To a mathematician a vector is a geometrical entity in space that can be defined perfectly well without establishing up a set of x,y,z,t axies.  In other words, mathematically we can work with vectors as entities without the need to determine their components in some basis.  My question is, can the laws of physics also be framed in this way? Of course we can write $$\vec{F} = m\vec{a}$$ but how useful is that unless we also talk about inertial frames of reference?  So, can we define frames of reference without  making the coordinates primary to our definition (or establishing basis vectors, which amounts to the same thing)?  [I don't believe this is a duplicate question.  The other fellow was asking for a book recommendation - I am asking about the framework for the laws of physics]
I am a graduate student in mathematics who would like to learn some classical mechanics. However, there is one caveat: I am not interested in the standard coordinate approach. I can't help but think of the fields that arise in physics as sections of vector bundles (or maybe principal bundles) and would love an approach to classical mechanics or what have you that took advantage of this.  Now for the questions:   Is there a text book you would recommend that phrases the constructions in classical mechanics via bundles without an appeal to transition functions?   What are the drawbacks to this approach other than the fact that it makes computations less doable? (if it does that)   Are there benefits to thinking about things this way, ie would it be of benefit to someone attempting to learn this material to do it this way?
Why aren't search engines indexing my content? Can we exempt downvoted accepted answers from getting the top spot? Tricks for getting lots of arrows? Any tips or tricks for getting arrows (besides cheating)? Or tricks for finding a skeleton spawn?  I've tried farming chickens and gravel for feathers and flint, but it takes a ridiculously long time. How do people get enough arrows to make using a bow actually feasible? I'm trying to do raster algebra with two rasters, but when I transform the rasters to array I can't do the operation because the files do not have the same extent  example:  from osgeo import gdal import numpy as np  IMG1 = gdal.Open('C:/IMG1.TIF') IMG2 = gdal.Open('C:/IMG2.TIF')  img1 = IMG1.ReadAsArray() img2 = IMG2.ReadAsArray()  img1+img2 &gt;&gt;&gt;&gt;&gt; ValueError: operands could not be broadcast together with shapes (5353,3352) (5548,3232)    So.. if I see the gt of every img show that img1 and img2 have the same size of the pixel (100) but different extent  g1 = IMG1.GetGeoTransform() gt1 &gt;&gt;&gt;&gt;&gt; (482084.999999999, 100.0, 0.0, 5652715.00001374, 0.0, -100.0) g2 = IMG2.GetGeoTransform() gt2 &gt;&gt;&gt;&gt;&gt; (475184.999999999, 100.0, 0.0, 5643715.00001535, 0.0, -100.0)   So.. I used this operation in order to get the intersect extent.. but I don't know how to apply  r1 = [gt1[0], gt1[3],gt1[0] + (gt1[1] * IMG1.RasterXSize), gt1[3] + (gt1[5] * IMG1.RasterYSize)] r2 = [gt2[0], gt2[3],gt2[0] + (gt2[1] * IMG2.RasterXSize), gt2[3] + (gt2[5] * IMG2.RasterYSize)] intersection = [max(r1[0], r2[0]), min(r1[1], r2[1]), min(r1[2], r2[2]), max(r1[3], r2[3])] I have very little background in physics, so I apologize if this question is painfully naive.  Consider the following thought experiment: an observer is in a closed room whose walls, floor, and ceiling are made entirely of mirrors, with a single light source in the middle of the room. When the light is on, the observer can see many copies of his reflection all over the place.  Suddenly, the light source turns itself off. Intuitively, I would expect the observer to "instantly" see darkness. However, I can't figure out why that is the case under the "particle" interpretation of light. There are obviously lots of photons already in the room from before. Furthermore, we know that they don't get "consumed" when they hit a wall, because otherwise the observer wouldn't see so many reflections of himself. Basically, when the light goes off, what happens to the photons already in the room?  I suspect the answer goes something like this: the photons in the room lose a little bit of energy every time they bounce off a mirror, but it's so minuscule that we can still see more reflections than our eye can resolve anyway. When the light goes off, however, it takes them a very small fraction of a second to bounce around the room enough times to diffuse completely, which our eye cannot detect.  Is that about right? If we had a theoretical "perfect reflector", would the light remain trapped in the room forever? If we had instruments that could measure such things very finely, would it take (slightly) longer for the light to go out in a room made of mirrors as opposed to a room made of, say, black cloth? Display Blank when Referencing Blank Cell in Excel 2010 Will I miss anything by watching Dragon Ball Z Kai instead of the other series? I've read that Dragon Ball Z Kai is an HD remaster and recut of Dragon Ball and Dragon Ball Z. However, Kai has  while Dragon Ball has  and Z has .  If I watch only Dragon Ball Z Kai, do I miss anything I would get by watching the other two series? And is anything censored in Kai that is not censored in the others? How to set security options for a PDF using pdflatex My questions is very simple. How can I set security setting for a PDF file created with PDFLatex?  I should set some password or print permissions, etc. Backup and restore settings in Texstudio Does someone know, how to save all the settings of Texstudio and restore them in a simple way?  