I'm currently reading through a textbook on Real Analysis, and the following is written:     "The great advantage of Leibnitz symbols is that, when they are properly   interpreted, we can treat the symbols d f, d y, dx, and so on, as if they represent members   of IR and carry out various algebraic operations, and the resulting formulas will have   meaning." - Patrick M Fitzgerald, Advanced Calculus, P114   In my physics courses, we very frequently say things like, "If you have a small amount of charge $dq$ in a small volume $dV$, we can define the charge density $\rho$ as $\frac{dq}{dV}$. However, I have been warned in the same classes that treatment of differentials in this manner is not mathematically rigorous and can lead to issues. What issues? I am not sure.  At the same time, in my Real Analysis course, we were told that $\frac{df(x)}{dx}$ should be treated as just notation, and not as a fraction.  And then I read the quoted text above, which appears to be merging the two concepts.  My question is this: How can differentials be treated? When is it rigorous to have a small amount of charge $dq$ be defined as $\rho dV$? My friends and I have grown skeptical of derivations of physics concepts when differentials were used, because we have been told to be skeptical. When is this warranted? When is it not? What do I watch for?
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?
SF Story about human adapting to alien environment This is a story that was read on BBC radio about thirty years ago, and this is about as right as I can remember it: The human pilot of a spacecraft is doing some reconnaissance on an alien planet. He crashes but survives. For some reason, no one will know where to look for him. He sees something in the distance that looks like a building of some sort. When he gets there, it seems to be a home. Very much like a human home. There's furniture, but it's strange and uncomfortable . There's food, but it's inedible. There's a faucet but what comes out of it seems more like acid. Everything looks as though it should be right, but it's not. He leaves the house, wanders and after a few days is desperate for water. But there isn't any water anywhere. He's close to death, crawling on the ground and then something happens. There are cracks in the soil and there appears to be water bubbling up from perhaps some underground stream. The water revives him. He goes back to the house and tries the food again. It's delicious and he eats enough to gain back his strength. Then he decides to try the water again. It's pure and wonderful. He takes a shower and the last line of the story is how refreshing the water is on his skin and something about him lifting his tail. Is the following equivalent to the axiom of choice? How can I find $\lim_{n\to \infty} a_n$ Let $f$ be entire and suppose that $\text{Im} f\ge0$. Show that $\text{Im} f$ is constant. How do you show that if $f$ is entire and if $\text{Im} f\ge0$, then $\text{Im} f$ is constant? Is this even true in the first place? I am currently using Ubuntu 12.04 because the latest upgrade 14.04 is not working in my system. I need to install the latest Adobe Flash Player 17 plugin, but it only allows me to install up to version 11.02, so media files cannot be opened. It says that Adobe Flash Player 17 is not compatible with my system. Function $f: \mathbb{R} \to \mathbb{R}$ that takes each value in $\mathbb{R}$ three times Does there exist a function $f: \mathbb{R} \to \mathbb{R}$ that takes each value in $\mathbb{R}$ three times? If not, how could I prove that such a function does not exist? Residue theorem with $\frac{1}{2\pi}\int_{0}^{2\pi}e^{\cos\theta}\cos(n\theta) d\theta$. The following integral should be doable using the residue theorum:     $$\frac1{2\pi}\int_{0}^{2\pi}e^{\cos\theta}\cos(n\theta) \,d\theta$$ Why is voltage added when batteries are connected in series but not in parallel? As I understand, the different voltage between two object is caused by the total different charged between two object. Thus, the higher extra electron between two opposite charged object, the higher voltage between them, isn't it? So if i have two battery or capacitor connect in parallel, the negative connect to negative, the positive connect to the positive. Caused the charge from each plate to be double. So that's as my expected, it must be double voltage, but ironically why it's out of my expected and the reality is vice-versa? My Tier 5 YMS visa is set to start on September 30th, 2016 however, I want to get there earlier.  Can I enter the UK as a tourist, if I explain to border control/immigration that I am entering as a tourist and do not plan to work or do anything really until my Tier 5 visa starts?  