$|z-1|=1$ is obviously a circle centered at $x=1$ and $y=0$ on z-plane. However, I am having a hard time visualizing it in the $w=z^2$. Is there an analytic way of getting the equation for $z$ in $w$-plane?   How should I do it?  Work done so far:   Nothing, i am not sure how to begin!  Thanks in advance!
How do i show that $w=z^2$ maps $|z-1|\leq 1$ onto the cardioid $R=2(1+\cos \theta)$?
Electromagnetic Tensor in Cylindrical Coordinates I understand that the  is given by  $$F^{\mu\nu}\mapsto\begin{pmatrix}0 &amp; -E_{x} &amp; -E_{y} &amp; -E_{z}\\ E_{x} &amp; 0 &amp; -B_{z} &amp; B_{y}\\ E_{y} &amp; B_{z} &amp; 0 &amp; -B_{x}\\ E_{z} &amp; -B_{y} &amp; B_{x} &amp; 0 \end{pmatrix}$$  where $\mu$, $\nu$ can take the values {0,1,2,3} or {$t,x,y,z$}.  So, for example  $$F^{01}=F^{tx}=-E_x$$  My question is, what would the following expression be?  $$F^{t\rho}=?$$ or $$F^{z\rho}=?$$  where $\rho=\sqrt{x^{2}+y^{2}}$ is the radial coordinate in ?  And more generally, how can we construct the Electromagnetic Tensor in cylindrical coordinates? Where $\mu$, $\nu$ now take the values {$t,\rho,\varphi,z$}. Why don't Muggle-born wizards use Muggle technology to fight Death Eaters? We know that wizards in the Harry Potter universe are subject to regular physical harm (e.g. witness injuries from being hit by Bludgers or just colliding with things when playing Quidditch).  Therefore, it's almost a certainty that an average wizard would be vulnerable to, say, a bunch of bullets.   There may likely be spell(s) to protect from projectiles - witness Dumbledore's  deployed when Voldemort sent a bunch of glass at him during the  at the end of Harry Potter and the Order of the Phoenix (according to , possibly an unusually powerful Protego or the Silver shield).   But:   those spells obviously take effort/energy/time to cast (not everyone's Dumbledore) AND more likely than not most Death Eaters wouldn't recognize a Muggle gun for a threat (or realize what kind of threat) till too late.   So, the guns should at best allow an over-matched OotP member (e.g. Harry or Hermione) to greatly equalize the power between any regular Death Eaters and themselves, and even perhaps help against Voldemort (who can't be killed with a bullet, but, at best, would need to expend magical energy and time into conjuring a shield, and, at worst, be stopped/interrupted/driven off).  So I'm looking for an explanation - ideally in-universe, but possibly just some statement by JKR - of why the Muggle-born OotP - who are likely very much at ease AND familiar with Muggle technology and live in the late 20th century - do not use guns. Or, for that matter, ANY Muggle technology of the late 20th century? Cue Arthur C. Clarke's "indistinguishable from magic" meme.  Even if they are on some kind of idiotic moral crusade to never kill their opponents (even the Jedi aren't this dumb), given the time-frame, non-lethal weapons would have already existed, such as rubber bullets and tasers.  NOTE: Please don't offer the "if they do it, Death Eaters would start using guns as well" theory. First, Death Eaters would have used the guns if they knew how to, anyway. Second, the guns, like any "secret weapon", could be reserved for strategic battles, say the defense of Hogwarts in Deathly Hallows.    The origin of the question stems from two things.  First, Harry's fascination with all things that are different in the magical world (e.g. moving photographs), based on his knowledge of the Muggle material world, as well as the somewhat amusing cluelessness of the "Muggle artifacts" expert, Mr. Weasley, regarding Muggle items and how they work.   Basically, from the books, it's clear that the Wizarding world wouldn't know details about guns or what they are for or how they work.  The second thing is it's clear, at least initially, that wizards are afraid of Muggles finding out about them. Hence all the efforts to cloak their society (basically, do you REALLY want every Muggle gunning for you, even with Magic)? The idea of how control over the Muggle world seems to be more indirect, by Death Eaters asserting mind-control over the government. This means there's an implicit understanding by Death Eaters that they don't stand a chance against armed Muggles even given the magical disparity.  Considering Harry and Hermione's overall adaptability and inventiveness (and brains in the latter case), it's mind-boggling that the idea of fighting Death Eaters with modern technology completely slipped their mind. I mean, it's a basic part of human myth (see Steel vs. Magic themes in Conan books, or Beowulf, or heck, A Yankee in King Arthur's Court). So it's not like the idea of using technology against magic would be impossible to come up with. The question begs itself - why not? Considering the fact that Rowling generally tried to be logical and consistent in building the HP universe, I feel like there must be some in-universe explanation for this that I just didn't notice.   (in-universe answers only unless there are some authoritative out-of-universe statements that are part of canon). What can I do to increase battery life on my Android device? What can I do to get better battery life on my Droid? Battery life seems awful. Less than 15 hours or so on normal usage. How to replace a string with a string containing slash with sed? I am looking for a way to replace a string in a file with a string that contains a slash by using sed.  connect="192.168.100.61/foo" srcText="foo.bar=XPLACEHOLDERX" echo $srcText | sed "s/XPLACEHOLDERX/$connect"   The result is:  sed: -e Expression #1, Character 32: Unknown option for `s' Automatically keep current version of config files when apt-get install Derive a formula to find the number of trailing zeroes in $n!