Let $f : \mathbb{R}^3 \rightarrow \mathbb{R}$. How can i show that       If $f(x,y,z)=0$ then $$\frac{∂x}{∂y}\frac{∂y}{∂z}\frac{∂z}{∂x}=-1 $$   Any help will be appreciated.
Ok guys, continuing my passage through edwards... here is the question... thanks for hints/solutions in advance:  Suppose $f(x,y,z)=0$ can be solved for each of the three variables $x,y,z$ as a differentiable function of the other two. Then prove that  $\displaystyle \frac{\partial x}{\partial y} \frac{\partial y}{\partial z} \frac{\partial z}{\partial x} = -1$  Verify this is the case for the ideal gas equation $pv =RT$ where (where $p,v,T$ are the three variables and $R$ is the constant).
Accidentally hit "toggle full screen", How do I exit? Is there a standard way to do an fopen with a unicode string file path? Provide two arguments to one option using getopts In below code when I give option r then getopts requires one arguments:  while getopts ":hr::l:" opt; do     case $opt in         r ) echo "Run Numbers - argument = $OPTARG " ;;         l ) echo "Latency range - argument = $OPTARG" ;;         h ) helptext             graceful_exit ;;         * ) usage             clean_up             exit 1     esac done   But I need to pass two arguments after -r option, instead of one. Is there an easy way to do this? regex to match word boundary beginning with special characters Can you connect to a LAN server without being on the same internet connection I'm using retextured snowballs and eggs for the different paintballs for two teams in a paintball map, so I can't replace the eggs with snowballs.  I have tried using /gamerule doMobSpawning false, but that still allows the chickens to spawn. I have also tried using /tp @e[type=Chicken] 0 0 0 on a repeating command block to quietly remove the chicken, but there's still a very noticeable frame where the chicken is visible. /effect @e[type=Chicken] minecraft:invisibility 10 1 false isn't fast enough, either.  I'm not using any plugins or mods.  tl;dr Without snowballs, how can I stop chickens from appearing for even a single frame. BitLocker issue. Recovery Key is required on each startup after installing secondary OS I'm using a laptop with TPM module with BitLocker-encrypted internal HDD, running Windows 7. Recently I've installed a Windows 8 on the external, eSATA drive. Windows 8 detected the Windows 7 and correctly replaced the bootloader with dual-boot option.  Unfortunately, I'm now forced to enter BitLocker Recovery Key each time I start the computer. The only workaround I've found was to disable TPM module in BIOS, and suspend BitLocker on the Windows 7 drive.  How to accomplish the following configuration:   Keep the BitLocker running on Windows 7 (internal drive)  Leave the Windows 8 (external, eSATA drive) completely unencrypted  Don't be bugged to enter Recovery Key each time during startup  No matter if the eSATA drive is plugged in, or not   Is it possible? I don't care if TPM is enabled and I just want the Windows 7 drive to be BitLocker-encrypted. Which commit has this blob? Given the hash of a blob, is there a way to get a list of commits that have this blob in their tree? How to trigger workflow from button in O365 I created a script editor webpart. I added a button which start a js function in the onClick event. I would like to start a workflow which is implemented on some list. Is this possible? I was wondering if it was possible to use any of the API's provided by SF to build an external (C#, Java) application to manage building changesets.   Building changesets is very tedious when trying to include 100's of items and I was hoping to streamline the process for our team by building an application I can load an excel file to with the required components and have it build the outbound changeset. How to code an automated bot that can browse and do operations on a webpage For the Pell's equation where $d=2$: $$x^2-2y^2=1$$ What are all the integer solutions to the equation. Apparantly there are infinitely many solutions, but how would I represent them in an expression? Question of an isomorphism of $\epsilon_ 0$ and a subset of the rationals. I don't know if this question is appropriated for this site. Anyway, I'm searching for an isomorphism of order $f:K \longrightarrow \epsilon_o $, such that $(K, \leq)$ is a subset(proper or not) of $(\mathbb{Q}, \leq)$ and $\epsilon_o = \sup\{\omega, \omega^{\omega}, \omega^{\omega^{\omega}}, ... \}$. Actually,  I'm not finding neither an isomorphism between $K$ and $\omega^{\omega}$.  Thanks in advanced. How can I obfuscate (protect) JavaScript? I want to make a JavaScript application that's not open source, and thus I wish to learn how to can obfuscate my JS code? Is this possible? How to remove snap completely without losing the Chromium browser? I am wondering whether or not there is a way to completely remove snap from Ubuntu 19.10 without losing the ability to install important applications like Chromium.   When I just recently updated to the newest Ubuntu release I realized that the installer programmatically reinstalled snap, although I had manually removed it before. Additionally the installer removed Chromium, which was installed via the repositories, and reinstalled it via snap.   As I don't want snap to be installed on my machines for various reasons my question is if anybody knows a safe way to remove it, and to get the Chromium DEB back to the sources?  Is there a PPA? Could I use a source of an Ubuntu flavor additionally, which didn't remove the Chromium Deb from its sources? When I install "Installer for Microsoft TrueType core fonts" (ttf-mscorefonts-installer) from the Software Center on 14.04, it gives a positive response--but after this I get a download error. It seems that the files actually providing the fonts are never retrieved. Is there any way to resolve this? Is Con modifier × level added to HP every level-up? Given fields $K\subseteq L$, why does $f,g$ relatively prime in $K[x]$ imply relatively prime in $L[x]$? I'm trying to remember a cartoon tv show where the main boy, a blonde girl, and another boy capture monsters. There was something about a 'veil' between their world and the monster world, but a 'secret society' of sorts stopped the monsters from crossing, or used them to battle the people who wanted to use them for evil led by a wise old lady. I'm pretty sure they trapped monsters into pieces of jewelry or clothing.   Not so sure about this part: and they had these 'supernatural powers' that were connected to their 'aura' and they could travel from place to place with it.   (Another brown haired girl joined because she found out? There was an arc where the blonde got jealous and the monster she controlled attacked the brown haired girl?)  I can't remember much of it, but I remember in the end the 'veil' was broken and monsters flew over the city. And the main boy and the blonde girl held hands... or kissed? Selecting features that do NOT intersect in ArcGIS Desktop?