How to derive $\int_{\Sigma}\delta(*F)\wedge\delta A =\int_{\Sigma}d\Sigma^{\mu}\delta F_{\mu\nu}\wedge\delta A^{\nu}$ for free EM?
Induced metric on a null hypersurface
Show that $A\in\mathbb{C}_n$ is normal $\iff$ $tr(A*A) = \sum_{i = 1}^n|\lambda_i|^2$, where $\lambda_1,...,\lambda_n$ are the eigenvalues of $A$. Can Tamron 16-300mm Macro Lens fit to Canon EOS 600D? I use Canon EOS 600D and I want a lens with a larger range. I found out Tamron 16-300 f/3.5-6.3 Di II VC PZD Macro Lens, I added it into my wishlist. I did some research on it and I'm scared that it couldn't fit to my Canon EOS 600D.  I'm still a beginner... I'm confused with EF, EF-S and APS-C/DX. Can someone please explain it to me? I have 238gb of timelapse photos which was taken over a period of twenty days and all the photos are split up into 20 folders. I would like to put this together into a timelapse, I tried using Premiere Pro but that is taking very very long, any tips on putting it together fast btw Im on a Mac, thanks. Workflow rule order of execution How to Create a Half-Turn Staircase? I would like to create a half-turn staircase without a landing. Is there an easy way to accomplish this without having to manually calculate and create each step individually?  Here’s an image similar to what I would like to achive:    I have found the Archimesh add-on including . While I can bend the created stairs, I haven’t found any way to get anywhere close to the image above. My stairs just always look out of shape and I don’t know how to get the outline of the staircase into a square.  I should mention that I only have very little experience with Blender modeling, so it’s well possible that I’m just missing some curve modification basics. Perhaps Archimesh is also the wrong approach for this in the first place? Python idiom to return first item or None I'm sure there's a simpler way of doing this that's just not occurring to me.  I'm calling a bunch of methods that return a list.  The list may be empty.  If the list is non-empty, I want to return the first item; otherwise, I want to return None.  This code works:  my_list = get_list() if len(my_list) &gt; 0: return my_list[0] return None   It seems to me that there should be a simple one-line idiom for doing this, but for the life of me I can't think of it.  Is there?  Edit:  The reason that I'm looking for a one-line expression here is not that I like incredibly terse code, but because I'm having to write a lot of code like this:  x = get_first_list() if x:     # do something with x[0]     # inevitably forget the [0] part, and have a bug to fix y = get_second_list() if y:     # do something with y[0]     # inevitably forget the [0] part AGAIN, and have another bug to fix   What I'd like to be doing can certainly be accomplished with a function (and probably will be):  def first_item(list_or_none):     if list_or_none: return list_or_none[0]  x = first_item(get_first_list()) if x:     # do something with x y = first_item(get_second_list()) if y:     # do something with y   I posted the question because I'm frequently surprised by what simple expressions in Python can do, and I thought that writing a function was a silly thing to do if there was a simple expression could do the trick.  But seeing these answers, it seems like a function is the simple solution. What laptop specs are important for intensive image editing work? How important is the following in purchasing a laptop that will be used for image editing:   processor speed amount of RAM integrated vs. dedicated graphics card HD speed (SSD; or 5400/7200/10000 rpm) What happened to the Yin half of Kurama? Did Naruto get both halves of Kurama after fighting Sasuke, or did the other half of Kurama die when Minato was killed? How can I easily clone my entire system? I am running an Ubuntu 14.04 LTS and I experienced some issues with my hard drive. I bought a new HDD of the same size (1TB), and I intend to move all my data (softs, work, pics, etc.) on the new HDD.  What is the easiest way to do that? What is the name of this theorem, and are there any caveats? For random variable $X$ that follows some distribution, $f(x)$ is the probability density function of that distribution if and only if $$\mathbb{E}[\phi(X)] = \int_{-\infty}^\infty \phi(x) f(x)dx$$ for all functions $\phi$.  Context: My professor used this in lecture to demonstrate a way to find the distribution of $cX$ given random variable $X$ that follows a specific distribution.   What is the name of this theorem? Have I missed any qualifications/caveats? I am in particular curious about whether "for all functions" is correct. My professor mentioned "positive, bounded, deterministic functions," but I am not sure what he meant by that. Where can I find a proof of this theorem? What is a pull up and pull down? Can some one explain this terminology please I think I understand it but not completely sure.   