Maybe an example will help.  $F(x,y)= x^2 + y^2$.  Now the gradient would look like $&lt;2x , 2y &gt; or  2xi + 2yj$ where $i$ and $j$ are unit vectors of the $x$ and $y$ axis.  This vector field spans every quadrant.  The gradient is sort of a place holder to build a vector field.    Certainly it does not point in one direction so in what sense is it pointing in the maximum direction? The maximum direction of what?    If you don't mind a little extra question on the side, but relevant, since it was the source of my original confusion.  Is directional derivative is a dot product with the gradient and a unit vector pointing in any direction you choose but if the angle between the gradient and the unit vector is 0 , the cosine is 1 so that means that unit vector was pointing in the direction of the gradient ?   Since the directional derivative is a number only  of what utility is it other than this observation ? I may be missing the boat on directional derivative.
I am trying to really understand why the gradient of a function gives the direction of steepest ascent intuitively.  Assuming that the function is differentiable at the point in question, a) I had a look at a few resources online and also looked at this   , a popular question on this stackexchange site. The accepted answer basically says that we multiply the gradient with an arbitrary vector and then say that the product is maximum when the vector points in the same direction as the gradient? This to me really does not answer the question, but it has 31 upvotes so can someone please point out what I am obviously missing?  b) Does the gradient of a function tell us a way to reach the maxima or minima? if yes, then how and which one - maxima or minima or both? Edit: I read the gradient descent algorithm and that answers this part of my question.  c) Since gradient is a feature of the function at some particular point - am I right in assuming that it can only point to the local maxima or minima?
What would happen to a star if a Dyson sphere lined with mirrors reflected a significant portion of the stars light back to the star I have looked for similar questions here on stack exchange. The closest example to this that I found is . That question assumed less than perfect absorption of material lining the inside of the sphere, and the effect of thermally re-radiating it inside the sphere.  A lot of the answers focused on that much of the thermal radiation would not be reflected back to the star, and is likely to be reabsorbed by the sphere at another location. On the other hand, this question asks what would happen if the Dyson sphere, using mirrors, deliberately refocus a significant portion of the rays directly back at the star. Using old version of numpy even with a newer version installed Convergence to function that is not measurable Suppose ($X,\cal M, \mu$) is not complete. Show that there is a sequence {$f_n$} of measurable functions on $X$ that converges pointwise a.e. on $X$ to a function $f$ that is not measurable. Was Tracy Mills a part of John Doe's plan from the beginning? In , John Doe is shown as being extremely meticulous and patient while carrying out his plan of murdering and/or torturing people in accordance with the seven deadly sins.  At the end of the film, it is revealed that John Doe had killed and decapitated Tracy Mills, the wife of Detective David Mills.  Was Tracy Mills his intended target from the beginning?  Did John Doe decide to kill Tracy only after Detective Mills and Somerset discovered his residence? I would like to install Ubuntu  for almost 120 users. But it's very difficult in using USB and drives.  Any suggestions? I have the following code that will remove lines with the pattern banana and 2 lines after it:  sed '/banana/I,+2 d' file   So far, so good! But I need it to remove 2 lines before banana, but I can't get it with a “minus sign” or whatever (similar to what grep -v -B2 banana file should do but doesn't):  teresaejunior@localhost ~ &gt; LC_ALL=C sed '-2,/banana/I d' file sed: invalid option -- '2' teresaejunior@localhost ~ &gt; LC_ALL=C sed '/banana/I,-2 d' file sed: -e expression #1, char 16: unexpected `,' teresaejunior@localhost ~ &gt; LC_ALL=C sed '/banana/I,2- d' file sed: -e expression #1, char 17: unknown command: `-' If $f$ is a continuous function with $\displaystyle \int_{0}^{x}f(t)dt\rightarrow \infty$ as $|x|\rightarrow \infty\;,$ Then show that every line $y=mx$ Intersect the curve $\displaystyle y^2+\int_{0}^{x}f(t)dt=2$  $\bf{My\; Try::}$ Put $y=mx$ in $$\displaystyle y^2+\int_{0}^{x}f(t)dt=2\;,$$ We get $$\displaystyle m^2x^2+y^2+\int_{0}^{x}f(t)dt=2.