I am trying to find gravitational potential energy by integrating the gravitational force:  $$\mathbf{F}(\mathbf{r}) = - \ G \frac{Mm}{|\mathbf{r}|^2} \ \hat{\mathbf{r}}$$  where $\mathbf{r}$ is the vector from the centre of the Earth with mass $M$ towards the point of a mass $m$, and $\hat{\mathbf{r}}$ is the unit vector pointing towards the vector $\mathbf{r}$.  The potential energy at a point of the distance $|\mathbf{r}|$ can be found from  $$U(\mathbf{r}) = -\int_{\infty}^{r} - \ G \frac{Mm}{|\mathbf{r}|^2} \hat{\mathbf{r}} \ \cdot \ d\mathbf{r}$$  Then, removing the minus signs the expression becomes  $$U(\mathbf{r}) = \int_{\infty}^{r}  \ G \frac{Mm}{|\mathbf{r}|^2} \hat{\mathbf{r}} \ \cdot \ d\mathbf{r}$$  However, look at the picture below that I drew.   Since I am integrating from infinity to a point r, I thought $d\mathbf{r}$ should be directed towards the opposite direction of $\hat{\mathbf{r}}$, but actually, if I calculate in that way I get  $$U(\mathbf{r}) = \frac{GMm}{r},$$ in which the minus sign is omitting.  From a mathematical perspective, either of $\hat{\mathbf{r}}$ and $\mathbf{r}$ direct towards the same direction, hence the scalar product should be a positive value.  But I still do not understand why $\hat{\mathbf{r}}$ and  $d\mathbf{r}$ should be considered the same direction whenever I am calculating. This is perhaps not accounting for the upper and lower bounds.  Can anybody explain please?
A solid sphere has charge $q$ and radius $R$. Find the potential at a point a distance $r$ from the center of the sphere where $r&gt;R$, using infinity as the reference point.  My attempt:  From Gauss' theorem we may deduce that $\displaystyle\mathbf{E}=\frac{1}{4\pi\epsilon_0}\frac{q}{r'^2}\hat{\mathbf{r}}$ where $r'$ is the distance of an arbitrary point from the center of the sphere provided $r'&gt;R$.  $V=-\int_\infty^\mathbf{r}\mathbf{E}\cdot d\mathbf{l}$  My question: What is $d\mathbf{l}$? Since we are traversing in the direction opposite to $\hat{\mathbf{r}}$, I think it should be $-dr'\hat{\mathbf{r}}$. But when I use it to find $V$, I get a sign error. Please help!
how can i display the latest items from another site collection I have a web application, with three site collections . and i want inside my team site to be able to display the latest pages that  have been added to my enterprise wiki site collection. so what are the available approaches which i can follow?  Thanks in advance for any help.  Regards The computer gave me this output in a window:  E: Encountered a section with no Package: header E: Problem with MergeList /var/lib/apt/lists/archive.ubuntu.com_ubuntu_dists_natty_main_binary-i386_Packages E: The package lists or status file could not be parsed or opened.   How can I fix this? I have a SNP table looks like this  A N N N N N N N N N N N N C N N N C N N N N N N N N N N N N N N N N N N N C N N N C N N N N N N N N N N N N N N N N N N C C N N N N N N N N N N C C N N N C C N N N N N N N N N N N N N N N N N   I want to count total number of N in each row? any idea how to do that I would like to install the restricted extras package, which I didn't select during the install. How can I do that?        is a meta-package that installs:         Support for MP3 and unencrypted DVD playback   Microsoft TrueType core fonts   Flash plugin   codecs for common audio and video files Meta-analysis of means and medians in R? I've been asked to review the literature for all studies that estimate the mean and median Blood Lead Level (BLL) in a certain country, and then perform a meta-analysis to come up with one overall value for the mean and median level.   Is such a thing typically done? I'm not interested in any "effect size," just a univariate measure.   I'd like to create a forest plot and do some of the standard tests of heterogeneity across studies. Is there a package and function in R that will allow me to do a meta-analysis of mean and median BLLs?  Just to clarify, this would be a separate meta-analysis for means, and a separate meta-analysis for medians. I have been thinking of getting a flash to improve my low light photography (particularly indoors). However, there seem to be a number of different models available at similar price points (budget of INR 8,000-12,000 ~$150-250). What features and functionality should I look for to choose a suitable flash?  I'm using a Canon 550D with the 18-55mm IS &amp; 55-250mm IS lenses. Metz seems to be one of the brands easily available in India (beside the OEM flashes). Currently, I am considering the Metz 36 AF-5, 44 AF-1 and 50 AF-1, which seem to be considerably cheaper than their Canon counterparts. Has user testing found that the "three horizontal bars" for main menu on mobile is commonly understood? How do I ensure a piece of code runs only once? Can a ring without a unit element have a subring with a unit element? I am trying to send an email using a Visualforce template and I want that template to include an icon. Is this possible? I am a beginner of ordinals and I don't know any powerful techniques in it. I come across with a problem about the first uncountable ordinal like this.  