I am a beginning physics graduate student. I am often bewildered by the strange properties of the  such as:   $\delta (a x)= \frac{1}{a} \delta (x)$ The derivative of $\delta (x)$    etc etc.  Such strange properties of $\delta (x)$ are mentioned in the first chapter of Arfken-Weber (7th ed.) without proof. Please suggest me a book from which I can learn the minimum essential mathematics of $\delta (x)$ function that is required to study physics. I am not looking for an advanced mathematical treatment but the book should have the proofs of the theorems stated.
I am looking for tutorials, articles or books containing theory of  in context of mathematical physics. Please suggest.
Can I connect a PC to a PoE switch? Replace \input{fileX} by the content of fileX automatically How to ask great questions As programmers we are daily faced with having to get requirements, test things out, etc.  It's practically half our job to know what questions to ask, and how to ask them to get the most concise, actionable answer possible, especially from reluctant users/managers/customers/etc.  How do you ask questions that get answers? If $x+y+z=xyz$, prove $\frac{2x}{1-x^2}+\frac{2y}{1-y^2}+\frac{2z}{1-z^2}=\frac{2x}{1-x^2}\times\frac{2y}{1-y^2}\times\frac{2z}{1-z^2}$ given that $x^2~,~y^2~,~z^2\ne1$  I came across this question in an ancient ($19$th century) Trigonometry book, and this is the method they use to prove the result (please note: I understand this method fully): Let $x=\tan A$ ,$~y=\tan B$ and $z=\tan C$ which is acceptable without any loss of generality. This means that we are saying that $$\tan A+\tan B+\tan C=\tan A\tan B\tan C$$ Consider $\tan (A+B+C)$: $$\tan (A+B+C)=\frac{\tan A+\tan B+\tan C-\tan A\tan B\tan C}{1- \tan A \tan B- \tan C \tan A - \tan B \tan C}$$ So if $~\tan A+\tan B+\tan C=\tan A\tan B\tan C~~$ then $~~\tan (A+B+C)=0$. Hence, let $~A+B+C=\pi$. Now consider $\tan (2A+2B+2C)$: $$\tan (2A+2B+2C)=\frac{\tan 2A+\tan 2B+\tan 2C-\tan 2A\tan 2B\tan 2C}{1- \tan 2A \tan 2B- \tan 2C \tan 2A - \tan 2B \tan 2C}=0$$ $$\implies \tan 2A+\tan 2B+\tan 2C=\tan 2A\tan 2B\tan 2C$$ $$\implies \frac{2\tan A}{1-\tan^2 A}+\frac{2\tan B}{1-\tan^2 B}+\frac{2\tan C}{1-\tan^2 C}=\frac{2\tan A}{1-\tan^2 A}\times\frac{2\tan B}{1-\tan^2 B}\times\frac{2\tan C}{1-\tan^2 C}$$ $$\therefore\frac{2x}{1-x^2}+\frac{2y}{1-y^2}+\frac{2z}{1-z^2}=\frac{2x}{1-x^2}\times\frac{2y}{1-y^2}\times\frac{2z}{1-z^2}$$ as required.  My question is, is there any other way of proving this without this rather heavy use of trigonometry? I also would prefer not to work through heaps of algabraic manipulation and expansion to obtain the required result, although if that's necessary I will put up with it ;) Thank you for your help. What causes the dimensions of a star increase when its hydrogen fuel is exhausted? What is the procedure for taking down Titans? Lebesgue measure of the set of zeroes of Brownian motion Let $B_t(w)$ be a Brownian associated with a measure $P(dw)$. Suppose that $X$ is a random set of times $t$ where $0 \le t \le 1$ such that $B_t=0$. Show that $X$ has Lebesgue measure zero $P$ almost surely. Unsealed meat in freezer Below is my situation:  Is completely (vacuum) sealing leftover meat in the freezer absolutely necessary? I would always just use a plastic bag to cover the meat and fold it around couple of times so it's almost as good as sealed (usually grocery plastic bag is much bigger than the piece of meat) and have never had a problem with it last a decade or so I have been cooking.  But recently I got this new roommate who insists on me using special small plastic bags with sealers to use and he is being very paranoid, obsessive about it.  So my question is: Is he right? I think my MacBook Pro battery was replaced by Apple's service team. How do I check that? I handed my MacBook Pro (15" Touch bar) to the service centre in order to have the keyboard replaced. Now, when I check the battery cycle count, it shows "1". Before it was at "82".   Also, I'm getting a battery life of only 4.5 - 5 hours now.   Is there a way to find if my battery was replaced or not? How do I invoke a private static method using reflection (Java)? Test coverage for setCookies method I am adding cookie to my page in the controller using  ApexPages.currentPage().setCookies(new Cookie[]{CookieName});  When am trying to get the code coverage, my test class is not covering the above line.  How to get the code coverage for the above setcookie method in the test class. Find $n$ such that $x^2 + x + 1$ is a factor of $(x+1)^n - x^n - 1$. I have to find the form of n i.e. whether n is even or odd and whether it is multiple of 2 or 3 such that:  $x^2 + x + 1$ is a factor of $(x+1)^n - x^n - 1$  What I tried:  $$x^2 + x + 1 = (x + 1)^2 - x $$  also,  $$(x+1)^n-x^n-1= \binom{n}{o}x^n + \binom{n}{1}x^{n-1} + ... + \binom{n}{n-1}x + \binom{n}{n}.1 - x^n - 1$$  $$=\binom{n}{1}x^{n-1} + ... + \binom{n}{n-1}x$$  after getting this far I tried with taking x and n common but still couldn't able to find the right direction.  