Thank you in advance! I’m studying a PhD in Theoretical Physics (quantum field theory, to be specific), but I would like to learn more about black holes (and related areas. Does anyone have any suggestions for textbooks/available notes on these subjects for people without much background in them? Obviously I’m very comfortable with mathematics and physics in general so I’m not looking for “pop science” material. How can I accept the Microsoft EULA agreement for ttf-mscorefonts-installer? After a recent update, ttf-mscorefonts-installer prompted me to accept its license agreement.     ┌─────────────────┤ Configuring ttf-mscorefonts-installer ├─────────────────┐  │                                                                           │   │ TrueType core fonts for the Web EULA                                          │                                                                               │ END-USER LICENSE AGREEMENT FOR MICROSOFT SOFTWARE                             │                                                                               │ IMPORTANT-READ CAREFULLY: This Microsoft End-User License Agreement           │ ("EULA") is a legal agreement between you (either an individual or a          │ single entity) and Microsoft Corporation for the Microsoft software           │ accompanying this EULA, which includes computer software and may include      │ associated media, printed materials, and "on-line" or electronic              │ documentation ("SOFTWARE PRODUCT" or "SOFTWARE"). By exercising your          │ rights to make and use copies of the SOFTWARE PRODUCT, you agree to be        │ bound by the terms of this EULA. If you do not agree to the terms of          │ this EULA, you may not use the SOFTWARE PRODUCT.                              │                                                                               │                                  &lt;Ok&gt;                                         │                                                                           │   └───────────────────────────────────────────────────────────────────────────┘    Screenshot:   For some reason my terminal will not allow me to accept, or for some reason I am pressing the wrong hotkey...  I've tried every letter on the keyboard and Enter among others...  I'm sure there is a very simple and obvious solution to this.  I've also just tried to remove the package completely however the terminal states that due to the package not being correctly installed, I should reinstall the package before removing it.  Very frustrating!  Essentially, because I cannot successfully install this package, I can't really ever upgrade my system because I always have to end up terminating the terminal with the license agreement (thus the upgrade fails). How to redirect stderr to a file How do I render a Twig template and load a CSS file from a controller? Is this considered a good practice for rendering a Twig template and loading a CSS file from a controller?   DefaultController.php  namespace Drupal\acme\Controller;  use Drupal\Core\Controller\ControllerBase; use Drupal\Core\DependencyInjection\ContainerInjectionInterface; use Symfony\Component\DependencyInjection\ContainerInterface; use Drupal\Core\Template\TwigEnvironment;  class DefaultController extends ControllerBase implements ContainerInjectionInterface  {   /**   * @var Drupal\Core\Template\TwigEnvironment   */   protected $twig;    public function __construct(TwigEnvironment $twig)   {     $this-&gt;twig = $twig;   }    public static function create(ContainerInterface $container)   {     return new static(       $container-&gt;get('twig')     );   }    /**    * hello    * @param  string $name    * @return string    */   public function hello($name)    {      $template = $this-&gt;twig-&gt;loadTemplate(       drupal_get_path('module', 'acme') . '/templates/hello.html.twig'     );      $markup = [       '#markup' =&gt; $template-&gt;render([ 'name' =&gt; $name ]),       '#attached' =&gt; [ 'css' =&gt; [           drupal_get_path('module', 'acme') . '/assets/css/acme.css'         ]       ]     ];      return drupal_render($markup);   } }   acme.routing.yml  acme_hello:   path: '/acme/hello/{name}'   defaults:     _content: '\Drupal\acme\Controller\DefaultController::hello'     _title: 'acme Title'   requirements:     _permission: 'access content'   hello.html.twig  &lt;div class="acme-hello-text"&gt;   &lt;h1&gt;Hello {{ name }}!&lt;/h1&gt; &lt;/div&gt; How to convert mediawiki syntax to latex? What book features people living in a tree who move up it in a bubble? Find this integral $$\operatorname I=\int\limits_{0}^{1}\dfrac{\arcsin{\sqrt{x}}}{x^4-2x^3+2x^2-x+1}\operatorname d\!x$$  My try: let $$f(x)=x^4-2x^3+2x^2-x+1$$ I found  $$f(1-x)=(1-x)^4-2(1-x)^3+2(1-x)^2-x+1=x^4-2x^3+2x^2-x+1=f(x)$$ so $$I=\int_{0}^{1}\dfrac{\arcsin{\sqrt{(1-x)}}}{x^4-2x^3+2x^2-x+1}dx$$ so $$2I=\int_{0}^{1}\dfrac{\arcsin{\sqrt{x}}+\arcsin{\sqrt{(1-x)}}}{x^4-2x^3+2x^2-x+1}dx$$ then I can't,Thank you very much Combinatorics That Looks Similar to Vandermonde's Identity Android - Call log/ history : Retrieve a dialed number when there was no network coverage I dialed a number when I had no network support but it does not appear in the call history of my android phone.  Is there any way to get it ? (system logs or whatever)  THanks a lot in advance, Jack What kind of "mathematical object" are limits? RSA: Fast factorization of N if d and e are known