How to calculate the work without displacement?
Why does holding something up cost energy while no work is being done?
I know that, whenever I want to see how some control sequence, say \mycontrolsequence, is defined, I can use \show\mycontrolsequence  then compile my code and look up the .log file. However, robust commands seem impervious to \show. For instance, if I compile the following code, \documentclass{article}  \newcommand\mycommand{foo} \show\mycommand  \DeclareRobustCommand{\myrobustcommand}{bar} \show\myrobustcommand  \begin{document} hello \end{document}  my log file contains  \mycommand=\long macro: -&gt;foo. l.4 \show\mycommand  \myrobustcommand=macro: -&gt;\protect \myrobustcommand  . l.7 \show\myrobustcommand   Note that, whereas the definition of \mycommand is exposed (foo), nothing transpires of \myrobustcommand's definition. Of course, in practice, I could always identify in which file a given robust command is defined and look up its definition there. However, finding the file in question may not be obvious. Hence my question: Is there an alternative way of peering into the definition of a robust command? A question about $f(x)\equiv 0$ Are all limits solvable without L'Hôpital Rule or Series Expansion What is the difference between "I am done" and "I am through"? Help with footer always to bottom Make rigid body collide with hair particles For any prime $p > 3$, why is $p^2-1$ always divisible by 24? How can a lens with a single focal length focus on more than one plane? Theorem. If the identity is written as the product of $r$ transpositions, $id=τ_1τ_2\dots τ_r$, then $r$ is an even number. Proof. We will employ induction on $r$. A transposition cannot be the identity; hence, $r &gt; 1$. If $r = 2$, then we are done. Suppose that $r &gt; 2$. In this case the product of the last two transpositions, $τ_{r−1}τ_r$, must be one of the following cases: $(ab)(ab) = id\\ (bc)(ab) = (ac)(bc)\\ (cd)(ab) = (ab)(cd)\\ (ac)(ab) = (ab)(bc),$ where $a$, $b$, $c$, and $d$ are distinct. The first equation simply says that a transposition is its own inverse. If this case occurs, delete $τ_{r−1} τ_r$ from the product to obtain $id=τ_1τ_2···τ_{r−3}τ_{r−2}$. By induction $r − 2$ is even; hence, $r$ must be even. In each of the other three cases, we can replace $τ_{r−1} τ_r$ with the right-hand side of the corresponding equation to obtain a new product of $r$ transpositions for the identity. In this new product the last occurrence of $a$ will be in the next-to-the-last transposition. We can continue this process with $τ_{r−2} τ_{r−1}$ to obtain either a product of $r − 2$ transpositions or a new product of $r$ transpositions where the last occurrence of $a$ is in $τ_{r−2}$. If the identity is the product of $r − 2$ transpositions, then again we are done, by our induction hypothesis; otherwise, we will repeat the procedure with $τ_{r−3}τ_{r−2}$. At some point either we will have two adjacent, identical transpositions canceling each other out or $a$ will be shuffled so that it will appear only in the first transposition. However, the latter case cannot occur, because the identity would not fix $a$ in this instance. Therefore, the identity permutation must be the product of $r − 2$ transpositions and, again by our induction hypothesis, we are done. Question: That is the proof written in the book, but unfortunately after more than $10$ times reading I don't understand the last two paragraph (i.e., from &quot;In each of the other three cases&quot;). Is there another simpler proof or could someone please clarify what is going on? Thank you. Are there fields and ordered fields of every infinite cardinality? On the top of my head, I cannot think of any fields of cardinality more than that of the reals. (It is known that the process of algebraic closure does not increase the cardinality of an infinite field.)  What is the simplest way to give an example of a field (and an ordered field) of a specific cardinality $\alpha$?  I see there is the "Field" of surreal numbers, but it is a proper class rather than a set (and hence do not have a cardinality as such). However, there seems to be some modified construction which gives proper fields with the cardinality of some strongly inaccessible cardinal. That's really embarrassing, however I need to ask it. I could not prove that the normed space    $\langle \mathscr{C} [0, 1], \| \cdot \|_1 \rangle $ is complete (as a metric space), where $\| f\|_1 = \int_0^1 |f(x)|dx$. I don't really know how to explicit a continuous functions as a limit of a Cauchy sequence. I tried to prove that the pointwise limit (of the Cauchy sequence) have just a finite number of discontinuities (I don't know if it's true) and then approximate it by a continuous function.  Thanks in advance. If $\alpha$ is an algebraic element and $L \subset K$ are both field, does the polynomial ring $L[\alpha]$ is also a field?  I am trying to prove that the ring of fraction $L(\alpha)$ is equal to $L[\alpha]$. To do this, I use an exercise done in class : Let $K$ a field, $\alpha \in K$ and $L$ a subfield of $K$. Then $L(\alpha)$ is the smallest subfield of $K$ containing $L$ and $\alpha$.  Is anyone could help me? Dictionary application needs to download dictionary "Apple Dictionary" before use If $x,y,z\in\mathbb R\setminus \{1\}$ and $xyz=1$, prove that $\frac{x^2}{(x-1)^2}+\frac{y^2}{(y-1)^2}+\frac{z^2}{(z-1)^2}\ge 1$. Let $z_1,z_2,\ldots,z_n$ be complex number such that $|z_i|&lt;1$ for all $i=1,2,\ldots,n$. Show that we can choose $a_i \in\{-1,1\}$, $i=1,2,\ldots,n$ such that $$\left|\sum_{i=1}^n a_iz_i\right|&lt;\sqrt3.$$ How do I remove the space between inline/inline-block elements? Why does a vertical line have a SLOPE (steepness) of undefined? The curve shows :   x goes 1 to right  y goes up by 3   the SLOPE is defined as the ratio of rise to the step  SLOPE = rise/step = 3/1 = 3  Now that :   Horizontal line has SLOPE of 0. Vertical line has SLOPE of undefined.   It makes sense for me to imagine horizontal line has SLOPE of 0 since there is no rise at all. But why the vertical has SLOPE of undefined.? Automate on-site plagiarism detection There have been many cases where on-site plagiarism has been coincidentally discovered by  good citizens either while looking for , or while answering questions merely to find out one of their  etc.   Most of the cases have been identified as verbatim copy-paste of the original answer.  I understand plagiarism is considered very offensive and can't be tolerated. It is very unprofessional and unfair. So here are my questions (and possibly a feature request):   Could verbatim plagiarized content be automatically detected?  Should it be pushed into a review queue similar to low quality posts for other users to confirm? Which words in a title should be capitalized? Are there any concrete rules that say which words (parts of speech) in a title should start with a capital letter? What would be a correct capitalization for the title of this question? How to add greek polytonic with setxkbmap I want to add greek polytonic keyboard layout under Lubuntu (so in LXDE). I already realized there is no easy tool to do it with the graphic interface, so I need to use setxkbmap executing something the like of:  setxkbmap pl,gr   Now, how to make setxkbmap set the greek polytonic variant?