This is apparently a classical differential problem yet I struggle to find the demonstration of the next affirmation: The restrictions to $S^2(1)$ of the orthogonal lineal applications of $R^3$ are the isometries of the unit sphere.
Got this as homework and I don't know how to tackle this. Help please!  Prove that the isometries of $\mathbb{S}^{n} \subset \mathbb{R}^{n+1}$, with the induced metric, are restrictions to $\mathbb{S}^{n}$ of the linear orthogonal transformations.
I was wondering why I couldn't delete my own question. Well actually, I have accepted the question itself to be wrong and even I feel that I have accepted a wrong answer, but when I try to delete my question, I get a message saying that,"This question can't be deleted, flag it for moderator attention". Why is this happening and how can I solve this? We are having issues with our exchange 2013 server. The server is running Windows Server 2008 R2 Standard.  The issue we're having is that our Exchange server isn't receiving internet emails. It can currently send mail out to internet addresses, and send and receive internal emails from the current email server. I have checked the settings of the receive connectors and everything seems correct. We have tried everything that we can think of.   We can send and receive email locally, and send out to a gmail account. However, we don't get the reply email from the gmail account to the inbox.    Thanks in advance!! Image is of the DNS settings on the exchange server. According to WolframAlpha,  but I don't know how I can prove it. What is the correct way to get a pgfkey value? Thunderbolt 2 docks have been quite expensive and they will be obsolete once I purchase the new Macbooks with USB-C ports. I currently use a 2014 MacBook Pro with Thunderbolt 2 ports, and I want to futureproof my purchase by acquiring a cheaper USB-C dock now.  The cheapest Thunderbolt 2 dock costs around $200 while USB-C docks cost around $30. Being able to use a USB-C dock with older Thunderbolt 2 Macbook Pros would not only be cheaper but also allows me to use the same USB-C dock if I decided to upgrade to a future Macbook.  For that to work I need a Thunderbolt 2 (male) to USB-C female adaptor, after some searching it doesn't seem to exist. Is there a reason why something like this hasn't been made? Or is the USB-C specification designed such that it is impossible? (hence businesses can still continue to sell expensive Thunderbolt 2 docks...)  I will accept an answer that meets the following requirements:   If there is such an adapter please give a link and some form of evidence that these would have 100% compatibility with USB-C docks. If there is no such adapter please explain:   Is it impossible to produce? If not, why hasn't a vendor created one yet? I've explained that its a better financial investment than buying a TB2 dock... surely there's a market for those still using TB2 MBPs wanting to upgrade in the next couple of years. Not only docks but it would open a gateway to cheaper alternatives to those hideously expensive TB2 peripherals. If I am travelling on a car at around 60 km/h, and I shine a light, does that mean that the light is travelling faster than the speed of light? The title says it all.  If I was on a bus at 60 km/h, and I started walking on the bus at a steady pace of 5 km/h, then I'd technically be moving at 65 km/h, right?  So my son posed me an interesting question today: since light travels as fast as anything can go, what if I shined light when moving in a car?  How should I answer his question? How do i prove that $(p \vee q) \wedge (\neg p \vee r) \rightarrow (q \vee r)$ is a tautology without using the truth table? Printing labels along with equation numbers Working with documents with huge amount of equations it becomes cumbersome to go back and remind yourself what you labeled a certain equation to (I try to name them in a smart way, but there are too many).  Is there a convenient way to have equations print the label along with the tag/number, so that I can easily check the label in the compiled document?  Something like this would be perfect:  x = y               (14.1) eq:my_label Viewing all my questions from Android app How do you express that your understanding of something is completely changed? When something blows your mind and changes your understanding of something that you've believed in for years, how do you express it (formally or informally)? Search field with or without a button? I can see two broad patterns as far as the design of the search field is concerned. One like we see on Google has a button that the user can hit to initiate Search. (They could also hit enter to do so)  The other pattern much like what is on this site (stackexchange) do not have a button. So the only way the user can initiate search is by hitting enter.   