I am having a hard time wrapping the exponential map around my head. So if $G$ is a Lie group we have this map : $exp : T_eG \rightarrow G $ My understanding of lie groups is that they represent stuff like rotations. So if I take $G=O(n)$ the rotations, then I know that $T_{I_n}O(n) = \text{skew symmetric matrices}$. So I take one skew symmetric matrix $S$ which as I understand it somehow represents an infinitesimal rotation. Then $exp(tS)$ is a curve for which the elements are rotations of $O(n)$. But what do these element mean? How to think intuitively about what this exponential map actually is?  Edit :  Sorry, but I don't see how the linked answers tell me how to think about the curve $\gamma(t)=exp(tX)$, and neither did OP since apparently they were not accepted...
I'm trying to understand an algorithm that tries to map points from a lie group to its lie algebra using the exponential map. The background is the representation of 3d coordinate transformations as a lie group.  I can imagine, how one can map points on a manifold (eg a sphere) to the tangent space (tangent plane) at some point x on the sphere. I can imagine to create the tangent space by computing derivatives at that point x. But I cannot imagine how the exponential of some matrix is giving me that tangent space?  How can I envision the exponential of a matrix? Is there some picture or analogy I can use as intuition?
"Oriented" vs. "orientated" What are the origins of the word orientated?  As far as I know, the correct spelling is oriented and orientated is not an alternative spelling but an error that is in common use.  Is it for example more commonly used in a certain country or by a certain people? Is there a reason for people choosing to say or write orientated instead of oriented? Is explicitly closing files important? Is there a vendor for animal parts? I'm looking to upgrade all my pouches, but some animals are hard to find, or the part needed drops rarely.  Is there a merchant somewhere that sells them outright? I have shards to spare. What is dependency injection? Allow flags on locked posts What Do the Gringotts Goblins Do With Muggle Money? ‘But you’re Muggles!’ said Mr Weasley delightedly. ‘We must have a drink! What’s that you’ve got there? Oh, you’re changing Muggle money. Molly, look!’ He pointed excitedly at the ten-pound notes in Mr Granger’s hand.   Chamber of Secrets - page 47 - UK Hardcover - chapter 4, Flourish and Blotts   So Gringotts exchanges Muggle money for Wizarding money. What do the Goblins do with the Muggle money after the exchange has been made? Is it possible they keep a stash of Muggle money on hand for reverse exchange purposes, or do they surreptitiously feed the money back into the Muggle economy?  What do Goblins do with Muggle money? "Outside" or "outside of"? I have a web application that's branded according to the user that's currently logged in. I'd like to change the favicon of the page to be the logo of the private label, but I'm unable to find any code or any examples of how to do this. Has anybody successfully done this before?  I'm picturing having a dozen icons in a folder, and the reference to which favicon.ico file to use is just generated dynamically along with the HTML page. Thoughts? What is the best way to remove and prevent leaf stains on concrete pavers? How can I dynamically unload a javascript file? How does Dijkstra's Algorithm and A-Star compare? How to only perform LTS upgrades? Nautilus doesn't launch on Ubuntu How can I install updates and shut down from the command line? I know that there is a simple way to shutdown the computer in command prompt, but is it possible to do system updates and then shutdown the computer in command prompt?   For example, when Vista needs to install updates, it will let me install them and shut down by clicking the Shut Down button (while relegating the normal shut down button into the menu). So is there a command to do this in command prompt? 3rd iterate of a continuous function equals identity function If $ f: \mathbb{R} \to \mathbb{R} $ is continuous, and $\forall x \in \mathbb{R} :\;(f \circ f \circ f)(x) = x  $, show that $ f(x) = x $.  The condition that $f$ is continuous on $\mathbb{R}$ is crucial for the proof. I can find functions such as $\displaystyle\frac{x-3}{x+1}$ that satisfies  $ (f \circ f \circ f)(x) = x $. I have tried to negate the conclusion to see if there's a contradiction, but got stuck. Does every linearly independent set of n vectors in $R^n$ forms a basis in $R^n$? Basically does a vector set that is linearly independent in $R^n$ automatically span $R^n$?  My initial thought is yes, but is there some counterexample that can disprove this? I've just executed a long-running process from the bash prompt. In hindsight, I wish I'd run time on it, or noted down the time at which I kicked it off.  Is there any way of getting this information retrospectively? The .bash_history doesn't seem to include timestamps.  In my particular case it's Mac OS X, but I'm interested in general Unix/Linux solutions.  To clarify, the process has now completed, and I'd prefer not to run it again unless absolutely necessary! proving of inequality $(n!)^2 \leq n^n\cdot n!\leq (2n!)$, where $n\in \mathbb{N}$ How can we prove the inequality $(n!)^2 \leq n^n\cdot n!\leq (2n!)$, where $n\in \mathbb{N}$  $\bf{My \; Try}$::  $1\leq n$  $2\leq n$  $3\leq n$  ....  ....  $n\leq n$  So $1\cdot 2 \cdot 3 ..n \leq n^n$  So $n!\leq n^n\Rightarrow (n!)^2 \leq n^n\cdot n!$  But  did not understand how can we prove $n^n\cdot n! \leq (2n!)$  Help me  Thanks "All our X" vs. "all of our X" Is the word "of" optional in this instance? Is either of these considered preferable to the other?        Taste all our delicious treats.   Taste all of our delicious treats. Similarity of real matrices over $\mathbb{C}$ $A  \underset{\mathbb{C}}{\sim} B \overset{\text{def}}{\iff} A=C^{-1}BC, \space C\in M_{n}(\mathbb{C})$ and similarly for $\underset{\mathbb{R}}{\sim}$.     I want to prove that $ A \underset{\mathbb{C}}{\sim} B$ for $A,B \in M_{n}(\mathbb{R})$ therefore $A \underset{\mathbb{R}}{\sim} B$.   My idea is that elementary divisors of $A,B$ over $\mathbb{C}$ are the same, and if $(x-z)^k$ is elementary divisor than $(x-\overline{z})^k$ is also elementary divisor $\implies$ $A,B$ have same elementary divisors over $\mathbb{R}$. But i think it's not clear.