I am taking a course in Mathematical Methods for physicsPhysics (Junior Level).   We are working from   > Mathematical Methods for physicists, George B.Arfken .    I just need online resources covering ( Matrices, Determinants, Vector Analysis, Tensors and differential forms and Vector spaces...etc) to study and understand from.
A week ago I  people on this site what mathematical background was needed for understanding Quantum Physics, and most of you mentioned Linear Algebra, so I decided to conduct a self-study of Linear Algebra. Of course, I'm just 1 week in, but I have some questions.    How is this going to be applicable to quantum physics? I have learned about matrices (addition, subtraction, multiplication and inversion) and about how to solve multiple equations with 3 unknowns using matrices, and now I am starting to learn about vectors. I am just 1 week in, so this is probably not even the tip of the iceberg, but I want to know how this is going to help me.  Also, say I master Linear Algebra in general in half a year (I'm in high school but I'm extremely fast with maths), what other 'types' of math would I need to self-study before being able to understand rudimentary quantum physics mathematically?
Find the general solution of $y'= a^{x + y}$ where y is the function How should I approach this problem? Should I put ln logarithm on both sides and move factor of variable of lnx on right side in front of the ln, or do somehing else? I understand general solving and types of solving these differential equations of first order, but I have no idea what my next step should be. Mechanism of Screenshot Basically I with my team are figuring out a way to block screenshots taken on our corporation's computers (Mac mostly). I have searched it extensively and found that there is something called Screen buffer which is being accessed by any screenshot taking software (eg. Snipping Tool etc). I tried to find the exact mechanism and the actual implementation of such softwares but with no luck. Also I found out that Netflix uses some kind of technology which corrupts screenshot image and blackens it. I did not find any actual mechanism or implementation to study it further. I also came across something called Hardware overlays but did not understand it properly in order to use it for actual implementation. I have considered a way to edit scancode maps in registry but I am looking for a more general way. I read a few articles for apple's FPS though. Can somebody please guide me through this? Some good resources for the same would be highly appreciated. Any particular or relevant forums or subs to discuss this are also welcome Sorry if this is the wrong place to ask this, I'm new to the site :/  My issue is this: when editing in vim, my syntax file highlights underscores and carets when found in non-math text, which is fine, except vim doesn't recognize that "align" is a math environment. What do I need to do to fix this? $\operatorname{Aut}(V)$ is isomorphic to $S_3$ I'm currently working my way through Harvard's online abstract algebra lectures (if you're interested, you can find them ).  The lectures come complete with notes and homework problems.  Of course, since I don't actually go to Harvard, I can't hand in the homework assignments to find out if I'm doing them correctly.  So I've decided to try to crowd source the grading of my solutions.  I'll post individual questions as I finish them and wait for comments.  I would like to get critiques of not just my reasoning but the style of my write ups as well.  Also, any alternative approaches to the problem would be welcome.  I've looked for forums dedicated to these kinds of OCW courses but have not been able to find any.  This surprises me.  It seems like, with the advent of these free online educational resources, an online meeting place for those who take advantage of them would be a natural offshoot.  So this might be a bit of an experiment.  Unless I'm the 873rd person to post something like this here.  If that's the case, sorry for being so long winded.  Anyway, on with the question.  This one is assigned in the 4th lecture.     Let $V$ denote the Klein 4-group.  Show that $\operatorname{Aut}(V)$ is isomorphic to $S_3$.   (I've moved my solution below to keep this question from showing up in the unanswered list.)  Thanks... Will getting a Software Testing Certification benefit me in my career? I have 2 years of experience in manual testing, and I am currently learning automation testing in my spare time. I was thinking that will getting a foundation/entry testing certification really help me in promotions and increase my salary or just increase my knowledge and nothing more? Even if I wish to try to get certified and pass the exams, is it too late or can I try after a few more years?  Kindly suggest! How to get battery status? How do I get the battery status on an iPhone? If $\forall x \in R, x^2-x \in Z(G)$, than $R$ is commutative Let $R$ be a ring such that for every $x\in R$ we have $x^2-x \in Z(G)$. Show that $R$ is a commutative ring.    My thoughts  What should I do? I could show that every $y \in R$ could be written in the shape $x^2-x$ but I don't know if that is true. We know that  $y \cdot(x^2-x) \ = \ (x^2-x)\cdot y$ for any $x,y \in R$, but that doesn't mean that $xy = yx$. Please give me a hint Intutitive meaning behind the formal definition of sufficient statistic? According to the definition of sufficiency, a statistic is sufficient for a parameter if the conditional distribution of $X$ given a value of statistic does not depend upon the parameter.   What I am trying to understand is how does conditional distribution of $X$ not being a function of the parameter fit into the intuitive meaning of sufficiency i.e statistic value holding the same amount of information as that of the respective sample. Can anyone can suggest where I can freely download ArcObjects SDKs for ArcGIS?  I am using version 10.2. Is there any research or data on horizontal vs. vertical navigational menus? I've seen many sites using both horizontal (top) navigation, vertical (left) navigation, and both across web sites for a long time. My understanding is that they're both conventions, but I would assume there must be some data out there supporting one route over the other.   My question is: are any studies or data for or against horizontal or vertical navigation (as suggested above) with regards to primary navigation (specifically on a smaller site or web app).  