I can find no official contact or information source on this with UK Visas Information, the British High Commission, etc. How do I *really* justify a horizontal menu in HTML+CSS? You find plenty of tutorials on menu bars in HTML, but for this specific (though IMHO generic) case, I haven't found any decent solution:  #  THE MENU ITEMS    SHOULD BE    JUSTIFIED     JUST AS    PLAIN TEXT     WOULD BE  # #  ^                                                                             ^  #    There's an varying number of text-only menu items and the page layout is fluid. The first menu item should be left-aligned, the last menu item should be right-aligned. The remaining items should be spread optimally on the menu bar. The number is varying,so there's no chance to pre-calculate the optimal widths.   Note that a TABLE won't work here as well:   If you center all TDs, the first and the last item aren’t aligned correctly. If you left-align and right-align the first resp. the last items, the spacing will be sub-optimal.   Isn’t it strange that there is no obvious way to implement this in a clean way by using HTML and CSS? Intuitive explanation of the tower property of conditional expectation I understand how to define conditional expectation and how to prove that it exists.  Further, I think I understand what conditional expectation means intuitively. I can also prove the tower property, that is if $X$ and $Y$ are random variables (or $Y$ a $\sigma$-field) then we have that  $$\mathbb E[X] = \mathbb{E}[\mathbb E [X | Y]].$$  My question is: What is the intuitive meaning of this? It seems quite puzzling to me.  (I could find similar questions but not this one.) I don't know if this is a famous problem but my professor assigned this for a research project and I am hitting a dead end( I don't know what to do next).   Note: In $\phi(n)$ ; $n=rows=columns$ and this function throws out total combination of matrices that satisfies the following rule:  Rule for this square binary matrix:  Every Row and Column has two 1's.  So,    $$\phi(2)=1$$ \begin{bmatrix}    1 1 \\    1 1 \\ \end{bmatrix} $$\phi(3)=6$$  I can't write out all the possible matrices for $n=3$ but this is one of them (btw i did this by hand): \begin{bmatrix}    0 1 1 \\    1 0 1 \\ 1 1 0 \\ \end{bmatrix}   According to a program made by my friend, he told me:   $$\phi(4)=90$$ $$\phi(5)=2040$$ Email searches in Outlook 2013 work as expected. I am trying to refine it by excluding certain folders by default. I was hoping I could uncheck certain folders in Control Panel > Indexing Options > Outlook. Unfortunately, the checkbox is for Outlook itself. I can either search Outlook or not. There is no refinement beyond that. Is there another way to exclude searching of specific folders? How do you increase MAXSYMLINKS In a python script, I am creating a bunch of symbolic links chained together.  example: link1->link2->link3->.......->somefile.txt  I was wondering how you can change the max number of symlinks to be greater than 20? How do I get the value of text input field using JavaScript? I want to create liveUSB on part of my flash stick. I made a partition with GParted and downloaded .iso file of Ubuntu 19 from the official website to mount with gnome-disk-utility.  I noticed that the iso file has "amd64" in the name. But I intend to use this USB not only on amd computers!  Is that important? Will the live USB work on PC with Intel/other processor? Run a shell script as another user that has no password I would like to run a script from the main ubuntu shell as a different user that has no password.  I have full sudo privileges, so I tried this:  sudo su -c "Your command right here" -s /bin/sh otheruser   Then I have to enter my password, but I am not sure if that script is now really running under that user.  How can I confirm that the script is really running under that user now? Prove $\mathbb{Z}_2 \times \mathbb{Z}_2$ is not isomorphic to $\mathbb{Z}_4$ Based on the table Encounter XP Multipliers from the Basic Rules' DM section, Page 57, when you're building encounters in 5E, you include an XP modifier depending on the number of creatures your PCs will be fighting: 1.5x for a pair, 2x for 3-6 and so on. However, the PCs only get the base XP as a reward.     This doesn't change the actual xp reward ... just your calculations of how difficult the encounter is.   Why is this? After acknowledging that battles vs multiple opponents are more difficult, why not reward them as such? A quick search for currency regex brings up . The problem I have in choosing one is that regex is difficult to verify without testing all the edge cases. Does anyone have a regex for U.S. currency that has been thoroughly tested? My only requirement is that the matched string is U.S. currency and parses to System.Decimal:  [ws][sign][digits,]digits[.fractional-digits][ws]   Elements in square brackets ([ and ]) are optional.  The following table describes each element.   ELEMENT             DESCRIPTION ws                  Optional white space. sign                An optional sign. digits              A sequence of digits ranging from 0 to 9. ,                   A culture-specific thousands separator symbol. .                   A culture-specific decimal point symbol. fractional-digits   A sequence of digits ranging from 0 to 9.