I need extra hints to solve this. How to make a surface with a transparent background in pygame jQuery Ajax POST example with PHP Trouble With Persistence After Making a USB Live Thumb Drive Using Pendrivelinux I created a bootable Ubuntu on my 32GB thumb drive (on a NTFS partition) and I selected a persistence of 4GB.  I've noticed that when I reboot from the initial booting off of the USB thumb drive, nothing is preserved.  No apps that were installed nor saved bookmarks.  Is there a way for me to manually force persistence? Story about human space explorers who run into alien telepath space crew I read a short story in an anthology around 1998 (the book could have been older than that) which went like this - A human space exploration scientific crew is exploring an alient planet (peaceful planet, as I recall) when they run into an alien spaceship (from a different planet which also happened to be there) - the aliens were telepaths.   The alien commander and first officer discuss the earthlings after a meeting and realize that the human crew are the descendants of a species (Banthu/Klanthu/Zantha, something like that) which had tried to violently take over the galaxy some 20/30/50,000 years ago and which had been beaten back by the combined might of the rest of the galaxy. They had supposedly been eliminated but they somehow managed to survive and are back in space only a few thousand years after they were completely beaten (which is something the aliens are amazed by).   The alien commander kills the human explorers via telepathy but realizes that there is no information available to locate their home planet and speculate that they will be meeting again soon given the past history of the humans.  I vaguely think there was also a fantasy short by L Sprague de Camp in the same anthology, but can't be 100% sure. Any thoughts/ideas? Thanks! I have read answers to questions like  but I couldn't find a general rule for using "you" or "yourself", "him" or "himself" that could explain things I see in some examples.  For example, I think that one normally say (according to examples I see in ):   You bring something with you. You have something in you. You have something on you. You pressed her to you.   But on the other hand one would say:   You draw attention to yourself. He killed himself. You can try it out for yourself.   As a general rule, it seems that the reflexive pronoun is used when the object is the same as the subject of the verb. But for prepositional phrases, it seems more complicated. Do some prepositions (like "with", "in", "on") always come with the accusative form of the pronoun, and some other (like "for") don't? What about "to", that seems to be followed by both in different circumstances? I have seen this a lot but i ran into this question today:    And the answer (while good) has 83 upvotes which seemed a little over the top.  I see dozens of answers of this caliber every week that only seem to get a few upvotes.    Was there some sort of SO policy that encouraged people to upvote more during this period?  Am I just imagining that answers from this period received more votes than comparable ones now? story on forest timescale I'm looking for the title of a short story by Ursula Le Guin about a scientist (?) who is given a drug that slows the timescale of his perception enormously, so that he can perceive that an alien (?) forest is actually behaving as a sentient being, reacting to current events, but at an enormously slow (from a normal human point of view) timescale. It might be in , but I don't remember the title (and it doesn't jump out at me from the list of titles) and don't have access to that collection right now. It is not &quot;The Word for World is Forest&quot;, nor &quot;Direction of the Road&quot; (from ), nor &quot;Vaster than Empires and More Slow&quot;, the latter two of which look at similar plant perception/time-scale issues. Update: it turns out that the description was right, but I had most of the details wrong: not Le Guin, not alien, not a scientist ... As per the following example, and as in  In bash, where has the trailing newline char gone?, I want to know "why" it happens    x="$(echo -ne "a\nb\n")" ; echo -n "$x" | xxd -p  # Output is: 610a62  # The trailing newline from the 'echo' command #   has been "deleted" by Command Substitution   I assume there must be some very significant reason for a shell action, namely Command Substitution, to actually delete some data from the command output it is substituting... but I can't get my head around this one, as it seems to be the antithesis of what it is supposed to do.. ie. to pass the output of a command back into the script process... Holding back one character seems weird to me, but I suppose there is a sensible reason for it...  I'm keen to find out what that reason is...