$$ We can write it as $$\displaystyle 2\int_{0}^{x}2m^2tdt+\int_{0}^{x}f(t)dt=2$$ So $$\displaystyle \int_{0}^{x}\left[2m^2t+f(t)\right]dt-2 =0$$ Now Let $$g(x) = \displaystyle \int_{0}^{x}\left[2m^2t+f(t)\right]dt-2.\;,$$ When $x\rightarrow \infty\;,$ Then $g(x)\rightarrow \infty.$ (Where $g(x)$ is an Continuous function.) Now I did not Understand How can i solve that, Help me Thanks I downloaded the July 30th update for Pokemon Go and when I open the app I'm prompted to login, But once I login, it redirects me to this page on Cookies::     If I click "Done" in the top left, it prompts me to login again! I've tried reinstalling the game, restarting my phone, etc. I can't much enable Cookies on the inline Safari browser. It doesn't make sense, and cookies are on in Safari (because I don't use Safari and they're enabled by default).  How can I fix this? Can the weak force create a bound state? My understanding is that the exchange of Z bosons could yield an attractive or repulsive "force" between two fermions. For most combinations the electromagnetic or strong interactions will take place.  However, if we have two neutrinos, or a neutrino and a charged lepton, only the weak interaction comes into play. Are these possible bound states? Some time ago, we announced that . This has gone well and we're extremely thankful to everyone for their support and feedback. Because it has been successful so far, we're going to be extending the program to the majority of our graduated sites. Well, we've pretty much already done that, but there are quite a few sites that have ads enabled, yet still display nothing. The thing is, we don't really have an abundance of clients offering products that a  would find relevant, but Amazon has stand mixers, a selection of super fine chinois, deliciously sharp cutlery and enough cast iron to fill the Gulf Of Mexico with . Beginning as early as this week (week of 5 February 2018), our ad team is going to be hand selecting affiliate ads to be displayed on select sites. The use of inventory will of course depend on the availability of relevant ads to display which means:  Some sites might see very few, if any ads. Some topics simply don't come remotely close to orbiting anything worth advertising. Furthermore, while graduated, some sites simply don't have enough activity for ads to make sense.  Since these are hand-selected, it could take a while for ads to appear initially.  These would of course yield to inventory from clients if clients had products that were relevant to any given topic.   Nothing changes when it comes to the . If you're used to occasionally seeing a community promotion ad, you'll just notice that you'll also occasionally see an ad for something that is likely to be of interest to someone interested in the topic that your site covers. In fact, to be clear, this is simply a continuation of the announcement we made in November of 2016. We just wanted folks to know that we're going to be continuing the program where there's good potential to show relevant advertisements, even if we don't have clients lined up with a demand for it. Where can I give feedback about a certain ad on [site]? Open a meta post on your per-site meta and let us know. Please remember, the bulk of advertisements are an effort to get something small back from folks that hit the sites through search engines and are extremely unlikely to ever become contributors. Their soufflé stopped falling, and we'll probably never see them again. Maybe we can get a few bucks once they realized that copper ions help egg whites stiffen properly and buy a bowl or a proper bloom whisk. However, you're the experts on these topics and we're very likely going to listen to you. What sites, specifically? Are any sites exempt? If you're a former Stack Exchange 1.0 site and / or subsequently joined the network under a special charter, nothing changes - don't panic :) For the rest, we're just extending what we're currently doing with ads on graduated sites to also include those where there's an opportunity to make sensible use of affiliate ads. If a site has less than 15k (average) daily visits, or fewer than roughly 50k combined total posts - it might not yet be time to invest in selecting ads to be shown there. But, part of this exercise is also seeing what works best. I don't really have feedback, just some suggestions Ads aren't just an opportunity to sell stuff, they're an opportunity to really show people that we understand a given topic. That, of course, makes suggestions from people