I've read some interesting things that are flipping me back and forth between liking horizontal more than vertical and vice versa. For example if you look at Jacob Nielsen's post on  you get the feeling that more important content should be on the left -- but you could take that as it being more important for the navigation (the user's escape routes) to be in that valuable space. How can phone companies detect tethering (incl. Wifi hotspot) When a 'blur' event occurs, how can I find out which element focus went *to*? Suppose I attach an blur function to an HTML input box like this:  &lt;input id="myInput" onblur="function() { ... }"&gt;&lt;/input&gt;   Is there a way to get the ID of the element which caused the blur event to fire (the element which was clicked) inside the function? How?  For example, suppose I have a span like this:  &lt;span id="mySpan"&gt;Hello World&lt;/span&gt;   If I click the span right after the input element has focus, the input element will lose its focus. How does the function know that it was mySpan that was clicked?  PS: If the onclick event of the span would occur before the onblur event of the input element my problem would be solved, because I could set some status value indicating a specific element had been clicked.  PPS: The background of this problem is that I want to trigger an AJAX autocompleter control externally (from a clickable element) to show its suggestions, without the suggestions disappearing immediately because of the blur event on the input element. So I want to check in the blur function if one specific element has been clicked, and if so, ignore the blur event. How is the zero flag set in terms of hardware? How can I find non-ASCII characters in MySQL? I'm working with a MySQL database that has some data imported from . The data contains non- characters (em dashes, etc.) as well as hidden carriage returns or line feeds. Is there a way to find these records using MySQL? Proving that an integral domain has at most two elements that satisfy the equation $x^2 = 1$. Export/Deploy Drupal site/page to static? (no cache) I wonder if there is an existing  module that can help in the subject. I am not looking for a caching module (like Boost or Varnish), but a way to export my whole site (or only certain pages) into static.  The idea is that I want my web server will be a very simple and light and will only serve static files (html/css/js) - no database on production, no php engine will be installed on server. All the Drupal engine will be on another server, export will be done each time adding/editing/changing a node or view, and a new set of html files will be created in a directory structure.  These files will later on be copied to the production site that will serve always and only static files, no .htaccess that will check requests if file exist etc.   Is there a module for that, or a workaround that this can be achieved with Drupal?  Thanks So I defeated a Team Valor gym earlier today using my highest cp Pidgeot. It turned grey, so I went to claim it with that Pidgeot, except it wouldn't let me. What determines which pokemon you are allowed to place at a gym? Is it just those who have participated in a battle can't? Is there a time period for it? What about training at a friendly gym? Does this impact anything? Light fixture with a switch that has an outlet I hope this explains it better: in this room there is a switch with one set of wires and only the black one went through the hole in it, and another black one in the other hole, and is hooked to the outlet by being wrapped around the screw terminals. The ground is cut short and not grounded.  Now on to the outlet: the wire from the switch is connected to the top screw - but it's on the side that says "white".   There are two more sets of wires that connect to the outlet as follows:   the two black ones are on the the other screw terminal on the bottom (one on the "white" side)  the grounds are grounded to the screw terminal below that  the white ones are connected by the one from the switch to the bottom (black side), with one of the white ones from the other set on the screw terminal, and the one from the other cable is on the top screw terminal on the "white" side.   Now to the light: there are the black and white that come out of the box, and that is the set that has the ground cut short.  The original light fixture was a single 660w/250v rated socket.  The one my friend put up was a fixture with two 660w /250v sockets.  At the light only black and white connected, not ground... It worked then made noise and this is where I'm at.  When I wire test them, the black one from the switch and the white ones from either sets nothing...  but all the other ones light up.....! Why didn't the Ministry use owls to find Sirius? Harry, before the Triwizard Tournament, was able to send owls to Sirius when he was in hiding (not at Grimmauld Place) just by telling them "just find him, will you?", and the owl would come back a few days later with an answer. This doesn't happen with Hedwig only, because, if I recall correctly, he uses school owls as well.  If finding Sirius was as easy as telling an owl "find him", and a 13-year-old wizard could do it, how come the Ministry had no clue as to where he was? They could've sent an owl with a summons and followed it from a distance.  Two theories, neither mentioned in the book, and both a bit unlikely:   He was living as a dog and -hopefully- the Ministry would not be able to recognize him. Unlikely because the owl would have got there anyway, so they would have guessed the dog was Sirius in disguise, not necessarily as an animagus, but maybe he could perform human-animal transfiguration (remember he was brilliant at school).. Kingsley Shacklebolt was already in charge of the hunt and Dumbledore told him Sirius was innocent and all that. Then, Kingsley would have prevented the Ministry from using any method that could've actually worked. Unlikely because they could've used owls before Dumbledore's tip, when Kingsley didn't know Sirius was innocent, and also because Kingsley was quite powerful and had the support of Crouch and others at the Ministry, so if he was in charge, they would probably have succeeded.   Your thoughts? An example of a norm which can't be generated by an inner product I realize that every inner product defined on a vector space can give rise to a norm on that space.  The converse, apparently is not true.  I'd like an example of a norm which no inner product can generate.