The equation  $$TdS=dU+PdV$$ is a combination of  $$dQ=dU+PdV$$ and  $$TdS=dQ.$$ But in some process that is very irreversible, $TdS&gt;dQ$, should we write  $$TdS&gt;dU+PdV$$ instead of the first equation? I guess the equation is different for reversible and irreversible process.
Combining the first and second law of thermodynamics we can get the following equation  $$TdS=dU-P_{ext}dV$$  First question: Is this equation applicable for irreversible processes such that that $dS≠\dfrac{dQ}{T}$?  Second question:If the system temperature $T_{sys}$ is smaller than the surrounding temperature $T_{sur}$, which temperature should we put in the equation?   I have this question because sometimes people use $T_{sur}$ instead of $T_{sys}$ (e.g. , ) but the equation is supposed to describe changes in the system.
How can I prevent my parents' divorce? I am a young teenager, younger than 16. (I do not want to give my age online)    My dad told me he wants a divorce, but I don't want my parents divorced. My dad says he can't stand my mom anymore.  My dad is older than 45 and my mom is older than 40.    If it happens, I would choose to stay with my mom. My dad can cook and drive, but my mom can't. There's a school bus in high school though.    What can I do to prevent this divorce? I'm trying to prove that if $\alpha(t)=(x(t),y(t))$ is a $C^2$ regular curve $(\alpha'\neq0)$ with constant and positive curvature, then $\alpha$ is on the circunference and if $\alpha$ is the circunference, then its curvature is constant.  I solved the second part since $\alpha$ is the circunference, then $\alpha(t)=(r\cos\theta,r\sin\theta)$, so $|\alpha'(t)|=r$, I need help in the first part.  Thanks a lot. Keeping the time stamp when syncing with Ubuntu One? I would like to keep the original time stamp of a file when synchronizing with Ubuntu One.  Now Ubuntu One is messing with my time stamps, and I no longer know when I last worked on the file.  Ubuntu One also seem to mess up time stamps on the original source host.  This really is a show stopper, and we are just about to implement Ubuntu One across the many users in the organization. General integral to find resistance My question is: is there a simple and truly general equation for the resistance between two electrical equipotential surfaces?.  Obviously, if so, what is it, and if not, why?  It would be very difficult to solve, granted, but I just want to see a calculus equation that is fully descriptive.  I have two frameworks under which this could be entertained, I'll write those out and then explain the motivation.  To start with, we need propose that the volume separating the two surfaces has a volumetric resistivity, $\rho$ in units of $(\Omega m)$.  Single Volume Framework  We can limit the discussion to a defined volume, then the surfaces reside in that volume or on the surface of it.  This volume may have a constant resistivity $\rho$ while everywhere outside the volume is completely electrically insulating.  Infinite Volume Framework  An alternative to the above approach that might make the task more or less difficult would be to replace a constant resistivity with a spatial dependence $\rho(\vec{r})$ and no longer require a boundary condition.  In that case we only have 3 mathematical inputs to the problem, which is the resistivity defined for all $\vec{r}$ and a definition of the two surfaces, $S_1$ and $S_2$.  Known Algebraic Analogs  The  that I find insufficient is:  $$R = \rho \frac{\ell}{A}$$  Where $l$ is the length of the restive material that is any shape which has translation symmetry over that length, and $A$ is the cross-sectional area.  Obviously, this is a rather simple equation that won't apply to more complicated geometry.  Even more  seem to give equations that fall short of what I'm asking.  For example:  $$R = \rho \int_0^l \frac{1}{A(x)} dx$$  I think it's obvious that an equation such as this is built upon a myriad of assumptions.  For a thought experiment, imagine that the area starts out as very small and then pans out to very large quickly.  Well, accounting for the larger area in the above sense underestimates the resistance, because the charge has to diffuse out perpendicular to the average direction of flow as well as parallel to it.  I have some reasons to suspect this might actually be rather difficult.  A big reason is that all the approaches I'm familiar with require the flow paths to be established beforehand, which can't be done for what I'm asking.  