In all my practice I have simply taken as given that   $$\frac{\partial^2 f}{\partial x \partial y}=\frac{\partial^2 f}{\partial y \partial x}$$  When does this not hold?
I've taken multivariate calculus and am wondering if I can see a specific function where the order of taking the partial derivative matters. I've been told that there are some exceptions where $ \dfrac{\partial ^2 f}{\partial x \partial y} \ne \dfrac{\partial ^2 f}{\partial y \partial x} $, so I'm curious to see what this looks like.  EDIT: And why would this true?
So I'm trying to prove the following two inequalities: $$\frac{\sqrt{n}}{2} \leq \phi(n) \leq n.$$ The upper bound we get from simply noting that $\phi(n) = n \prod_{p | n}\left( 1 - \frac{1}{p}\right)$ and the fact that $(1 - \frac{1}{p}) \leq 1$. But how can we get the lower bound? I tried using the same expression for $\phi(n)$ but it seems to not really give me the inequality. Can you help? Does Professor X have any responsibility for the state of the world? Throughout the course of Logan, it's obvious that Professor Xavier is having issues controlling his powers, and at the end of the movie, just before      X-24 kills him   he claims to "remember" what he did and expresses regret. We also hear about an incident in Westchester that killed several mutants, that sounds like Xavier may have been involved. Is Xaiver the reason there are no more mutants? Is that why he's sorry, because he knows he killed them all? How do I resolve unmet dependencies after adding a PPA? How can we communicate short messages long-distance? I'd like to play Dragon Age Inquision in english instead of the language of my operating system, but it seems to default to that and I can't find a way to change it. I already tried setting the Origin client language to english, but that didn't change anything.  Where can I change the language for Dragon Age: Inquisition? I want to block Facebook (the Facebook site) for concentrating purposes, but I don't want to break the share or like buttons on other pages (like a good article or a nice clip on YouTube I want to like or share). How can I do that? I use both Windows and Linux. Play a short sound in iOS Downloading PlayOnLinux I'm downloading PlayOnLinux to run Paint Tool SAI, but I cannot open PlayOnLinux. When I click on it to open it, a dialog box comes up that says: "PlayOnLinux cannot find curl (from cURL) You need to install it to continue". How do I install this?   I'm referring to this article: test class with Batch and Callout I have a problem with FeedItems in my Test class, it's very weird behaviour Does someone explain to me why this happens?  Callout Class  public with sharing class Callout {  public static void runProccess(Case[] records) {     Database.executeBatch(new CalloutBatch(records), 10); }  public static void send(Case[] records) {     Case[] toUpdate = new Case[]{};       for (case record : records) {         HttpRequest req = new HttpRequest();         req.setEndpoint('callout:test/SOME_URL');         req.setMethod('POST');         req.setHeader('Content-Type', 'application/json');         req.setBody('');         Http http = new Http();         HTTPResponse res = http.send(req);          Map&lt;String, Object&gt; response = (Map&lt;String, Object&gt;) JSON.deserializeUntyped(res.getBody());          toUpdate.add(new Case(Id = (Id) response.get('Id'), Status = (String) response.get('Status')));     }      update toUpdate;  } }   CalloutBatch Class  global with sharing class CalloutBatch implements Database.Batchable&lt;Case&gt;, Database.AllowsCallouts{  private final Case[] records;  global CalloutBatch (Case[] records) {     this.records = records; }  global Iterable&lt;Case&gt; start(Database.BatchableContext BC) {     return records; }  global void execute(Database.BatchableContext BC, Case[] scope) {     Callout.send(scope); }  global void finish(Database.BatchableContext BC) { } }   Test Class  @IsTest  private class CalloutTest { @IsTest static void shouldReaciveData() {     String status = 'Closed';     Case caseRecord = new Case(Type = 'Web', Status = 'New');     insert caseRecord;      //insert new FeedItem(ParentId = caseRecord.Id, Visibility='AllUsers', Type = 'TextPost', Body = '');      System.assertEquals(0, [SELECT Count() FROM Case WHERE Id = :caseRecord.Id AND Status = :status]);      Test.startTest();     Test.setMock(HttpCalloutMock.class, new CalloutTestMock(caseRecord.Id, status));     Callout.runProccess(new Case[]{caseRecord});     Test.stopTest();      System.assertEquals(1, [SELECT Count() FROM Case WHERE Id = :caseRecord.Id AND Status = :status]); } }   Mock Class  public with sharing class CalloutTestMock implements HttpCalloutMock {  private Id recordId; private String status;  public CalloutTestMock(Id recordId, String status) {     this.recordId = recordId;     this.status = status; }  public HTTPResponse respond(HTTPRequest req) {     HttpResponse response = new HttpResponse();      response.setStatusCode(200);     response.setBody('{"Id": "' + recordId + '", "Status": "' + status + '"}');      return response; } }   This code works, but when I uncomment insert FeedItem in Test Class I receive an error message:     System.CalloutException: You have uncommitted work pending. Please commit or rollback before calling out   Class.Callout.send: line 23, column 1   Class.CalloutBatch.execute: line 20, column 1   FeedItem doesn't have an impact on this functionality, (I don't have any Process Builders, triggers etc. on FeedItem object) Customize equation numbering for Equation environment? I want to manually label equation number ? For example I want to label this equation 3.1 instead of (1) which automatically pops up. Is it also possible in split environment as well or not ?      \begin{equation}     \log \mu_{ijk}= \lambda + \lambda_i ^I + \lambda_j ^J+\lambda_k ^K+\lambda_{ij} ^{IJ}+\lambda_{ik} ^{IK}+\lambda_{jk} ^{JK}+\lambda_{ijk} ^{IJK} \label{eq:3.1}     \end{equation} I showed up here to write a perfectly normal rant about misaligned pixels, and there's a big fat checkbox cluttering up the "Ask" page:    WHAT THE EEEEK IS THIS NEW ABOMINATION? The first homology group $ H_1(E(K); Z) $ of a knot exterior is an infinite cyclic group which is generated by the class of the meridian. I was writing test classes for my apex Rest Service that I had built for an Ipad app.  I noted some very interesting things while experimenting. Before doing anything, I just did a google search and found an interesting blog post by Jeff.    @RestResource(urlMapping='/v.9/member/*/results/*')  global with sharing class MemberRestSvc {  @HttpGet  global static ReturnClass doGet() {  String[] uriKeys = RestContext.request.requestURI.split('/'); // get the member name from the uri String memberName = uriKeys.get(uriKeys.size()-3);  // do awesome programming stuff here &amp; catch any exceptions try {    List&lt;Contact&gt; contacts = [Select Id From Contact where member_name__c = :memberName];   return new ReturnClass('true', 'Query executed successfully.', contacts);  } catch (Exception e) {   return new ReturnClass('false', e.getMessage(), null); }   }   global class ReturnClass {  global String success; global String message; global List&lt;Contact&gt; records;    global ReturnClass(String success, String message, List&lt;Contact&gt; records) {   this.success = success;   this.message = message;   this.records = records;  }  }  }   And here is the test class  @isTest private class Test_MemberRestSvc {  static { // setup test data   }  static testMethod void testDoGet() {  RestRequest req = new RestRequest();  RestResponse res = new RestResponse();  req.requestURI = 'https://cs9.salesforce.com/services/apexrest/v.9/member/me/results/today';   req.httpMethod = 'GET'; RestContext.request = req; RestContext.response = res;  MemberRestSvc.ReturnClass results = MemberRestSvc.doGet();  System.assertEquals('true', results.success); System.assertEquals(10, results.records.size()); System.assertEquals('Query executed successfully.', results.message);  }  }   I just commented a few lines and ran and it. I still achieved enough test coverage so my question is why do I need to simply instantiate the Rest request and response objects since I am directly calling the class?    @isTest   private class Test_MemberRestSvc {    static {     // setup test data     }  static testMethod void testDoGet() {  /*RestRequest req = new RestRequest();  RestResponse res = new RestResponse();  req.requestURI =    'https://cs9.salesforce.com/services/apexrest/v.9/member/me/results/today';   req.httpMethod = 'GET'; RestContext.request = req; RestContext.response = res; */Commented the code meant for REST apex intialization and merely calling the method!!  MemberRestSvc.ReturnClass results = MemberRestSvc.doGet();  System.assertEquals('true', results.success); System.assertEquals(10, results.records.size()); System.assertEquals('Query executed successfully.', results.message);  }  }   Can someone explain this behavior? Also what are the best practices for writing Test classes for apex Rest services? Recursive grep vs find / -type f -exec grep {} \; Which is more efficient/faster? I stumbled upon this "relation" (is the name correct?):  $$ \lim_{m \to \infty} \lim_{n \to \infty} \cos^{2n}(m! \pi x) = \begin{cases} 1,&amp;x\text{ is rational}\\ 0,&amp;x\text{ is irrational}\end{cases} $$  How is it called and why is it so? I'm really not asking for a proof since I fear it would be too complicated for me to understand, but rather for an "intuition". can't create a jobs user profile Flow On Schedule I"ve created a flow that I'd like to run monthly on the 7th day of each month. How can I do this with scheduled apex. I have no experience writing apex from scratch. Any suggestions or starting off points appreciated.  Thanks,  Kris What medium should be used for long term, high volume, data storage (archival)? Regular expression: match start or whitespace Can a regular expression match whitespace or the start of a string?  I'm trying to replace currency the abbreviation GBP with a £ symbol. I could just match anything starting GBP, but I'd like to be a bit more conservative, and look for certain delimiters around it.  &gt;&gt;&gt; import re &gt;&gt;&gt; text = u'GBP 5 Off when you spend GBP75.00'  &gt;&gt;&gt; re.sub(ur'GBP([\W\d])', ur'£\g&lt;1&gt;', text) # matches GBP with any prefix u'\xa3 5 Off when you spend \xa375.00'  &gt;&gt;&gt; re.sub(ur'^GBP([\W\d])', ur'£\g&lt;1&gt;', text) # matches at start only u'\xa3 5 Off when you spend GBP75.00'  &gt;&gt;&gt; re.sub(ur'(\W)GBP([\W\d])', ur'\g&lt;1&gt;£\g&lt;2&gt;', text) # matches whitespace prefix only u'GBP 5 Off when you spend \xa375.00'   Can I do both of the latter examples at the same time? Concrete Mathematics - The Josephus Problem I've been going through  and have a question on the inductive proof for the . Recurrent relation:  $J(1) = 1$ $J(2n) = 2J(n) - 1$ $J(2n+1) = 2J(n) + 1$  Closed form hypothesis:  $J(2^m + l) = 2l + 1$, for $m \ge 0$ and $0\le l&lt;2^m$  Inductive Proof (for the even case):  $J(1)=1$ $J(2^m+l)= 2J(2^{m-1}+l/2)-1=2(2l/2+1)-1=2l+1$  How do the authors get from $2J(2^{m-1}+l/2)-1$ to $2(2l/2+1)-1$?