I found the following statement in  youtube lecture:     Let $A\subset \mathbb{R}^n$ be open and convex and let   $f:A\to\mathbb{R}^n$ be continuously differentiable vector field.      Then $f$ is conservative iff the Jacobian of $f$ is symmetric on $A$.   I couldn't find a citable reference for this (for $n\ne3$).  Does someone have one?  Update: I'm still looking for a citable reference.
I want to determine whether a certain n-dimensional, simply-connected vector field is conservative. I can very easily calculate its Jacobian in numerical form.  In 3 dimensions, the problem is easy: the Jacobian is a 3x3 matrix, and the curl can be obtained from its elements:  [jacobian[2,1] - jacobian[1,2],  jacobian[0,2] - jacobian[2,0],  jacobian[1,0] - jacobian[0,1]]  A curl of zero would mean that the vector field is conservative.  But what if my Jacobian has 100 elements? The curl is only defined for 3 dimensions, but is there a formula for some high-dimensional generalization?
Are there other schools for witches and wizards in the rest of the world? In the books they only speak of 3 schools, but what about Asian or American people?  And even more, were they affected by the events happening in Europe? $\operatorname{Ext}$ and injectives, respectively projectives Calculating square roots using the recurrence $x_{n+1} = \frac12 \left(x_n + \frac2{x_n}\right)$ Let $x_1 = 2$, and define $$x_{n+1} = \frac{1}{2} \left(x_n + \frac{2}{x_n}\right).$$ Show that $x_n^2$ is always greater than or equal to $2$, and then use this to prove that $x_n − x_{n+1} ≥ 0$. Conclude that $\lim x_n = \sqrt2$.   My question:  So I know how to do this problem but I don't know how to prove that $x_n &gt;0$ for all $n\in \mathbb{N}$. What documents to show if I'm a US citizen with no US passport to re-enter the US? Suppose I am a naturalized citizen of the US, but also a citizen of another country, "XYZ". I then boarded a plane and left the US to another country (maybe not necessarily to XYZ), without first getting a US passport, but using my old, but valid XYZ passport.   In this case what would be the best documents to have for me to gain entry back into the US as a US Citizen without a passport?  Would I need to contact the US Embassy for help in order to be able return?  If the airline does let me aboard to return to the US, even without the US passport, what are the best documents could I show at the border to prove I am a US citizen but that I just don't have a passport, for whatever reason? Are there any Combinatoric proofs of Bertrand's postulate? I feel like there must exist a combinatoric proof of a theorem like: There is a prime between $n$ and $2n$, or $p$ and $p^2$ or anything similar to this stronger than there is a prime between $p$ and $(\prod_p p) + 1$ (Euclid's theorem).  I was trying to prove one by the Sieve on this grid      1     2     3     4 ... p   p+1   p+2   p+3   p+4 ... 2p  2p+1  2p+2  2p+3  2p+4 ... 3p ........................... ........................... ........................... p^2 p^2+1 p^2+2 p^2+3 p^2+4 ...   but it did not work.  Do any good arguments like this exist? I don't expect anything as strong as Prime Number Theorem or even Bertrand, but surely a direct combination proof can prove that there are lots of primes? Count and limit the number of files uploaded (HTML file input) how to set camera to the same view as I turn my cursor? I want to turn the camera to get the view I want, but everytime I turn the camera, I need to go to render window to check if it's the view I want. I turn the view with my mouse cursor, but that does not affect the actual camera view in rendering, can I set the camera view the same as I turn my cursor? Proving that a normal, abelian subgroup of G is in the center of G if |G/N| and |Aut(N)| are relatively prime. Counting nodes in a tree in Java First of all, I swear this is not homework, it's a question I was asked in an interview. I think I made a mess of it (though I did realise the solution requires recursion). Here is the question:  Implement the count() method which returns the number of nodes in a tree. If a node doesn't have either a left or right child, the relevant getXXChild() method will return null  class Tree {    Tree getRightChild() {     // Assume this is already implemented   }    Tree getLeftChild() {     // Assume this is already implemented   }    int count() {     // Implement me   } }   My reason for asking the question is simply curious to see the correct solution, and thereby measure how bad mine was.  Cheers, Tony What is the safest way to clean up /boot partition? I have 200 MB assigned for the /boot partition.  