What is the cause of the surface tension of the liquids? What is the cause of the surface tension of the liquids? How to know the direction of tension force on the free surface of the liquid?   I know that the surface tension is the force acting normally per unit length of the imaginary line drawn on the free surface of a liquid, but don't have any idea about the cause of the surface tension. More over I don't know the direction of the tension force i.e. where it towards liquid or away from the liquid.  Any idea or help will be greatly appreciated!
Why is surface tension parallel to the interface? A text says:     The surface tension of a liquid results from an imbalance of intermolecular attractive forces, the cohesive forces between molecules:   A molecule in the bulk liquid experiences cohesive forces with other molecules in all directions.   A molecule at the surface of a liquid experiences only net inward cohesive forces   If this is the case: Why is surface tension considered or experienced as a force parallel to the interface while it is so obvious that it must be perpendicular to it?
sudo apt-offline install not updating working with procedure  to update Ubuntu 18.04 offline machine, the sudo apt-offline install /path_to_downloaded_package_folder do not apply any updates/upgrades. The beginning of the massage I get is :   sudo apt-offline install ./updates_path/ No changelog available (Y) Yes. Proceed with installation (N) No, Abort. (C) Display changelog (?) Display this help message. What would you like to do next:  (y, N, ?)y Proceeding with installation ERROR: Path for keyring is invalid: /etc/apt/trusted.gpg gpgv: Signature made Mon 06 Jan 2020 08:17:27 AM MST gpgv:                using RSA key 3B4FE6ACC0B21F32 gpgv: Good signature from "Ubuntu Archive Automatic Signing Key (2012) &lt;ftpmaster@ubuntu.com&gt;" gpgv: Signature made Mon 06 Jan 2020 08:18:52 AM MST gpgv:                using RSA key 3B4FE6ACC0B21F32 gpgv: Good signature from "Ubuntu Archive Automatic Signing Key (2012) &lt;ftpmaster@ubuntu.com&gt;" gpgv: Signature made Mon 06 Jan 2020 08:18:25 AM MST gpgv:                using RSA key 3B4FE6ACC0B21F32 gpgv: Good signature from "Ubuntu Archive Automatic Signing Key (2012) &lt;ftpmaster@ubuntu.com&gt;" gpgv: Signature made Thu 26 Apr 2018 05:38:40 PM MDT gpgv:                using RSA key 3B4FE6ACC0B21F32 gpgv: Good signature from "Ubuntu Archive Automatic Signing Key (2012) &lt;ftpmaster@ubuntu.com&gt;" security.ubuntu.com_ubuntu_dists_bionic-security_InRelease synced. security.ubuntu.com_ubuntu_dists_bionic-security_Release synced. security.ubuntu.com_ubuntu_dists_bionic-security_Release.gpg synced.   What am I doing wrong? How to find the sum of all the numbers that can be formed using the digits 2,3,3,4,4,4.. What should be the way of doing this type of problem?? Please guide Prove that $\left | \frac{a-b}{a+b}+\frac{b-c}{b+c}+\frac{c-a}{c+a} \right | < \frac{1}{8}.$ Let $a,b,$ and $c$ be the lengths of the sides of a triangle. Prove that $$\left | \dfrac{a-b}{a+b}+\dfrac{b-c}{b+c}+\dfrac{c-a}{c+a} \right | &lt; \dfrac{1}{8}.$$   The best idea I had was to expand the fractions to get something nicer. So we get $\dfrac{a-b}{a+b}+\dfrac{b-c}{b+c}+\dfrac{c-a}{c+a} = \dfrac{a^2 b-a^2 c-a b^2+a c^2+b^2 c-b c^2}{(a+b) (a+c) (b+c)}.$ Then I would try to relate this to side lengths of a triangle to prove the desired inequality but that is the step where I am unsure. Concurrently interleaving execution of Lua and TeX in LuaTeX The usual way to interleave Lua and TeX in generating documents is by invoking Lua from Tex through the \directlua facility and generate Tex from Lua using tex.print().  For example, consider the Plain LuaTeX:  \directlua{     es=""; someluatex = function () tex.everypar="PAR!" end     tex.print("one", es, "two")     someluatex()     tex.print("three", es, "four") } five\bye   Looking at the document LuaTeX generates for that input, we see that someluatex() is run before the Tex backend does anything with the output: the PAR!s will be inserted before "one" and "two", as well as after "three".  Is there best practice for ensuring that TeX code issued by some Lua code is issued before some other LuaTeX code?  There are several ways to do this: with some code I am working with I am trying the following scheme:   Wrap the code executed by the \directlua in a coroutine, call it myco; At the point we want to ensure the Tex so far tex.print-ed is executed, use:  tex.print("\\directlua{coroutine.resume(myco)}") coroutine.yield()  If everything is in order, the Lua will resume after the Tex is processed.   This is not ideal, however: we must ensure that the TeX backend is in the right state to understand the \directlua command, which may not always be the case.  Is there a better way to achieve this? Prove that $a^2(b+c-a)+b^2(a+c-b)+c^2(a+b-c) \leq 3abc.