So maybe this will result in two interconnected calculus equations.  Motivation  I had an interest in , and it occurred to me that I can't quickly and simply write down the equation for resistance between two points.  The unique thing about Squishy Circuits is that it calls for two types of dough, one that conducts and one that is mostly insulating.  However, the recipes aren't perfect and because of that, the young children who play with these circuits regularly encounter the limits of conductor and insulator definitions.  If you make your conductor dough too long and/or too thin, you will encounter dimming of the light you connect with it.  Similarly, a thin insulator layer will lead to a lot of leakage current which also dims the light. Maximum value of $\sin A+\sin B+\sin C$? What is the maximum value of $\sin A+\sin B+\sin C$ in a triangle $ABC$. My book says its $3\sqrt3/2$ but I have no idea how to prove it.   I can see that if $A=B=C=\frac\pi3$ then I get $\sin A+\sin B+\sin C=\frac{3\sqrt3}2$. And also  maximum is attained for $a=b=c$. But this does not give me any idea for the proof.  Can anyone help? Custom post type - posts list - white screen of death i am getting a weird error - white screen in the list of posts for a specific Custom post type (just for that one)   tried deactivating all the plugins tried checking for error (debugging = true)   Still nothing the page just doesnt echos anything... (nothing in source too)  I am talking about such a url in the admin:     Here is register_post_type part i am using:    function register_submodelcpt() {     $labels = array(         'name'                  =&gt; __('Sub Models', THEME_NAME),         'singular_name'         =&gt; __('Sub Models', THEME_NAME),         'add_new'               =&gt; __('New Model', THEME_NAME),         'add_new_item'          =&gt; __('Add new Model', THEME_NAME),         'edit_item'             =&gt; __('Edit Model', THEME_NAME),         'new_item'              =&gt; __('New Model', THEME_NAME),         'all_items'             =&gt; __('All Sub Models', THEME_NAME),         'view_item'             =&gt; __('Watch Model', THEME_NAME),         'search_items'          =&gt; __('Search Models', THEME_NAME),         'not_found'             =&gt;  __('No Models found', THEME_NAME),         'not_found_in_trash'    =&gt; __('No Models found in trash', THEME_NAME),          'parent_item_colon'     =&gt; '',         'menu_name'             =&gt; __('Sub Models', THEME_NAME),      );      $args = array(         'labels'                =&gt; $labels,         'public'                =&gt; true,         'publicly_queryable'    =&gt; true,         'show_ui'               =&gt; true,          'show_in_menu'          =&gt; true,          'query_var'             =&gt; true,         'rewrite'               =&gt; array('slug' =&gt; 'submodels'),         'capability_type'       =&gt; 'post',         'has_archive'           =&gt; true,          'hierarchical'          =&gt; true,         'menu_position'         =&gt; 5,         'menu_icon'             =&gt; get_stylesheet_directory_uri().'/images/cpt/subcars.png',                     'supports'              =&gt; array('title', 'thumbnail', 'revisions', 'page-attributes')     );      register_post_type('submodelscpt',$args); } add_action('init', 'register_submodelcpt');   Did anyone encounter such an issuse? can you think of a reason this might happen?   Another weird thing when i change this:    To this:     The posts list loads correctly... So many interesting questions get asked on Stack Overflow and then immediately closed (or worse yet, closed and re-opened for days on end) because they're subjective, argumentative, and fundamentally unanswerable due to our lack of omniscience: without the ability to see into the future and/or know the secret motivations of companies and their developers, we are left to make wildly varying (un)informed guesses. The obvious solution to this is to add a  feature to the Ask a Question page. Similar to the current &quot;Related Questions&quot; feature, this could be used to answer otherwise-unanswerable questions before they're asked, thus saving everyone time and frustration. I propose a simple system whereby a web cam is aimed at an ordinary  device, long recognized as one of the most accessible and accurate fortune-telling mechanisms available to modern society. A simple  could be placed under this rig and triggered in response to the user's typing. Once focus leaves the Title input field, a short delay will be triggered, followed by a bit of logic to capture the 8-ball's result and feed it through a simple OCR routine to produce the result.  