When does $d$ mean a differential and when does it mean a derivative?
How to understand the notion of a differential of a function
Let $f,g:X\to Y$ be maps of spaces such that $f\simeq g$. Is it true that the mapping cones $\operatorname{cone}(f)$ and $\operatorname{cone}(g)$ are homotopy equivalent?  Can we write down an explicit homotopy equivalence? It ought not to be too hard to get a map $\operatorname{cone}(f)\to \operatorname{cone}(g)$ from the existence of a homotopy $H:X\times I\to Y$, but I haven't been able to find a sensible candidate, despite trying for quite a while. Edits that shadow a suggested edit inconsistently reject but include the suggestion Prove the following equality using mathematical induction: Prove the following equality using mathematical induction: For any integer $n \ge 1$  $$\sum_{i=1}^{n} \frac{1}{i(i+1)} = \frac{n}{n+1}$$  I understand for the base base I need to have $n=1$. If I substituted $n$ for 1 and $i$ for 1 I would get   \begin{align} \sum_{i=1}^{1} \frac{1}{1(1+1)} &amp;= \frac{1}{1+1} \\ \sum_{i=1}^{1} \frac{1}{2} &amp;= \frac{1}{2} \end{align}  As for the inductive step, I inputted $n +1$ but it did not work for me because honestly I did not know what to do with the $i$ and what I should do next.  How would I go about doing the next step and an explanation for $i$ would be helpful. Automatic labeling of enumeration items Question involving the Cauchy-Goursat Theorem Show $\int_C \frac{dz}{(z-z_0)^n} = 2\pi i$ when $n = 1$ and 0 when $n \gt 1$   I'm not entirely sure how to approach this one.  Any help would be much appreciated. Should we improve time to first byte or Page Speed Insights score for SEO? Google posted last summer that "page speed" will impact the search result rankings also on mobile devices. But I think there is a little confusion about the term "page speed".  The test tool which Google offers is called "Page Speed Insights". Many people we are talking with, are thinking that the overall score of this tool for a particular site for example Desktop: 46/100 and Mobile: 45/100 is used for the ranking algorithm.  BUT I found a very interesting article in German language which states that only the TTFB (time to first byte) value is used and this should be optimized not all suggestions which "Page Speed Insights", further called PSI offers. Here is the link to the German article:   In short terms it says that a very large site in this case "The New York Times" has a low overall score of Mobile:41 and Desktop:45. As they are used to sell reader memberships they should  offer a great user experience. So, they should optimize their site. But when I call this site it is amazing fast. The overall PSI score does not reflect this behaviour. But when I believe this article the TTFB for this site is perfect. Also when I call it here from Austria it states 0.881sec.  I really like to know on which value we should focus our work for optimization not to fall in the ranking. Because as many of you already said it is not useful to implement all suggestions from the PSI site for one domain.  So, it would be real  if you could clarify things here or post links to official posts from Google. What I found about this, Google always is talking about "page speed" but what exactly is this? Advantages/Disadvantages of Partitioning a Drive I have a 100GB hard drive in my Win XP Pro desktop. I use an external 250GB drive for backups, and am wondering if there would be any advantages to partitioning my internal hard drive.   I understand that some people like to keep their OS and programs on one partition, and their data on another partion on the same hard drive.  I'd like people to list the advantages, or note any disadvantages.  Thanks. Viruses in wine? For example, you're doing a research for a car to purchase. You get a list of cars with engine number, city where a car was assembled, and a color. What's missing is make, model and price. What is a terminal? How do I open a terminal window? How do I use the terminal? Cannot start XAMPP on Ubuntu 11.10 I have a query to insert a row into a table, which has a field called ID, which is populated using an AUTO_INCREMENT on the column. I need to get this value for the next bit of functionality, but when I run the following, it always returns 0 even though the actual value is not 0:  MySqlCommand comm = connect.CreateCommand(); comm.CommandText = insertInvoice; comm.CommandText += "\'" + invoiceDate.ToString("yyyy:MM:dd hh:mm:ss") + "\', " + bookFee + ", " + adminFee + ", " + totalFee + ", " + customerID +  ")"; int id = Convert.ToInt32(comm.ExecuteScalar());   According to my understanding, this should return the ID column, but it just returns 0 every time. Any ideas?  