Wikipedia says a conservative vector field is a vector field that is the gradient of some function. For this vector field I think a potential function could be arctan(y/x), and thus every line integral of this vector field over a closed path is 0. As I calculated the line integrals directly, I found they were not 0. Then I checked if $\frac{\partial Q}{\partial x} = \frac{\partial P}{\partial y}$ and it turns out $\frac{\partial Q}{\partial x} \neq \frac{\partial P}{\partial y}$. Umm... why is that?
My question is really ''what is the definition of a conservative vector field''?  I've consulted 3 textbooks that all say a vector field $\vec{F}$ is conservative by definition if there exists a scalar potential $\phi$ such that $\nabla \phi = \vec{F}$. Then, they go on to talk about connected domains, path independence and the equality of mixed partials and how they are all related.   In particular, they emphasize that e.g. in $\mathbb{R}^2$ given $\vec{F} = \bigl&lt;F_1,\,F_2\bigr&gt;$, if $\frac{\partial F_1}{\partial y} = \frac{\partial F_2}{\partial x}$ on a simply-connected domain, then $\vec{F}$ is conservative on that domain.  However, without fail, all of them then offer the example of $\vec{F} = \bigl&lt; \frac{-y}{x^2+y^2},\, \frac{x}{x^2+y^2} \bigr&gt;$, pointing out:   it's line integral is not path independent, even though $\frac{\partial F_1}{\partial y} = \frac{\partial F_2}{\partial x}$   and this is explained by pointing out that the domain is not simply connected (if the path contains the origin).  That much makes sense to me: we cannot conclude $\vec{F}$ is conservative based on the partial derivatives, because the domain is not simply connected - totally consistent with what has been presented.  What none of them address is why $\vec{F} = \bigl&lt; \frac{-y}{x^2+y^2},\, \frac{x}{x^2+y^2} \bigr&gt;$ is not conservative when there exists a potential $\phi = \arctan(y/x)$ such that $\nabla \phi = \vec{F}$. None of the texts mention any necessary conditions on the scalar potential. So is the existence of a scalar potential the definition of a vector field being conservative or not?
I'm on a Sprint Samsung Galaxy S3 with Android 4.3.  I would love to become a power user of the dictation feature to dictate text messages and emails. More often than not, however, I have to hand-edit the dictated text not because of text comprehension issues but because of punctuation, capitalization and spacing problems.  These are the "commands" I've found to work. What is the full list?  "period"     --&gt; .   (works fairly reliably) "comma"      --&gt; ,   (works fairly reliably) "new line"   --&gt; \n  (works about 50% of the time)   I've not gotten capital or anything like that to work, or acronyms. If you know how it works, please share. Can't delete the filter called My Filter I'm unable to get rid of a filter that goes by the name of "My Filter".  This is the default name you get when you click on New Filter.  There's a clickable "Delete this filter", which leads to a confirmation question on whether you really want to delete it.  Clicking "Yes" leads you back to the "New Filter" Screen,  with a new filter named "My Filter".    I don't know whether this is the same filter I was just trying to get rid of,  or whether it got rid of the old one, and now is making a new one with the same old name.  Either way, I'm stuck. Rank of a Matrix Sum I have $3\times3$ matrices such that S=A+B. I know there is an inequality connecting rank of the matrices A, B and its sum S. Could you write down that here? It will be a great help for me.  Means equation  or inequality connecting rank(S), rank(A) and rank(B). When to use an object pronoun or a possessive adjective before a gerund I have embedded a Python interpreter in a C program.  Suppose the C program reads some bytes from a file into a char array and learns (somehow) that the bytes represent text with a certain encoding (e.g., ISO 8859-1, Windows-1252, or UTF-8).  How do I decode the contents of this char array into a Python string?  The Python string should in general be of type unicode—for instance, a 0x93 in Windows-1252 encoded input becomes a u'\u0201c'.  I have attempted to use PyString_Decode, but it always fails when there are non-ASCII characters in the string.  