Let $X$ be a set of uncountable cardinality. Using the Principle of Well Order we have a well ordering $\le$ on X(and $&lt;$ means $\le$ but not euqual). By adding an element, which we denote by $∞$, and introducing the convention that $x &lt; ∞$ for all $x ∈ X$, we will assume that X has a maximum with respect to $\le$.   We define $ω_1 = \text{min} \{ x ∈ X : \{y ∈ X : y &lt; x \} \  \text{is uncountable}\} $  Clearly such a $ω_1$ exists. we define $0 = \text{min} \ X$ and intervels $[0,x], [x,y), (a,b]$ etc. in the usual sense.  Here are my questions:  (1)For any countable $A ⊂ [0, ω_1)$ there is an $x &lt; ω_1$ so that $A ⊂ [0, x]$.(I don't know how to make use of the countability and uncountablity here)   (2)Equipped with the topology generated by open intervals,  $[0, ω_1]$ is compact.   (3)A famous application of the first uncountable ordinal is to find an example of a Borel measure (with respect to the topology in (2)) that is not Radon. So how to construct a finite Borel measure $µ$ on $[0, ω_1]$ which is not a Radon measure?  Any solutions or elementary references will be appreciated! Make 15 lines / ticks with Photoshop, each of them rotated of 20° Unable to open any graphical app with "sudo" Tried to find some answer here: , but it didn't solve my problem.  $ sudo gedit Failed to connect to Mir: Failed to connect to server socket: No such file or directory. Unable to init server: Could not connect: Connection refused.  (gedit:4574): Gtk-WARNING **: cannot open display:   only gedit in terminal works fine. This is my local computer, not a remote system. Is there any canon information in Harry Potter that proves one way or another whether a wizard always owns only one wand as a matter of routine?  What I am talking about is a wizard buying and owning one wand and also a second wand (for backup, or simultaneous use, or separate purpose, or ceremonial+work wand), akin to people buying more than one pair of boots, guns or hammers.  A proof would be either canon info showing someone owning 2 wands simultaneously; or a statement showing why that does not happen except in rare exceptions (see below).    NOTE: I am discounting special situations such as winning an extra wand in a duel ala Harry (e.g. the second wand's obtainment wasn't intended/planned), or someone winning a Death Stick from prior owner (in which case it was intended as a replacement wand and not an extra one); or taking another's wand for a time being, ala Voldemort and Lucius Malfoy (since it was more of a "temporary replacement because main wand was useless for the purpose due to paired cores" - far from a normal situation).  Also, I meant wands intended for use - e.g. if Malfoy would have a (imagined by me) collection of rare antique wands, that's as irrelevant as someone having flintlocks in a display case. Up my street and down the lane Possible Duplicate:          Except where there is obvious difference in elevation e.g. on a sloping road, how do native speakers decide whether to use up or down in phrases like up the road when referring to a location along the same road? This is a nice problem on series convergence that I recently stumbled upon. Given a non-negative sequence of real numbers $(a_n)$ such that $$\sum_{n=1}^\infty a_n &lt; \infty,$$ show that there exists a non-decreasing sequence of non-negative numbers $b_n$ such that $$b_n \to \infty \quad\text{ and } \quad \sum_{n=1}^\infty a_n b_n &lt; \infty.$$ In other words, for every convergent series with non-negative terms, there is another convergent series with "substantially larger" terms.  I have a solution (see below), but maybe someone else has a different, simpler, and/or more elegant solution. My computer name in System Preferences &gt; Sharing is set to &quot;archos&quot;, but is showing as &quot;iphone&quot; on Terminal. It just started doing this after I loaded Xcode for doing iPhone development: Last login: Mon Nov  7 14:46:55 on ttys001 iphone:~ travis$   Any ideas what could be causing this? Dictionary application needs to download dictionary "Apple Dictionary" before use I'm repeatedly being asked to download the "Apple Dictionary" when I open the dictionary application.     Dictionary application needs to download dictionary "Apple Dictionary" before use.   This doesn't happen every time I open it. I haven't pinned down specific usage cases where this message appears, but I think it may be every time after I reboot my computer.  Is there anyway for the Dictionary application to remember that I have "Apple Dictionary" permanently? This has only started happening after I upgraded from OS X Mountain Lion to macOS Sierra. Good tools for drawing schematics I was going to add a bit of information to my post on a previous day using schematics and some instructions. What programs are being employed for this purpose?  I mostly want to see what others are using and that I can easily use to give descriptive schematics.  In a perfect world, and I know this is a case of me wishing, it would be:   Free. Extremely easy to draw schematics in. Allows simple production of waveforms for the inputs/outputs. I have been trying to install the latest version of java-8. However, I found out that the  due to changes in the java licence.  How should I update my java to 8.0_201 or 8.0_211 now?  Thanks!