I was trying to prove  $$d(xy) = x(dy) + y(dx)$$  earlier this morning and I used this  to help me understand the task.   I understood the entirety of the post for my calculus class, apart from one step.   Considering an area of a rectangle with dimensions $x$ and $y$ $xy$ makes sense.   Likewise, the area of a rectangle with dimensions $(x+\Delta x)(y+\Delta y)$ giving    $$A_1=(x+\Delta x)(y+\Delta y)=xy+x \Delta y+y\Delta x+\Delta x \Delta y$$  made sense too.   I was confused however about the subtraction of the two areas that gives  $$x \Delta y+y\Delta x+\Delta x \Delta y$$  but the small approximation of $\Delta x $ and $\Delta y$ very small, ensuring that  $\Delta x \Delta y$ is negligible afterwards made sense.  Why is there the need to subtract the areas as part of the proof for this differentiation property?
I've seen $d(xy) = y(dx) + x(dy)$, but I don't understand the principle behind it and memorizing it is lame.  Can anyone explain what is going on here?  For example from physics, $$F = {{dP} \over {dt}}$$ $$F = {{d(mv)} \over {dt}}$$ $$F = {{v(dm) + m(dv)} \over {dt}}$$ Since $$dm = 0$$  $$F = {{m(dv)} \over {dt}}$$ Since $$a = {{dv} \over {dt}}$$ $$F = ma$$
Prevent Use of the Back Button (in IE) I keep getting this warning whenever I try to run sudo apt-get update.  W: Duplicate sources.list entry http://archive.ubuntu.com/ubuntu/ precise-updates/main i386 Packages (/var/lib/apt/lists/archive.ubuntu.com_ubuntu_dists_precise-updates_main_binary-i386_Packages) W: You may want to run apt-get update to correct these problems   Below is the output from /etc/apt/sources.list file:  deb http://archive.ubuntu.com/ubuntu precise main restricted deb-src http://archive.ubuntu.com/ubuntu precise main restricted  deb http://archive.ubuntu.com/ubuntu precise-updates main restricted deb-src http://archive.ubuntu.com/ubuntu precise-updates main restricted  deb http://archive.ubuntu.com/ubuntu precise universe deb-src http://archive.ubuntu.com/ubuntu precise universe deb http://archive.ubuntu.com/ubuntu precise-updates universe deb-src http://archive.ubuntu.com/ubuntu precise-updates universe  deb http://archive.ubuntu.com/ubuntu precise multiverse deb-src http://archive.ubuntu.com/ubuntu precise multiverse deb http://archive.ubuntu.com/ubuntu precise-updates multiverse deb-src http://archive.ubuntu.com/ubuntu precise-updates multiverse  deb http://archive.ubuntu.com/ubuntu precise-security main restricted deb-src http://archive.ubuntu.com/ubuntu precise-security main restricted deb http://archive.ubuntu.com/ubuntu precise-security universe deb-src http://archive.ubuntu.com/ubuntu precise-security universe deb http://archive.ubuntu.com/ubuntu precise-security multiverse deb-src http://archive.ubuntu.com/ubuntu precise-security multiverse   How do I fix it? Renaming Files and Directories Recursively using rename/find Is it ok to leave a slow cooker unattended? Force a Blender operator to always start with default values I'm writing an operator for which it is crucial that every time it gets called all the properties are set to their default values.   The ops.object.delete has 1 param. use_global. It is always false, no matter what the user put in last time it got called. I would like to replicate exactly that.   (For example: I want to modify delta scale. The default val for my_factor is 1, so nothing happens until the user alters that number. If the user calls the operator again, it will have stored whatever number the user has used last, however I want it to always start with 1.) A story about the world split into three realms This is a series I read partially something like 15 years ago (I think). It is set in a universe with magic of some sort - I think the magic was derived from geological processes (?). One of the things I remember - which was the reason I abandoned the series - was that it grew ever more gory; when I stopped reading, one of the main protagonists was flayed alive.   I only remember fragments - there were hints that the universe had been magically split into three or four parts, three relatively normal parts and The Void. I remember there was a powerful person/ruler called Tensor, and a kind of monstrous race with powerful magic, called 'Lorsk', which originated in the void and had evolved from humans that had been trapped there.  I'm a bit ambivalent about reading the series again - it seemed to be a never-ending death spiral of soap operatic proportions, but I'd like to see if my impression from then was actually right. How do I find out the number, type, resolution &amp; refresh rate of additional monitors I can attach to my MacBook or MacBook Pro?  