Is the equation $$\nabla_a\sqrt{-g}=0$$ correct? Here $\nabla_a$ is the Levi-Civita connection, and $g$ is the determinant of metric $g_{ab}$. Apparently, we have $\nabla_ag_{bc}=0$, but I am not sure whether the up-mentioned equation is correct or not.
This question, , seems to indicate $$\nabla_a \sqrt{g}=0.$$ Why is this the case? I've always learned that $$\nabla_a f= \partial_a f,$$ hence surely $$\nabla_a \sqrt{g}=  \partial_a\sqrt{g} \neq 0. $$  Where's the hole in my logic?
I declared random list using TikZ package and can get items randomly. But at same time I need to get all items in sequential order. How can I do it? Is it possible print \mylist items declared in noncomma variat?  \documentclass[a4paper,14pt]{extreport} \usepackage{tikz}   \def\mylist{{one}{two}{three}{four}{five}}   \pgfmathdeclarerandomlist{mynum}{\mylist} %Define the list   \begin{document} \pgfmathrandomitem{\mynum}{mynum} \pgfmathresult~\mynum\\  \pgfmathrandomitem{\mynum}{mynum} \pgfmathresult~\mynum\\  \pgfmathrandomitem{\mynum}{mynum} \pgfmathresult~\mynum\\  \pgfmathrandomitem{\mynum}{mynum} \pgfmathresult~\mynum\\  \pgfmathrandomitem{\mynum}{mynum} \pgfmathresult~\mynum\\ \end{document} What's the most efficient way to select the last n rows in a table without changing the table's structure? What's the most efficient way to select the last n number of rows in a table using mySQL? The table contains millions of rows, and at any given time I don't know how large the table is (it is constantly growing). The table does have a column that is automatically incremented and used as a unique identifier for each row. Power Supply Capacity Formula Given that I am building a home computer from scratch and I have several components (MB, CPU, HDD, Video, Sound, etc) which have specific power requirements, is there a formula for me to calculate the size/capacity of the Power Supply I will need? I want to be able to expand the setup later on also... I gave my friend  as a brainteaser; while her attempted solution didn't work, it raised an interesting question.  I flip a fair coin repeatedly and record the results.  I stop as soon as the number of heads is equal to twice the number of tails (for example, I will stop after seeing HHT or THTHHH or TTTHHHHHH).  What's the probability that I never stop?  I've tried to just compute the answer directly, but the terms got ugly pretty quickly.  I'm hoping for a hint towards a slick solution, but I will keep trying to brute force an answer in the meantime. Perfect set without rationals How do I turn on workspaces? (Why do I only have one workspace?) It appears that Unity on Raring only comes with a single virtual desktop (aka workspaces) by default, and CCSM's Wall plugin has no way to add more. Is this configurable at all? Create a table with LaTeX Is ¬¬(¬¬P → P) provable in intuitionistic logic? Should the first word after a colon be capitalized? Which of the following is correct?   For example: This. For example: this. The interaction of TeX and Lua can be quite delicate at times.  While the former has category codes and tokens, the latter has escape sequences and datatypes.  Mixing the disparate concepts of these two worlds can lead to unexpected behaviour which favours common errors, known as gotchas. In the community wiki answer below you can find an overview of the common mistakes that people make when getting to grips with LuaTeX. Can I start a new Clash of Clans village on my iPhone separate from my account on my iPad? If I download Clash of Clans on my iPhone, can I start a new village despite my account on my iPad? I want to make a village for my younger brother. Deleting a middle node from a single linked list when pointer to the previous node is not available Is it possible to delete a middle node in the single linked list when the only information available we have is the pointer to the node to be deleted and not the pointer to the previous node?After deletion the previous node should point to the node next to deleted node. What are the blue dots next to App names in iOS and macOS? They disappear once you open the apps and then return to the home screen, but I can't tell what they indicate.  Are they apps that haven't been opened since an app update? Or that it thinks have never been opened (not accurate in my case, but that could be a function of a "restore" confusing the phone). How do you calculate the likelihood of drawing certain cards in your opening hand? In Magic, at the start of the game, you draw 7 cards. How would you calculate the likelihood of drawing a specific card in your opening hand?  For example, let's say I have a 60 card deck, and I'm running 4 . What is the percent chance that I will have at least one Bird in my opening hand? Is there an easy way to show which spheres can be Lie groups? SQL Server nvarchar(max) vs nvarchar(n) affects performance Disable cache for some images I have county data in which the &quot;FIP&quot;s (actually State &quot;FIP&quot;s and County &quot;FIP&quot;s combined). I have data imported as CSV with the sames &quot;FIP&quot;s data titled 'STCounty'. However, the shapefile &quot;FIP&quot;s is listed as a Real and the CSV came in as an Integer and the Join function does not seem to work.  The drop boxes are greyed out/empty. So the bigger questions is how do you change the attribute type in a table? I tried to create a new column in the shapefile (county data) and set it to an integer and then multiply the &quot;FIP&quot;s*1, but got a column full of NULL. I am new to QGIS and have a lot of experience in MapInfo, so the join concept is not new to me, just the syntax in QGIS. How can I efficiently drive an LED? I understand that I can not connect an LED directly to a battery because it will draw too much current. Thus, there must be something else in the circuit to limit the current. What options are there? Are some methods more efficient than others? I have a Samsung 960 Pro SSD with Ubuntu 18.04  is it safe to put the swap file on there or would it hurt it in the long run,  I heard some people saying that new SSDs don't suffer from that issue anymore, is this true?