that do that stuff every day quite valuable. Post on your per-site meta and our community / ad critters will take a look. Questions? Comments? Kvetches? Let us know in an answer or comment below. If anything about the plan changes, we'll update this post to let folks know. Labels and nesting gather and align I have nested gather and align according to  \begin{gather}        \sum\limits_{i=1}^n i + \text{expression} = \dfrac{n(n-1)}{2} \label{l1}\\        \begin{align}             a+\text{expression}&amp;=b \\ %\label{l2} doesnt work             c&amp;=d+\text{expression} \\             e+\text{long expression}&amp;=f-\text{expression}        \end{align}  \end{gather}   to get an output looking like this:   The problem is that it is not possible to label all equations separately. The problem is identical to that discussed in , except  that in my problem the first line needs to be centered.  Thanks for any help. I’m looking for a word I’m struggling to remember that refers to a job or role which is inconsequential and usually created in order to install someone in a company, not based on their merit.  An example would be creating a vague new role for a friend’s son. The word I’m looking for refers to the actual job itself - it’s a noun that would replace the ‘job’ (e.g “they put so and so’s cousin in a ____”).   I believe the word may not be a strictly formal or technical word but I don’t think it was slang. Avoid linebreaks in LaTeX console / log output (or increase columns in terminal) Is it possible to tell the LaTeX command not to use line breaks in its console / log output?  This would make it easier to automate the processing of the log file, e.g. extracting the names of missing graphics files like in this output  ! LaTeX Error: File `figures/triggers/muontriggers/notmu18_medium_mu10_loose/2d map_abs_eta_phi.pdf' not found.   which are otherwise cut into pieces and unnecessarily difficult to grep etc. Allow a user to read some other users' home directories Similar to  on the bugs in Minecraft, I would like a complete list of what error codes exist in Minecraft and how they occur, as well as how they can be resolved. I have a NOPASSWD line in /etc/sudoers (edited with visudo)  gatoatigrado    ALL=(ALL) NOPASSWD: /bin/set-slow-cpufreq   However, the output is,  gatoatigrado@coral:~&gt; sudo -n /bin/set-slow-cpufreq sudo: sorry, a password is required to run sudo   This kind of command works on an OpenSuSE machine, but not on Ubuntu 11.10. What am I doing wrong?  Note: I cannot find any relevant system log messages, e.g. via tail -f /var/log/syslog.  edit  Here is /etc/sudoers.  Defaults    env_reset  # things I've tried copying from an opensuse machine Defaults always_set_home Defaults env_keep = "LANG LC_ADDRESS LC_CTYPE LC_COLLATE LC_IDENTIFICATION LC_MEASUREMENT LC_MESSAGES LC_MONETARY LC_NAME LC_NUMERIC LC_PAPER LC_TELEPHONE LC_TIME LC_ALL LANGUAGE LINGUAS XDG_SESSION_COOKIE"  root    ALL=(ALL:ALL) ALL gatoatigrado ALL=NOPASSWD: /bin/set-slow-cpufreq %admin ALL=(ALL) ALL %sudo   ALL=(ALL:ALL) ALL monitor files (à la tail -f) in an entire directory (even new ones) I normally  watch many logs in a directory doing tail -f directory/*. The problem is that a new log is created after that, it will not show in the screen (because * was expanded already).  Is there a way to monitor every file in a directory, even those that are created after the process has started? I have answered 'No' to Siri initial prompt, but I can still see it in my process list. I've tried to kill it, but it loads again immediately. How can I unload and remove Siri from my Mac? I'm using LyX on Linux Mint 14 ...  Using BibTeX with the apalike style and some modifications in the preamble, I've managed to get the bibliography as it should be except for one thing. When I cite a work with multiple authors it gets printed with the separator "and" which isn't correct for my language (Danish). Curiously it's displayed correctly in the user interface of Lyx - for example "(Wittgenstein og Waismann 2003 - which led me to believe everything was ok but in the pdf the separator "and" is used both in the reference and in the bibliography.  Is there any way to fix this using BibTeX, or can it only be done with biber 1.5, as suggested in the last comment here:   I really hope their is because getting biber to work using the following guide has thus far proven to be impossible:   Help much appreciated ... I really hope I won't have to relapse into setting my master thesis up in LibreOffice. What should we do about Stack Overflow-specific questions on Meta SE?