EDIT:  When I run:  "INSERT INTO INVOICE (INVOICE_DATE, BOOK_FEE, ADMIN_FEE, TOTAL_FEE, CUSTOMER_ID) VALUES ('2009:01:01 10:21:12', 50, 7, 57, 2134);last_insert_id();"   I get:  {"You have an error in your SQL syntax; check the manual that corresponds to your MySQL server version for the right syntax to use near 'last_insert_id()' at line 1"} I am trying to understand what mathematics is really built up of. I thought mathematical logic was the foundation of everything. But from reading a book in mathematical logic, they use "="(equals-sign), functions and relations.  Now is the "=" taken as undefined? I have seen it been defined in terms of the identity relation.  But in order to talk about functions and relations you need set theory.  However,  seems to be a part of mathematical logic.  Does this mean that (naive) set theory comes before sentential and predicate logic? Is (naive)set-theory at the absolute bottom, where we can define relations and functions and the eqality relation. And then comes sentential logic, and then predicate logic?  I am a little confused because when I took an introductory course, we had a little logic before set-theory. But now I see in another book on introduction to proofs that set-theory is in a chapter before logic. So what is at the bottom/start of mathematics, logic or set theory?, or is it circular at the bottom?  Can this be how it is at the bottom?  naive set-theory $\rightarrow$ sentential logic $\rightarrow $ predicate logic $\rightarrow$ axiomatic set-theory(ZFC) $\rightarrow$ mathematics  (But the problem with this explanation is that it seems that some naive-set theory proofs use logic...)  (The arrows are of course not "logical" arrows.)  simple explanation of the problem:  a book on logic uses at the start: functions, relations, sets, ordered pairs, "="  a book on set theory uses at the start: logical deductions like this: "$B \subseteq A$", means every element in B is in A, so if $C \subseteq B, B \subseteq A$, a proof can be "since every element in C is in B, and every element in B is in A, every element of C is in A: $C \subseteq A$". But this is first order logic? ($(c \rightarrow b \wedge b \rightarrow a)\rightarrow (c\rightarrow a)$).  Hence, both started from each other? How to upload files to server using JSP/Servlet? As most of you know, in the Microsoft Store, there are . This means that on Windows I can emulate Ubuntu and have the Ubuntu command line directly on Windows.  The question is very easy: what can’t I do from the Ubuntu command line emulated in Windows that I can do on a proper Linux-based Ubuntu? Is it useful to download this Ubuntu application, install it, and work just with it instead of the real OS? Is it possible to install all the development libraries? Can I write (or not) device drivers? In other words: what are the limits?  What features of Ubuntu Linux-based are missing from Ubuntu-on-Windows?  New answers are welcome: I know that every software is always changing by improving characteristics and features! Design for Facebook authentication in an iOS app that also accesses a secured web service How to get a list of all the labels in a LaTeX document in WinEdt I am writing a large document using LaTeX and WinEdt.  I have labeled each chapter, section and so on.  I would like to print out a list of just the labels (so that I don't duplicate one later on, and so that I have it handy in order to write \refs).    Is there a way to do this? Using strtok with a std::string I have a string that I would like to tokenize. But the C strtok() function requires my string to be a char*. How can I do this simply?  I tried:  token = strtok(str.c_str(), " ");    which fails because it turns it into a const char*, not a char* Let $f$ be a real valued, continuous and differentiable function on the real numbers with $f'$ uniformly continuous. Show that $g_n(x)$ with $$g_n(x)=nf(x+1/n)-nf(x)$$ is uniformly convergent to the derivative of $f$.   I cannot figure out how to solve this problem. If anyone could help me for this problem I would be very grateful. C# How do I click a button by hitting Enter whilst textbox has focus? I'm working with a WinForm app in C#, after I type something in a textbox I want to hit the Enter key but the textbox still has focus (flashing cursor is still in textbox), how can I achieve this?