In Loring Tu's Introduction to Manifolds, the author states:      "The tangent space of $S_n$ [the space of symmetric $n\times n$   matrices] at any point is canonically isomorphic to $S_n$ itself,   because $S_n$ is a vector space."   There are several questions on Math stackexchange, but in all of those that I've seen, the answers are either incomplete, leaving aside details that shouldn't - specially for someone who is learning this by themselves - or they are too terse and confusing. Also, they usually assume that we can differentiate the determinant. When Tu writes the sentence above, he still hasn't talked about what's the differential of a determinant.  So, I'm looking for a complete proof, with an effort from the author to be as clear as possible, and without using the differential of the determinant. This last condition is not binding.   Thanks.
I'm self-studying Guillemin and Pollack, but I'm stuck on Problem 3 of section 2. It says that if $V$ is a vector subspace of $\mathbb{R}^N$, then $T_x(V)=V$ if $x\in V$.  If $x\in V$, then since $V$ is a manifold, there is a local parametrization $\phi: U\to V$ where $U$ is open in $\mathbb{R}^k$. Without loss of generality, we can require $\phi(0)=x$. Then $T_x(V)$ is defined to be the image of $d\phi_0$ on $\mathbb{R}^k$.   An arbitrary element of $T_x(V)$ looks like $$ d\phi_0(v)=\lim_{t\to 0}\frac{\phi(0+tv)-\phi(0)}{t}=\lim_{t\to 0}\frac{\phi(tv)-x}{t} $$  but this doesn't seem very useful to show $T_x(V)=V$. What is the right approach?
How can I recover deleted photos from an SD Card? My technophobe mother returned from a European trip to discover she'd erased all but 15 images from a nearly full 2GB card. As I am the computer guy in the family, she posted the card to me and asked that I make something magical happen.  The photos were taken with and deleted by a Nikon Coolpix. (I don't know the model.)  I am a Linux guy, but there is a Windows Vista box at my disposal.  What is my best strategy to try to recover something? Does it make sense to Favourite a Question you created? Is this just a redundant act, or are there differences? Review counts in top-bar and /review don't match So far, I thought that the number shown in the orange box in the top-bar was the number of items available for review. However, when I acually visit /review, the numbers don't match:    Note that top-bar shows 15 items (?) but the queues only add up to 8.  Since I can't think of a reason for this being by design (which, as experience shows, is not a good indicator for "this is not an SE feature) I'm tentatively labelling this a (UI) bug. How to backup 20+TB of data? We have a NAS server at the company I work for that is being used for storing photography sessions. Each session is approximately 100gb. Over the last couple of years this server has accumulated 10+ TB of data, and we are increasing the amount of photoshoots exponentially. I estimate that by the end of next year we will have 20+ TB stored on this NAS. We are currently backing this server up to tape using LTO-5 tapes with Symantec BackupExec. Since the size of this server has grown, full backups of this server are not completing overnight. Does anyone have any suggestion on how to backup this amount of data? Should we be backing it up to tape? Are there any other options which may be better? Using Inclusion/Exclusion to solve $x_1+x_2+x_3=15$ with $x_1,x_2\leq 5$ and $x_3\leq 7$ for non negative integers $x_1,x_2,x_3$ I want to solve for:     Number of integers solutions to the equation $x_1+x_2+x_3=15$ with $x_1,x_2\leq 5$ and $x_3\leq 7$ for non negative integers $x_1,x_2,x_3$   How can this be done using the principle of inclusion/exclusion? Efficiently select beginning and end of multiple contiguous ranges in Postgresql query I've got about a billion rows of data in a table with a name and an integer in the range 1-288.  For a given name, every int is unique, and not every possible integer in the range is present--so there are gaps.  This query generates an example case:  --what I have: SELECT * FROM ( VALUES ('foo', 2),               ('foo', 3),               ('foo', 4),               ('foo', 10),               ('foo', 11),               ('foo', 13),               ('bar', 1),               ('bar', 2),               ('bar', 3)      ) AS baz ("name", "int")   I'd like to generate a lookup table with a row for each name and sequence of contiguous integers.  