$ How to nest code within a list using Markdown Regular expression to match balanced parentheses I need a regular expression to select all the text between two outer brackets.  Example: some text(text here(possible text)text(possible text(more text)))end text  Result: (text here(possible text)text(possible text(more text))) How to capture all std outputs Use of "Or", inclusive or exclusive? My wife and I are playing a game where you roll dice and move so many spaces in a grid "vertically or horizontally".   In the use of English it is very common to say, this or the other when it comes to making a choice  (exclusive or). Now I know that "or" can also be inclusive, for example "she couldn't read or write", or can be clearly used as an xor "you either come or not" making the statement true for only one of the options but not both.   My issue is where it is not clear whether it is an inclusive/exclusive or, the best example being our game.   I argue that you can move in either direction (inclusive) and the normal use of this conjunction in English should be inclusive unless specified otherwise.   Is this right? Uniform continuity of $f(x) = x \sin{\frac{1}{x}}$ for $x \neq 0$ and $f(0) = 0.$ For the $f(x) = x \sin{\frac{1}{x}}$ for $x \neq 0$ and $f(0) = 0,$ my text book asks the following questions.  (b) Why is $f$ uniformly continuous on any bounded subset of $\mathbb{R}$?   (c) Is $f$ uniformly continuous on $\mathbb{R}$??  The graph for the function is this.    For the question (b), if I take subset between $[0.2,0.6]$ or the subset where the slope is steep, I don't think the function is uniformly continuous because I think for a given $\epsilon&gt;0$, there is no unique $\delta &gt;0$ for the bounded subset. Therefore, it also cannot be uniformly continuous on $\mathbb{R}.$ However, the questions sounds like the function is uniformly continuous and the book says that it is uniformly continuous. The answer on the book says something but I need more explanation. Thanks. The Start menu no longer shows up when I press the Start button Is it possible to change your spawn point? Is it possible to change the location you beam to on a planet? I've been to a couple of planets that have no usable resources near the spawn point and everything is a good distance East or West. It would be nice to be able to always beam down to there instead. What are good books/other readings for elementary set theory? I am looking to expand my knowledge on set theory (which is pretty poor right now -- basic understanding of sets, power sets, and different (infinite) cardinalities). Are there any books that come to your mind that would be useful for an undergrad math student who hasn't taken a set theory course yet?   Thanks a lot for your suggestions! Change the Fill Color of a cell if the value of that cell equals the value of another specific cell I want to change the fill color of cell C5 if it has the same value as cell A5.   Can this be done? I know how to format cell C5 if it has a certain value - but not when it is to be compared to the value of another cell. Ignore .DS_Store and Icon files in a folder with Cocoa NSFIleManager Is the matrix diagonalizable for all values of t? Will my software work on the next version of Windows I have some software and I'd like to know if it works on the next version of Windows (Windows 10). Which are the word orders that can be found in English? Is the Doctor Who episode 'Blink' the first use of the phrase "timey-wimey"? How to count derangements? I was going through .Suppose that there are n persons who are numbered 1, 2, ..., n. Let there be n hats also numbered 1, 2, ..., n. We have to find the number of ways in which no one gets the hat having same number as his/her number. Let us assume that the first person takes hat i. There are n − 1 ways for the first person to make such a choice. There are now two possibilities, depending on whether or not person i takes hat 1 in return:  A) Person i does not take the hat 1. This case is equivalent to solving the problem with n − 1 persons and n − 1 hats: each of the remaining n − 1 people has precisely 1 forbidden choice from among the remaining n − 1 hats (i's forbidden choice is hat 1).  B) Person i takes the hat 1. Now the problem reduces to n − 2 persons and n − 2 hats.  From this, the following relation is derived: !n = (n - 1) (!(n-1) + !(n-2))  I have a hard time understanding point B and the equation derived. I was trying to understand with an example of 4 people named A, B, C and D and the corresponding hats A, B, C and D. Everyone of them are supposed to wear any hat except their own matching hat i.e. A can wear B, C or D.  If suppose A wears B hat so we will be left with only 3 people and 3 hats. So I can understand "(n-1)!(n-1)" part but what i didn't understand is part B(marked above). Can anyone help?