Disclaimer - I'm not a mathematician, I'm a dirty physicist.  My work often involves performing calculus on various things without thinking about what I'm doing too much (I leave the proof of various identities etc for the pure mathematicians to worry about).  However I've often noticed that mathematicians get a little upset when I do tricks such as treating the differential $\frac{\text{d}y}{\text{d}x}$ as if it were a fraction.  The simplest example I can think of is how I think about the chain rule: $$\frac{\text{d}y}{\text{d}x} = \frac{\text{d}y}{\text{d}u}\frac{\text{d}u}{\text{d}x}$$ In my head, I imagine the $\text{d}u$ terms cancelling, which is why this works.  Indeed, this is how I'll explain to others how to use the chain rule when asked about it.  My question is the following:  Is is dangerous to think about differentials in this way?  After all, one of the very first examples of calculus I've ever seen (back when I was a baby in high-school) was the derivative of $y=x^2$ calculated from first principles in the following way:     \begin{align} y&amp;=x^2\\ y+\delta y&amp;=(x+\delta x)^2\\ y+\delta y&amp;=x^2+2x\delta x+\delta x^2\\ \require{cancel}\cancel{x^2}+\delta y&amp;=\cancel{x^2}+2x\delta x+\delta x^2\\ \delta y&amp;=2x\delta x+\delta x^2\\ \frac{\delta y}{\delta x}&amp;=2x+\delta x\\ \text{Now let }&amp;\delta x\rightarrow0\text{ leaving}\\ \frac{\delta y}{\delta x}&amp;=2x\\ \frac{\text{d}y}{\text{d}x}&amp;=2x \end{align}   And to me, this is just treating $\frac{\delta y}{\delta x}$ as a fraction.  I know that technically you're doing $0/0$ if you think about it, but are there any examples where treating $\text{d}y/\text{d}x$ is really inappropriate?
In the book Thomas's Calculus (11th edition) it is mentioned (Section 3.8 pg 225) that the derivative $dy/dx$ is not a ratio. Couldn't it be interpreted as a ratio, because according to the formula $dy = f'(x)dx$ we are able to plug in values for $dx$ and calculate a $dy$ (differential). Then if we rearrange we get $dy/dx$ which could be seen as a ratio.   I wonder if the author says this because $dx$ is an independent variable, and $dy$ is a dependent variable, for $dy/dx$ to be a ratio both variables need to be independent.. maybe?
Given a string file path such as /foo/fizzbuzz.bar, how would I use bash to extract just the fizzbuzz portion of said string? In Ubuntu 18.04.1, I would like to use my left Super-key only to simply open the Show Applications menu (aka Dash) like a left click on the dock-icon (with the 9 dots) does and preferably close it again without showing me any open programs.   I can open it with Super + A and even close it again with Super R, but Super L appears to do nothing at all.   In Xfce, something similar was possible by adding a custom keyboard shortcut for Super L and setting the command to xfce4-popup-whiskermenu, which would open and close a menu with favourite applications. Unfortunately, I don't know the command that opens and closes the Show Applications menu.   Is there any way to add such a shortcut in the new Gnome Shell? How can see call history in iPhone one month back, i also tried with tried party software but that is not perfect , is the any solution from apple? I've read about Fermat's little theorem and generally how congruence works. But I can't figure out how to work out these two:   $13^{100} \bmod 7$ $7^{100} \bmod 13$   I've also heard of this formula:  $$a \equiv b\pmod n \Rightarrow a^k \equiv b^k \pmod n $$  But I don't see how exactly to use that here, because from $13^1 \bmod 7$ I get 6, and $13^2 \bmod 7$ is 1. I'm unclear as to which one to raise to the kth power here (I'm assuming k = 100?)  Any hints or pointers in the right direction would be great. Suppose $ T \subset \mathbb{C} $. Show that the corresponding set $ S \subset \Sigma $ is  a. a circle if $ T $ is a circle. b. a circle minus (0, 0, 1) if $ T $ is a line.  Here we are defining $ \Sigma $ to be the Riemann sphere, given by the set: $$ \Sigma = \left \{(\xi, \eta, \zeta) : \xi^{2} + \eta^{2} + (\zeta - \frac{1}{2})^{2} = \frac{1}{4} \right \} $$  To take a point from $ \mathbb{C} $ to $ \Sigma $ we can use the following:  $$ \xi = \frac{x}{x^{2} + y^{2} + 1}; \eta = \frac{y}{x^{2} + y^{2} + 1}; \zeta = \frac{x^{2} + y^{2}}{x^{2} + y^{2} + 1} $$  We define a circle on $ \Sigma $ to be the intersection of a plane of the form $ A\xi + B\eta + C\zeta = D $ with $ \Sigma $. We also know the converse of this problem is true, that the intersection above yeilds a set in $ \mathbb{C} $ with the following property:  $ (C - D)(x^{2} + y^{2}) + Ax + By = D $. As you can see, when C = D, then an equation for a line is yeilded, otherwise it is a circle.   I really am at a  loss about how to solve this problem. The only thing I can think to do is to pick 3 points on a circle or radius $ r $ with center $ z_{0} $, use these points to find two vectors in $ \Sigma $, take their cross product to get a normal vector, use this normal vector to get a plane. Once I have the plane in form $ A\xi + B\eta + C\zeta = D $ then I could prove that the circle I had chosen corresponds exactly with $ (C - D)(x^{2} + y^{2}) + Ax + By = D $. Is there not an easier, less computation way to do this? What is the most efficient way to write the old-school:  StringBuilder sb = new StringBuilder(); if (strings.Count &gt; 0) {     foreach (string s in strings)     {         sb.Append(s + ", ");     }     sb.Remove(sb.Length - 2, 2); } return sb.ToString();   ...in LINQ? "All the good people" vs. "all of the good people" I have trouble on this question: "Characterize the groups G for which the map $$\iota: G \to G$$ sending $$x \mapsto x^{-1}$$ for all $x \in G$, is an automorphism of $G$." From  post I have seen that $G$ must be abelian. Can anyone propose another characteristic of $G$ that I can derive from the given conditions besides abelian? Thank you so much!! Where and why do I have to put the "template" and "typename" keywords? Copy skype chat history to text files? I'd like to convert my Skype chat history to a collection of text files for archival purposes.  