The official specifications of that motherboard says it only supports up to 8GB but it was released back when only 2GB sticks were available. What, if any, would prevent it from working with 4GB sticks now that 4GB sticks are available?  My general question is with motherboards in general... what would prevent a motherboard from working with bigger sticks if its all just the same type (DDR2 for example and same speeds) of memory? Scatterplot with too many points How to run a command that involves redirecting or piping with sudo? How to set ctrl+c to copy, ctrl+v to paste and ctrl+shift+c to kill process in xfce4-terminal? Hello I can't find the option to reassign shortcuts in xfce4-terminal 0.6.3. I'd like to reassign ctrl+c to copy, ctrl+v to paste and ctrl+shift+c to kill process. I know I can do that easily under gnome-terminal but since I'm using xfce I would like to avoid installing all the dependencies for gnome-terminal. Any idea on how to achieve that ? I'm experimenting with VPNs, so I wanted to install pptpd.  john@desktop:~$ sudo apt-get install pptpd [sudo] password for john:  &lt;snip&gt; The following extra packages will be installed:   bcrelay The following NEW packages will be installed:   bcrelay pptpd 0 upgraded, 2 newly installed, 0 to remove and 11 not upgraded. Need to get 90.5 kB of archives. After this operation, 442 kB of additional disk space will be used. Do you want to continue [Y/n]? y   And everything runs fine until I get down to here:  WARNING: The following packages cannot be authenticated!   bcrelay pptpd Install these packages without verification [y/N]? n E: Some packages could not be authenticated   No authentication?  I'm no security expert, but I'm fairly sure that's a bad thing.  So I .  john@desktop:~$ apt-cache policy pptpd pptpd:   Installed: (none)   Candidate: 1.3.4-3ubuntu1   Version table:      1.3.4-3ubuntu1 0         500 http://us.archive.ubuntu.com/ubuntu/ natty/main i386 Packages john@desktop:~$ apt-cache policy bcrelay bcrelay:   Installed: (none)   Candidate: 1.3.4-3ubuntu1   Version table:      1.3.4-3ubuntu1 0         500 http://us.archive.ubuntu.com/ubuntu/ natty/main i386 Packages   They're coming from Ubuntu.com and not being properly authenticated?  What's going on here? Reese's boyfriend seems to constantly be asleep. When I try and talk to him, she warns me that he's been hard at work all night, but I can't actually enter the shop at night.   Is there any way to talk to him or any benefit for doing so? I've tried to talk to him daily for a week (thinking it was something akin to talking to Sable) but I've had no luck. Trying to come in after-hours doesn't work. Is it possible for an x86 processor to match an ARM processor in terms of performance per watt? From my personal experience with my tablet, and from the benchmarks and articles I've read, it always seems ARM processors, as seen in virtually all mobile devices, deliver incredible performance for the amount of power they consume.  It really seems like ARM is inherently more power-efficient than x86.  To me, it looks like that this is because x86 is a  while ARM is a . However, Intel Atom processors deliver very good performance relative to their power consumption as well.  In addition, having studied computer architecture, it is my understanding that in modern processors, the complex x86  is internally translated into simpler  that use an internal instruction set that is closer to the simpler ARM or MIPS ISAs than x86 proper.  As such, the ISA as presented to the programmer is little more than an interface to issue commands to the processor, rather than a representation of the actual low-level operations the chip performs.  Then again, implementing this translation layer requires additional silicon space on the chip...  That said, assuming that they are implemented using the same semiconductor process, is ARM inherently more efficient than x86? With a properly designed , is it possible for an x86 processor to deliver the same performance per watt as an ARM processor?  Remember, I'm looking for a well-written technical answer in the spirit of Stack Exchange and not mere speculation. How do I enforce a password complexity policy? 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?