$ Possible Duplicate:          I know that I have to find the number of factors of $5$'s, $25$'s, $125$'s etc. in order to do this. But how can you derive such a formula for any number $n$? Influence of charged particle's own electric field on itself I read this in my textbook: A charged particle or object is not affected by its own electric field.  Since I find this completely unintuitive and my mind is yelling "wrong! wrong! how could a particle even distinguish between its own field and the external fields?" I would really like to hear an explanation about why this is true, if it is.  (A corollary question that comes to my mind after thinking about this is... What about gravity and mass considering the analogous situation? And the other two forces?) How does a magnet work? Drawing wavy, wiggly lines in QGIS? Is there a QGIS function or plugin to draw wiggly line?   I have used Spline Tool to manually draw some waves, but it is time consuming.  If possible, I would like to draw something like:    Inkscape Function Plotter ( sin(x) curve). Product rule intuition Can anyone think of an intuitive explanation of the product rule? I'm not asking for a derivation. To me it seems like quite an un-untuitive result, as apposed to the chain-rule (which is ironically harder to derive). Why does the air flow faster over the top of an airfoil? I understand the common explanation of lift, which describes the airflow over the top of the wing as moving faster than the air below the wing.  However, I don't quite understand why the air moves faster.  I've read some explanations citing a circulation effect among others, but haven't found a good, clear explanation for the exact cause.  Can someone help me out? Install MySQLdb (for python) as non-compressed egg The install instructions are:  $ python setup.py build $ sudo python setup.py install # or su first   This gives me an .egg file. How do I tell the install to dump the files as a normal, uncompressed library?  Thanks! How do I add CSS & JavaScript to ? I want to add the following code in my  section of the front page using a Drupal 8 sub theme of Bartik.   Can anybody help how to do it? What is the best way to do it in this new twig version of theme? Where is page.tpl?  &lt;script type="text/javascript" src="http://ajax.googleapis.com/ajax/libs/jquery/1.3.2/jquery.min.js"&gt;&lt;/script&gt; &lt;script type="text/javascript" src="stepcarousel.js"&gt;&lt;/script&gt; How precisely do I need to post close to 7:00 (somewhere) to get hero of time? Can I kill an indestructible creature with -X/-X and sufficient damage marked on it? Can I hit a 4/4 indestructible creature with a  (3 damage) and give it -1/-1 with  to kill it? (Does it matter which way around I cast those?)  I read  here and it says, damage is removed in cleanup step, so it should be on the creature still after both spells.     702.12. Indestructible   702.12a Indestructible is a static ability.   702.12b A permanent with indestructible can’t be destroyed. Such permanents aren’t destroyed by lethal damage, and they ignore the state-based action that checks for lethal damage (see rule 704.5g).   702.12c Multiple instances of indestructible on the same permanent are redundant. I have been using and making LED flashlights for years now and I am VERY familiar with all of the circuitry and technology inside of them, but I just got my first LED house light bulb that runs off of 110v AC electricity.  It is very similar to a flashlight - it has a large heatsink, and 1 or more LEDs inside of it (brand and specs of emitters, I do not know).  The part that I am most curious about is how the electricity from the socket gets to the emitter.  In a flashlight the power comes from a DC battery through a current regulator (LED driver) and into the LED itself.  This is not difficult because the voltage of common LED emitters is about the same voltage as Lithium batteries (3.7v, 7.4v, etc) which makes the current limiting relatively simple coming from a one or two cell light.  How does this work for a high voltage AC bulb?  Is there a transformer and rectifier inside of EVERY bulb (I sure hope not), or are the emitters manufactured to run off of 110v AC power and just use half of the wave cycle? What is a loss function in decision theory? My notes define a loss function as the 'cost' incurred when the true value of $\theta$ is estimated by $\hat\theta$. What kind of cost is it talking about? monetary cost? or is it something related to errors? What is serial voting and how does it affect me? I just noticed that I lost a bunch of points from my reputation score, and I used the "reputation" tab on my user profile page to try and track down the cause.  During my investigation, I noticed there was an unusual event of type "reversal". In the normal place of a question title, it says "voting corrected".     What does this mean, and what caused it?  Did I do something wrong? Why did I lose all of that reputation? Is the system punishing me for leaving too many good answers?  Is there any way to earn it back? Matrices commute if and only if they share a common basis of eigenvectors? How can I set default value in ActiveRecord?  I see a post from Pratik that describes an ugly, complicated chunk of code:   class Item &lt; ActiveRecord::Base     def initialize_with_defaults(attrs = nil, &amp;block)     initialize_without_defaults(attrs) do       setter = lambda { |key, value| self.send("#{key.to_s}=", value) unless         !attrs.nil? &amp;&amp; attrs.keys.map(&amp;:to_s).include?(key.to_s) }       setter.call('scheduler_type', 'hotseat')       yield self if block_given?     end   end   alias_method_chain :initialize, :defaults end   I have seen the following examples googling around:    def initialize      super     self.status = ACTIVE unless self.status   end   and    def after_initialize      return unless new_record?     self.status = ACTIVE   end   I've also seen people put it in their migration, but I'd rather see it defined in the model code.  Is there a canonical way to set default value for fields in ActiveRecord model?