Ideally, I'd like to have a command (e.g. script or app) so that I could do this on a regular basis.  Any one have any idea on how to go about doing this? Something's been bugging me about SO for a while now, that is the inaccuracy of "length of time" strings.  The best example I can give at the moment is .  If you look at the length of time he's been registered, it says 1 year.  He joined the site September 2008, which is only 3 months off two years.  It's the same with answers - you can give an answer 50 seconds before anyone else, but at certain times page views will show both as "5 minutes ago" or whatever.  That's not quite as bad, but with answers being sorted "randomly by votes" by default you could lose out on votes or being marked as the correct answer to the slower answer even if they're both the same.  There's probably a better method amongst these, IMO:   "Vulgar fraction" unicode characters     Member for 1½ years   answered 5¼ minutes ago asked 3⅓ hours ago  Different units   Member for 21 months (at least it's better than "1 year") Answered 306 seconds ago (ugly, I know) Asked 64 minutes ago (also ugly)  Combining units   Member for 1 year 9 months (fits easily on the profile screen) Answered 5 minutes 6 seconds ago (probably too long) Asked 3 hours 20 minutes ago (also long)  Rounding up   I personally prefer option 1, unicode fraction characters.  "Member for 21 months" is pretty good too because it's easy to quickly work out.  "Asked 452 seconds" ago is horrible though.  EDIT: I'm completely aware of the timestamp hover because I am forced into using it so much. It's a great feature, but it's not really in your face so not everyone knows about it. I still think there's a better "glancing" solution than having to hover over the time string. Why can't two series-connected diodes act as a BJT? Another form of the question is: will joining two diodes with wires (pn-np) make a transistor-equivalent?  I read that they are not equivalent, but why? How to find a computer name in a LAN from the IP address? My network includes machines running Linux and others running Windows. And my machine is running Linux. Evaluating this integral $ \small\int \frac {x^2 dx} {(x\sin x+\cos x)^2} $ The question:      Compute$$ \int \frac {x^2 \, \operatorname{d}\!x} {(x\sin x+\cos x)^2} $$   Tried integration by parts. That didn't work.  How do I proceed? How do I drop an object to the sand? Creating additional Virtual Host Ubuntu Server Arduino Negative Voltage Reading with Positive Ground I am trying to measure voltage &amp; current off a solar charge controller from panel, battery &amp; load simultaneously with my arduino. I have managed to achieve this by tying all the negatives together and using it as GND, however, when I do that the charge controller doesn't function properly and when I switch to off-mode the loads still run. After checking the instruction manual further, I found out that the positives are shared common between Pan, Bat, Load. This was confirmed with a multimeter.   So now what I am thinking of doing is using the positive from the battery as the GND reference and then measuring all other voltages (through voltage dividers with respect to the GND). The voltages should then range between -17V to 0V.   However, I have read that the analog input pins are not able to read negative voltages, and with respect to the positive GND, the Analog inputs will all be negative.   EDIT: I was thinking of using an amplifier to invert positive ground to a negative ground, meaning I could measure all the positive sides of the voltages. But after getting answers from you guys it seems I may not have to use an amplifier and a voltage divider may suffice. I have currently included a 2 resistor voltage divider between each analog input, +5V source, Vmeasure which you can see in the diagram below.  According to my calculations, if this works, @-17V I should see 0.6V @ analog input. @0V I should see 4V at analog input.   I'd appreciate feedback on this circuit, and below in the second image I've also included a piece of another method suggested by @jonk (thanks), and would be interested in knowing what the benefits are of method B) over method A)  Thanks!  Ps. the ameters are there in place of ACS712 current sensors which also go into analog input pins.   Method A)        &ndash; Schematic created using   Method B (EDIT: This method has worked, I am still yet to test method A) thanks @Jonk)) Can a possessive S be attached to nouns that are not human beings (or animals)? Can a possessive S be attached to nouns that are not human beings (or animals)?  For example, instead of saying "the back of the chair", can I say "the chair's back"?   I remember learning that not everything can get a possessive s, but this issue is not clear to me. Refreshing Formula Fields Dynamically (without save) I have an "invoice" custom object which has a couple of summarising formula fields in. I also have a complicated and dynamic visualforce page for building the invoice objects.  As the user is completing this form, different areas are dynamically updated to show "latest" values. This is all fine where I am copying data about or performing bespoke lookups and maths, but - in what is really the simplest case - a formula field, I can find no way to get the value to update, without performing some weird cyclic save + reload of the whole object.  This is inappropriate for a couple of reasons (the object should not be saved until the invoice is complete + accepted) and the other option I have found online is to spoof the value by duplicating my formula field maths into the custom controller and re-executing them there - also unacceptable from a code/logic location/duplication perspective.  Is there anyway to "force" an object/page to recalculate it's formula fields, either through standard or extended controllers? Android – Listen For Incoming SMS Messages