Each such row would contain:  name   --  the value of the name column start  --  the first integer in the contiguous sequence end    --  the final value in the contiguous sequence span   --  end - start + 1  This query generates example output for the above example:  --what I need: SELECT *  FROM ( VALUES ('foo', 2, 4, 3),               ('foo', 10, 11, 2),               ('foo', 13, 13, 1),               ('bar', 1, 3, 3)      ) AS contiguous_ranges ("name", "start", "end", span)   Because I have so many rows, more efficient is better.  That said, I only have to run this query once, so it isn't an absolute requirement.  Thanks in advance!  Edit:  I should add that PL/pgSQL solutions are welcome (please explain any Fancy Tricks--I'm still new to PL/pgSQL). I have a third-party library which consists mainly of a large number of static (.a) library files.  I can compile this into a single .a library file, but I really need it to be a single .so shared library file.  Is there any way to convert a static .a file into a shared .so file?  Or more generally is there a good way to combine a huge number of static .a files with a few .o object files into a single .so file? Interview: How to ensure that a thread runs after another? Exercise on local and global symmetry for a given Lagrangian density SED: insert text after the last line? This  inserts a tag to the beginning of a file:  sed -i "1s/^/&lt;?php /" file   How can I insert something to the end of each file with sed? Cropping Shapefiles in QGIS? I need to clip a large and very detailed shapefile in QGIS showing environmental data similar to a temperature map.   Is there something like the very simple crop tool like in Adobe Photoshop where you can drag a rectangle over an area of interest and crop the shapefile to this area only?  I do have an additional shapefile of neighborhoods/districts that I could also use for cropping but I do not know how to specify "extract a shp/kml only within the borders of district X, row xxxxx in the attribute table of districts.shp"  How can I extract the data primitively within a rectangle of choice or per district?  I need smaller pieces from the large map that has been provided by an environmental geo-institute to be implemented into an mobile Application (Android) to view the quite heavy data in a lighter way.   I made some screenshots of the data and posted them here: .  You can see the districts and the data layers. I need to have every district as a separate file / separately export every dataset per district to KML. Keep items in inventory on death in Minecraft I once played a vanilla Minecraft adventure map (called "Herobrine's Mansion") that prevented the player from losing their items on death. Which commands can be used to prevent the player from losing their items on death (in vanilla Minecraft)? What are weapon damage dice? Following discussion , what constitutes a weapon damage die?  Is it any damage die dealt by the weapon? For example, consider a  longsword.     While the sword is ablaze, it deals an extra 2d6 fire damage to any target it hits.   What are the weapon damage dice? Just the 1d8 from the longsword? Or do the 2d6 Fire damage also count? Since arrays and hashes can only contain scalars in Perl, why do you have to use the $ to tell the interpreter that the value is a scalar when accessing array or hash elements? In other words, assuming you have an array @myarray and a hash %myhash, why do you need to do:  $x = $myarray[1]; $y = $myhash{'foo'};   instead of just doing :  $x = myarray[1]; $y = myhash{'foo'};   Why are the above ambiguous?   Wouldn't it be illegal Perl code if it was anything but a $ in that place? For example, aren't all of the following illegal in Perl?  @var[0]; @var{'key'}; %var[0]; %var{'key'}; How can I have a different input method for each app window on windows 8. When I change input language to ENG all app window also uses this method and I don't want that. Like windows 7 that has a different input method for this purpose. thanks PHPUnit skips all tests How to get battery status? How do I get the battery status on an iPhone? Multi-window: prevent the out-of-focus app from pausing? How can I get admin access to a Mac without knowing the current password? My Father died last year and I inherited his Mac Mini. My wife has retired and we are trying to set up the mini for her use but the changes we want to make require admin access and we do not have the information my father used to set up the machine.  Is it possible to reset the admin information without having the old information? If so, how would I go about it?   I am not a Mac user usually so any guidance will have to be pretty simple! Cannot change lock screen background on Ubuntu 18.04