Has anyone done any research or analysis of this pattern ? What I have noticed is that search fields without the explicit search button (that normally is a magnifying glass) have "Search" as placeholder text, left aligned, whereas when there is no button the placeholder is usually a "Magnifying glass icon" with "Search" all left aligned. How should I detect unnecessary #include files in a large C++ project? Proving that these two sets are denumerable. (a) $S_k=\{A\subset\mathbb{N}: |A|=k\}$ for $k\in\mathbb{N}$  (b) $S = \bigcup_{k=1}^\infty S_k$   Work: For (a), I am not too sure about what approach I should use. I think finding a bijective function between the $S_k$ and $\mathbb{N}$ is hard to do . I also thought about using the Schroeder-Bernstein theorem and find injective functions $f:\mathbb{N}\rightarrow S_k$and $g:S_k\rightarrow\mathbb{N}$ in order to prove that $S_k$ and $\mathbb{N}$ are numerically equivalent. However, I have a hard time finding and proving these injective functions. I haven't really looked at (b) as I have yet to complete (a). Is there a way to display the number of messages for any label in the Gmail label list? The Gmail sidebar displays the list of our labels (or some of them, according to settings). Next to every label name there is a number showing how many unread messages we have for that label.  Is there a way to display also the total number of messages for each label?  Example:  [to answer] (0/36) [to do] (0/55) [waiting answer] (0/7) cooking (2/48) family (6/352) friends (1/1828) website (2/412) work (12/12801) What would happen if you used a dead person's hair in Polyjuice Potion? Polyjuice Potion is changing based on someone's current physical state. It shows tattoos (or lack of) and such from DH when Ron changed to Harry.     "I knew Ginny was lying about that tattoo,” said Ron, looking down at his bare chest.    What would happen if someone used hair of a dead person, taken from their body after death, and drank the potion? Would they die as well? Explanation behind Second Derivative of a Parametric Equation Formula I am looking to find out how the second derivative formula works.  I can blindly apply it but I don't have a grasp of what is going on or why. The first derivitive has the formula $\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}$. I told this was because if you divide them you cancel out the dts and get dy/dx. The second derivative has the formula $\frac{d^2y}{dx^2} = \frac{d}{dx} \left(\frac{dy}{dx}\right) = \frac{ \frac{d}{dt}\left(\frac{dy}{dx}\right) }{\frac{dx}{dt}}$. Is this something that can be explained with only a knowledge of calculus or is this something I would have to take on faith until I learn more. Any help would be greatly appreciated. I am looking for a pipe symbol that can be made to vary in height automatically.  One of the common ways of writing "such that" in mathematics is with the pipe symbol (|). The pipe can be represented in LaTeX directly as the character | or as \vert, but neither of these expands vertically in math presentation mode. I can adjust it manually this way:  \[\mathcal{H}=\left\lbrace h(x)=\sum_{i=0}^{k-1}a_{i}x_{i}\mod m\ \Bigg\vert      \ a_{i}\in\mathbb{Z}_{m^{k}}\right\rbrace\]     but I am looking for a true variable-sized option, parallel to \lbrace etc.  The amsmath package provides \lvert and \rvert, but I do not find them expanding as expected, and the amsmath documentation specifies that they should be used as binary delimiters (one on each side of something else), rather than alone. Difference of the \dots* I came to wonder, what is the difference between the following?  \dotsc \dotso   As well as between:  \dotsb \dotsi \dotsm What's the term for denying something so as to raise doubt? Topical example:  'There was a scuffle outside the local pub. There appears to be no connection with religious extremists.'  ...but now you're wondering if there was. It's a nasty tactic used to raise fear and doubt - deliberately calling attention to something by explicitly denying it.  What's the term for this? Messages on Macs don't work I've tried every suggestion I could find on Apple forums. Deleted .plist files, removed message caches, reset passwords, coordinated sign on/offs between my iPhone and my Macs, reset iPhone and reinstalled Mavericks on my Macbook Air. All pointless exercises.   I suspect it all started because I jailbroke my iPhone using Pangu. It may or may not have been the culprit, but my problems started a couple of days after I did it. All my phone conversations went green and stopped synchronizing with Messages apps on my Macs. Trying to send a new message from Macs would result in either "Your message could not be sent" error with a warning icon next to it, or:     XXX is not registered with iMessage.   So I restored my iPhone to factory defaults using iTunes. That didn't help immediately, but in a day or two, my conversations with other iPhone users went blue. So I am happy that works.  So now I sacrificed my Macbook Air to try and make messages work on it. I reinstalled Mavericks several times, each time trying different approaches: not immediately signing into iCloud, not associating an iCloud profile with login profile, etc. Nothing helps. I launch Messages app, it asks me to sign in (and I do, successfully), and then nothing goes through.   Sending messages to myself also works inconsistently. Yesterday, whatever I sent to myself (using phone number) would appear delivered to the same window, but not on my phone. Today I get "... not registered with iMessage" error.