I think pull down is where you place a resistor between +V and the other component and pull up is where you place the resistor between 0v and the component.    If I am completely wrong then let me know! I've got a Windows 10 Computer (home edition), which has 4GB of Ram and an i5 processor. I need to run an old 32 bit accounting programme. The vendor said it won't release a 64 bit version (although a 64bit version exists, it's missing key features from the older version).  The programme installs fine, but when I click on the programme icon, it says the programme won't work on a 64 bit system.   What options do I have?    I've done a fair bit of research, but it seems to run up dead ends.   1) WoW64     Apparently, this is supposed to let you run 32 bits on 64 bit Windows. However, I am not sure how to install it. I think it's for web servers only. However,  says it can work on Windows 7, but doesn't explain how to set it up (or if it will work on Windows 10).  2) XP mode     A 32bit Windows XP emulator. However, it seems to be for Windows 7 only. Also according to this , you need Windows Professional.   3) Virtual Box Machine    Let's you emulate Windows. Seems overkill for my purposes. Plus I would need to buy a 32 bit Windows license. I think it would run slowly with 4GB of Ram.  4) Partition drive into 32bit and 64 bit versions Apparently, you can use the free tool  to split the drive into two and then install separate 32bit and 64 bit versions.   Again, seems overkill and it looks like I would need buy a 32 bit Windows 10 license.   Are these my only options?     Images  Installation    First run (From the error message I guess it's a 16-bit program)   Clicking on programme While CtrlX works fine in vim under windows, CtrlA selects all (duh).  Is there a way to increment a number with a keystroke under windows? Hint on how to prove the identity $\sum_{0 \leq k \leq n } \binom{k}{m} = \binom{n + 1}{m + 1}$ What is the correct way to pluralize an acronym / initialism? For example, if I wanted to write the equivalent of  There are many automated teller machines in this city.  Would it be  There are many ATMs in this city.  or  There are many ATM's in this city. (could get confused with possessive form or contraction).  or just  There are many ATM in this city. (assuming the final s is included in Machines represented by M).  Maybe something else? In Shakespeare's Macbeth, the Third Witch says to Banquo: "Thou shalt get kings, though thou be none" (1.3.65); if I want to quote that line and write it in the third person singular, how should I conjugate the verbs "shall" and "be" correctly? Picklist dependent on 2 controlling picklist I know that each new tab on Google Chrome have its own process, but I have 14 chrome.exe processes running actually as I can see it on the Task Manager, and only two tabs are opened on Google Chrome.    Could it be due to Chrome extensions? What other reason can open more chrome.exe process and how to avoid it? What language is the Ubuntu Software Center written in? Just been wondering, since elementary AppCenter is written with Vala and Gtk+3.  Also, what GUI wrapper does it use? How do we know what type of gauge field to add to a theory? I've been watching  and in one lecture, he discusses a very simple gauge theory.  We have a complex scalar field $\phi(x)$ with Lagrangian  $$\mathscr{L} = \partial_\mu \phi^* \partial^\mu \phi - V(\phi^* \phi)$$  and we want to do the gauge transformation $\phi(x) \to e^{i\theta(x)} \phi(x)$.  But the derivative term in the Lagrangian is not invariant if $\theta$ varies from place to place, so we add a new vector field $A_\mu$, and define it to transform like $$A_\mu \to A_\mu + \partial_\mu \theta$$ under the gauge transformation.  Then we change the Lagrangian to use covariant derivatives $D_\mu = \partial_\mu + iA_\mu$ instead of ordinary ones; now when the gauge transform is done all the derivatives of $\theta$ cancel out, and the result is invariant.  My question is: was there any freedom in choosing to add a vector field, with that specific gauge transformation law?  Are there any other ways we could get gauge invariance here—by using a different transformation law, or with a different type of field (say, a scalar, tensor, or spinor) with some other transformation law?  More generally, how can we tell what type of field is needed and what its gauge transformation law should be, to get some particular Lagrangian to be invariant under some particular gauge transformation?