This is an attempt at a canonical QA, as per the Meta QA -   to try prevent the myriad "same but different" questions we get on every possible combination of computers, cable &amp; monitors There are perfectly good resources out there that Ask Different should not have to duplicate the efforts of. Old usernames stick to Community Wiki posts Can I easily travel in Italy by bus/train without any prior reservation? I am planning a trip to Italy this coming January. My plan is to land by plane in Roma and then go to France by train/bus stopping in some Northern Italian cities on the way, enjoying the architecture, the atmosphere, the art and and the food.   I want to do it at my own pace, without heavy scheduling. Ideally, I would just decide which cities to visit, and take a train whenever I feel it. I often do this here in South Korea where I am living, taking advantage of the "standing seats" as it is called here, that is travelling (for a cheap price and no reservation) standing on the romm between two cars. From my experience, it is also doable in Taiwan (at least in some trains).  In France, regional trains usually does not need any reservation, so you can buy a ticket and take the train whenever you want.  Can I expect a similar system in Italy, that is a way to make short travels between close cities without reservation?   Example of "short travel": Pisa - Firenze. When to throw an exception? Find the exact value of the infinite sum $\sum_{n=1}^\infty \big\{\mathrm{e}-\big(1+\frac1n\big)^{n}\big\}$ How can we find the exact value of the infinite sum  $$ \displaystyle\sum_{n=1}^\infty \left\{\mathrm{e}-\Big(1+\frac1n\Big)^n\right\}? $$  This problem appears in:  , p.114. java.lang.ClassNotFoundException: org.apache.struts2.dispatcher.ng.filter.StrutsPrepareAndExecuterFilter Is there a portable version of Chrome for Ubuntu? Given a function:  function x(arg) { return 30; }   You can call it two ways:  result = x(4); result = new x(4);   The first returns 30, the second returns an object.  How can you detect which way the function was called inside the function itself?  Whatever your solution is, it must work with the following invocation as well:  var Z = new x();  Z.lolol = x;  Z.lolol();   All the solutions currently think the Z.lolol() is calling it as a constructor. I'm reading  and I'm trying to understand the proof under "Degeneracy in One Dimension".  Here's what it says:     Considering a one-dimensional quantum system in a potential $V(x)$ with degenerate states $|\psi_1\rangle$ and $|\psi_2\rangle$ corresponding to the same energy eigenvalue $E$, writing the time-independent Schrödinger equation for the system:       $$-\frac{\hbar^2}{2m}\frac{ \partial^2\psi_1}{ \partial x^2} + V\psi_1 =E\psi_1$$       $$-\frac{\hbar^2}{2m}\frac{ \partial^2\psi_2}{ \partial x^2} + V\psi_2 =E\psi_2$$   Multiplying the first equation by $\psi_2$ and the second by $\psi_1$ and subtracting one from the other, we get:       $$\psi_1\frac{d^2}{d x^2}\psi_2-\psi_2\frac{d^2}{d x^2}\psi_1=0$$   Integrating both sides       $$\psi_1\frac{\partial \psi_2}{\partial x}-\psi_2\frac{\partial \psi_1}{\partial x}=constant$$   In case of well-defined and normalizable wave functions, the above constant vanishes, provided both the wave functions vanish at at least one point, and we find: $\psi_1(x)=c\psi_2(x)$ where $c$ is, in general, a complex constant. So, the two degenerate wave functions are not linearly independent. For bound state eigenfunctions, the wave function approaches zero in the limit $x\to\infty$ or $x\to-\infty$, so that the above constant is zero and we have no degeneracy.   Here are the parts I don't understand:   How does integrating $\psi_1\frac{d^2}{d x^2}\psi_2-\psi_2\frac{d^2}{d x^2}\psi_1=0$ yield $\psi_1\frac{\partial \psi_2}{\partial x}-\psi_2\frac{\partial \psi_1}{\partial x}=constant$.  Without having explicit formulas for $\psi_1(x)$ and $\psi_2(x)$, how do we just apparently pull them both out of the integral? I don't understand how the constant in the last equation vanishes just because $\psi_1$ and $\psi_2$ vanish at infinity. Even if it does vanish, how does that imply $\psi_1=c\psi_2$? What are the left and right ideals of matrix ring? How about the two sided ideals? What are the left and right ideals of matrix ring? How about the two sided ideals? A simple algebra that is not semisimple Add anchors to the how-to-ask and how-to-answer pages What are the minimum requirements to run Windows 10? Mass retagger badge I've been busy retaggin questions since I got my retag privileges.  Shouldn't there be an incentive to retag wrongly tagged questions, other than the urge to do things right?  I'm mean, the only "badge incentive" for retagging are "Firs edit" (Organizer) and "Creation of a tag used 50 times" (Taxonomist). The first one is easy to get, and the second one, right now in SO is almost impossible to get without cheating, that's editing 50 questions with an unlikely tag. Taxonomist I guess, could encourage wrongly tagging.  Is this really a problem?  We could have an "Tagged 200 question" badge, so people go for that badge, instead of taxonomist.