So far as I'm aware, SO would be the first major Q&amp;A site to implement such a feature...  Examples:            ...another UV resurrection... Does Windows still rely on MS-DOS? I am just about to start an OS course and as an Apple user I am not very familiar with the underlying details of Windows OS. I was wondering, is MS DOS still used with Windows running on top or is ONLY Windows used now as the OS? I was a little confused because I read somewhere that MS-DOS is used for booting but Windows has all other OS capabilities built into it and thus is used for all other OS operations... Does anyone know of a nice (and preferably free) way to integrate Prolog and C#?  Im looking to create a Prolog dll or similar to call from my managed code, and retrieve an answer once all the processing has been complete. Im looking for it to be predominantly one sided (c# calls Prolog).  I have seen  which talks about Prologs real world usage but I was wondering if anyone had either any experience with c# &amp; Prolog? or a nice tutorial/article? I read a story once about a man who escapes from his pursuers by doing an uncontrolled jump into hyperspace. If I'm not mixing my stories up he was blind but could see through the eyes of other sighted people/animals nearby. He was able to see through his pet but it dies and he thinks he's finished - he's blind and lost in hyperspace but there's a rat on board which he eventually catches by turning off gravity and then he can see through the rat's eyes. He works out how hyperspace operates and can hop back to where he came from.   Can anyone remember the name of the story and the author? I enjoyed it when I was a kid and would like to read it again. What's the simplest way to wrap extruded text around a cylinder? What's the simplest way to wrap extruded text around a cylinder that has a specific radius (for example, the label on a can of baked beans)?  I've tried using the Simple Deform modifier, but that (a) doesn't allow you to choose a radius and (b) requires the text to be converted to a curve.  I'm trying to model a cylinder with radius of 30, which has a line of embossed text that runs around the curved side, facing outward.  Thanks! Story about lost colonized planet sent into orbit around "free" star Is soaking beans 24 hours unrefrigerated safe? In educational material for the Russian language and linguistic literature the need to put accents on Cyrillic vowels often arises. However when I tried to do this with luaLaTeX the accents where misplaced.  \documentclass[11pt]{article} \usepackage[utf8]{inputenc} \usepackage{fontspec} \setmainfont{Linux Libertine:style=Regular} \begin{document}  Е́сли вы изу́чите  \'{Е}сли вы из\'{у}чите  \'{E}сли вы из\'{y}чите % Here "E" and "y" are Latin glyphs, but this workaround does not                         % work with all Cyrillic letters e.g. Я, Ю \end{document}     The last line would looks nice, but unfortunately these are Latin glyphs.  Is this a problem with the font used (Linux Libertine) or with LaTeX? Photoshop resize circularly Is there a way to do this circular resize of a pattern in photoshop (flat image) ? (I only want to resize the green disk, without changing the size of the interior white disk and without losing details of the green disk, just flatting them) Incorrect icons on hot network questions Extracting the function from InterpolatingFunction object What is the difference between a "link function" and a "canonical link function" for GLM Replace arbitrary characters in the middle of an IP address string with sed I need to find and replace either one or two numerical characters in strings in a file.  The strings are IP addresses of the form:   10.xx.y.z    Where xx can be one or two characters.    I want to replace the xx with the single character 0, so I have  10.0.y.z    preserving the values of y and z.   The string may appear multiple times in the file.  What is the sed invocation to do this? Was the blue fire from Crimes of Grindlewald Fiendfyre? In Crimes of Grindlewald, Grindelwald uses a powerful blue flame that takes on the shape of monsters. Was this Fiendfyre?