Here is an example that fails:  #include &lt;Python.h&gt; #include &lt;stdio.h&gt;  int main(int argc, char *argv[]) {      char c_string[] = { (char)0x93, 0 };      PyObject *py_string;       Py_Initialize();       py_string = PyString_Decode(c_string, 1, "windows_1252", "replace");      if (!py_string) {           PyErr_Print();           return 1;      }      return 0; }   The error message is UnicodeEncodeError: 'ascii' codec can't encode character u'\u201c' in position 0: ordinal not in range(128), which indicates that the ascii encoding is used even though we specify windows_1252 in the call to PyString_Decode.  The following code works around the problem by using PyString_FromString to create a Python string of the undecoded bytes, then calling its decode method:  #include &lt;Python.h&gt; #include &lt;stdio.h&gt;  int main(int argc, char *argv[]) {      char c_string[] = { (char)0x93, 0 };      PyObject *raw, *decoded;       Py_Initialize();       raw = PyString_FromString(c_string);      printf("Undecoded: ");      PyObject_Print(raw, stdout, 0);      printf("\n");      decoded = PyObject_CallMethod(raw, "decode", "s", "windows_1252");      Py_DECREF(raw);      printf("Decoded: ");      PyObject_Print(decoded, stdout, 0);      printf("\n");      return 0; } The Safari app has disappeared from my iPhone? I can't find the safari icon on my iphone 4?  I have tried resetting home screen layout and that doesn't work.  My restrictions button is 'on' and it asks for a password for this button.  I don't have a password.  Can you help? Thanks, How can I get the source directory of a Bash script from within the script itself? How do I get the path of the directory in which a  script is located, inside that script? I want to use a Bash script as a launcher for another application. I want to change the working directory to the one where the Bash script is located, so I can operate on the files in that directory, like so: $ ./application Mavericks unusually high RAM usage I thought Mavericks was suppose to utilize RAM memory better, thus run smoother.  Running ML on Mac mini 2012, with 16GB RAM...always running smoothly even with over a dozen apps running.  Now, after installing Mavericks, my Mac is SLOW, eating up RAM like crazy! Looking over usage, of the 16GB of installed memory...on average it uses 10-14 GB RAM when barely running a few apps!!!   There is this kernel_task that is always running with over 1GB of RAM and most apps are not taking that much RAM, so what is using up all my RAM?! File cache? It shows very high in that new description, with about 1/3 of my total RAM Fill cylinder cap with quads Is there any way of converting the triangle fan found on a cylinder cap into quads. Collection of short stories; in one a spy attempts to escape detection on Mars's moon by moving faster than the spaceship can scan Buy a second seat in order to bring an extra bag? Find highest level of a hierarchical field: with vs without CTEs Let $s_n$ be a sequence defined as given below for $n \geq 1$. Then find out $\lim\limits_{n \to \infty} s_n$.   \begin{align}   s_n = \int\limits_0^1 \frac{nx^{n-1}}{1+x} dx  \end{align}  I have written a solution of my own, but I would like to know it is completely correct, and also I would like if people post more alternative solutions. Chromatic Number The smallest number of colors needed to color a graph $G$ is called its chromatic number, and is often denoted $\chi(G)$.  Show that if graph $G$ is simple, then $$ \chi \ge \frac{V^2}{V^2 - 2E}. $$ How to add readable text to a light and dark image? I'm trying to add text to a photo of a city but the colors in the background make it difficult to read light and dark text. I've tried several light and dark colors but they all looked overpowered by the colors in the image. What colors could I use or what can I do to the text to make it show but also make it so you can still see the background?  Here is the image I'm trying to use: I've been trying to change the language of my LaTeX document by replacing \usepackage[swedish]{babel} with \usepackage[english]{babel} but I only get this message:  ! Package babel Error: You haven't loaded the option swedish yet.   