Whenever I try to update the kernel, I receive an error message that basically states /boot is full.  What can I do to cleanup /boot and remove/backup the older kernels? Problems with Curve Modifier In summer, should I close curtains during the day? how to pass value like 100,102,103 to view params I am trying to zoom plot numbers.Plot numbers like 1 10-12 13/P road 20(A) 100&amp;102 and etc.I am able to zoom to all combination plot numbers except 1,2,3.when I pass such value its trowing error:  1 feature types requested, but found 2 view params specified  geoserver sql view is:  select * from plotboundary where nmindar='%nmindar%' and plno='%plno%'   default values and validation are  Name  | Default Value   | validation  nmidar| Hanagawadi I A  | ^[\w\d\s]+$  plno  |  100,102&amp;103    | ^[ A-Za-z0-9_ *@.\/#&amp;+-\,()-_]*$   ajax function:   $.ajax({             url: 'http://XX.168.1.XX:8089/geoserver/XX/ows?service=WFS&amp;version=1.0.0&amp;request=GetFeature&amp;typeName=XX:vwplot&amp;maxFeatures=5000&amp;outputFormat=text/javascript',             contentType: 'application/json',             dataType: 'jsonp',             jsonpCallback: "parseResponse",             data: {                 viewparams: "nmindar:" + nmindar + ";plno:" + plno             }   Can any one tell me how to pass value to viewparams in geoserver, value includes comma. For example as mentioned above "100,102&amp;103" is a single value(plot number).Here   comma is not a separator. PS4 Controller not connecting to PS4 and won't reset? When is 6 a square in Zp To find when $6$ is a square I divided $6$ in to $2$ and $3$ by The Legendre symbol.  $$\frac{6}{p} = \frac{2}{p} \times \frac{3}{p}$$  I know that we can multiply the cases of when $2$ and $$ are squares because then it is (1)(1) by Legendre.   $2$ is a square when $p \equiv 1 \pmod{8}$   $3$ is a square when $p \equiv 1 \pmod{12}$  so $6$ is a square when $p \equiv 1 \pmod {24}$ ($96$, but the $lcm(8,12)= 24$)   but how do I find the cases when $2$ and $3$ are both NOT squares? because then $6$ would be a square then as well, $(-1)(-1) = 1$ by Legendre. Can a curve object modifier created geometry be used as boolean factor This question is related to .  Let me use an exemple to illustrate what I mean.  We've created a text object (made of curves). It can't be used in a boolean modifier as it won't appear in the list of usable objects in the modifier because it's a curve object.  But if we add a remesh modifier on the text, geometry is created from the shape.  The object could be used as boolean if we convert it to mesh. The remesh can't be applied on curve object.  My question is : Can we dynamicaly use the remesh modifier created geometry as real geometry or make the boolean modifier consider the text modified object as a real mesh?  The idea behind this is to be able to change the text (or any curve 3D shape) without having to make it a real mesh.  (don't hesitate to edit my poor english to make it understandable :) ) Why didn't Thanos come to Earth to retrieve Tesseract by himself? In the Avengers movie, why didn't Thanos personally come to Earth instead of Loki? Or, if Loki's skills were required, why didn't he come with Chitauri? He knew that humans were primitive; there wouldn't be any threat and he could personally secure Tesseract.  AFAIK, the movie didn't display much. Is there any other source answering this? Last night, I migrated my MacBook's operating system from 10.10 "Yosemite" to 10.11 "El Capitan". Everything seemed to go smoothly. Happily, regarding my TeX-related work, I am able to use the GUI front ends "TeXworks" and "TeXshop" just fine since completing the OS upgrade. I use MacTeX2015, with all the latest updates installed.  Oddly, though, since completing the upgrade I can no longer run any MacTeX command-line utilities such as texdoc and kpsewhich. (Actually, as I just discovered, I can no longer run pdflatex either from a command line. However, I never do run pdflatex from a command line prompt, so that's not an urgent problem for me -- at least not for now.)  The suggestions given in the posting  do not seem to be applicable to this case, as they all seem to revolve around setting up /usr/texbin correctly -- /usr/texbin is no longer allowed under MacOSX 10.11. The document  by Herb Schulz deals in depth with adjusting settings in the GUI front ends, but it appears not to touch on what to do about /usr/texbin, which appears to be the now-missing link to TeX's command line utilities.   Pointers and solutions most welcome! Servlet returns "HTTP Status 404 The requested resource (/servlet) is not available" Elmah: How to get JSON HTTP request body from error report