I've read in an other question at this forum that compiling enough would solve the problem, but that doesn't work for me. Does anyone know what I should do? How do I edit the boot order in the GRUB menu to have Windows first? I have a dual boot on my computer of Windows 7 and Xubuntu 12.10. The Linux boot is 1st and the Windows 2nd. I would like to reverse the boot order in the GRUB menu. I need to know the steps to go about editing GRUB. I do most of my work on Windows and need it to be 1st so I can boot up and not watch it. Mail vs. mail what is the difference How to diagnose computer lockups and freezes? I built a desktop computer a couple years back with the following specs:  CPU: Intel  Yorkfield 2.5GHz 6 MB L2 Cache LGA 775 95W Quad-Core Processor BX80580Q9300 Motherboard:  LGA 775 NVIDIA nForce 680i SLI ATX Intel Motherboard Video Card: Two  GeForce 8600GT SCC 256 MB 128-bit GDDR3 PCI Express x16 SLI Supported Video Card PSU: SeaSonic S12 Energy Plus  ATX12V V2.3 / EPS12V V2.91 SLI Certified CrossFire Ready 80 PLUS Certified Active PFC Power Supply Memory: Two G.SKILL 4 GB (2 x 2 GB) 240-Pin DDR2 SDRAM DDR2 800 (PC2 6400) Dual Channel Kit Desktop Memory   Since its inception, the machine has periodically locked up, the regularity having varied over the years from once a day to once a month. Typically, lockups happen once every few days. By &quot;lockup&quot; I mean my computer just freezes. The screen locks up, I can't move the mouse. Hitting keys on my keyboard that normally turn LEDs on or off on the keyboard (such as Caps Lock) no longer turn the LEDs on or off. If there was music playing at the time of the lockup, noise keeps coming out of the speakers, but it's just the current frequency/note that plays indefinitely. There is no BSOD. When such a lockup occurs I have to do a hard reboot by either turning off the computer or hitting the reset button. I have the most recent version of the NVIDIA hardware drivers, and update them semi-regularly, but that hasn't seemed to help. I am currently using Windows 7 x64, but was previously using Windows Server 2003 x64 and having the same lockup issues. My guess is that it's somehow video driver or motherboard related, but I don't know how to go about diagnosing this problem to narrow down which of the two is the culprit.  Additional information re: cooling Regarding cooling... I've not installed any after-market cooling systems aside from two regular fans I scavenged from an older computer. The fan atop the CPU is the one that shipped with it. One of the two scavenged fans I added it located at the bottom tower of the corner, in an attempt to create some airflow from front to back. The second fan is pointed directly at the two video cards.  SpeedFan installation and readings Per studiohack's suggestion, I installed , which provided the following temperature readings:  GPU: 63C GPU: 65C System: 76C CPU: 64C AUX: 36C Core 0: 78C Core 1: 76C Core 2: 79C Core 3: 79C   Update #3: Another Lockup :-( Well, I had another lockup last night. :-( SpeedFan reported the CPU temp at 38 C when it happened, and there was no spike in temperature leading up to the freeze. One thing I notice is that the freeze seems more likely to happen if I am watching a video. In fact, of the last 5 freezes over the past month, 4 of them have been while watching a video on Flickr. Not necessarily the same video, but a video nevertheless. I don't know if this is just coincidence or if it means anything. (As an aside, each night before bedtime my 2 year old daughter sits on my lap and watches some home videos on Flickr and, in the last month, has learned the phrase, &quot;Uh oh, computer broke.&quot;)  Update #4: MemTest86 and 3DMark06 Test Results: Per suggestions in the comments, I ran the MemTest86 overnight and it cycled through the 8 GB of memory 5 times without error. I also ran the 3DMark06 test without a problem (see my scores at ). So... what now? :-) Any further suggestions on what to check? Is there some way to get a stack trace or something when the computer locks like that? Resolution I have never did figure out the particular problems, but based on the suggestions here and elsewhere, I'm presuming it was a motherboard issue. In any event, I recently upgraded my system, buying a new motherbeard, PSU, CPU, and RAM, and that new rig has been working splendidly the past several weeks. I am using the same graphic cards as in the old setup, so I think it's safe to reason that they weren't the cause of the problem. How should sockpuppets be handled on Stack Exchange? A high-reputation, highly-active user has already been dinged once for a sockpuppet, and I just dinged him again. He probably has another sockpuppet as well.   What are the guidelines for handling users who game the system in this fashion? Specifically,   How can I be sure I'm looking at a sockpuppet? When should sockpuppets be considered a problem? How should moderators handle problematic sockpuppets once they've been identified? I was suspended for sockpuppetting and feel this is unfair