Why does the fact that $\int_0^1\int_0^1 \frac{x^2-y^2}{(x^2+y^2)^2} \,dy \,dx = \frac{\pi}{4}$ and $ \int_0^1\int_0^1 \frac{x^2-y^2}{(x^2+y^2)^2} \,dx \,dy = -\frac{\pi}{4}$ doesn't contradict Fubini's Theorem? Does the tangent function have something to do with that matter?
I have a function defined as follows: $f(x,y)= \dfrac{x^2-y^2}{\left(x^2+y^2\right)^2}$, if $(x,y)\neq (0, 0)$ and $f(x,y)=0$ if $(x,y)=(0,0)$.  Now, $$\int_0^1\int_0^1 f(x,y)\,\text{d}x~\text{d}y=-\frac{\pi}{4}$$ and $$\int_0^1\int_0^1 f(x,y)\,\text{d}y~\text{d}x=\frac{\pi}{4}.$$   The question I have is this: why does this not contradict Fubini's theorem?    thanks.
How do I monitor the computer's CPU, memory, and disk usage in Java? Proof of orthogonal matrix In the following question we regard vectors in $\Bbb R^n$ as columns and define the dot product in the usual way which means that $x · y$ = $x^{t}y$  a) If A is an n × n matrix show that $A_{ij}$ = $e_i \cdot A e_j$ where $e_i$, $i =1, \dots , n$ are the standard basis vectors in $\Bbb R^n$  b) Show that a matrix $P$ is orthogonal if and only if ($Px$) · ($Py$) =$x · y$ for all $x, y$ ∈ $\Bbb R^n$  Working -   I got part (a) alright but have no idea of where to start with part (b) except to use the information that an n x n matrix $P$ is orthogonal if $P^{-1}$= $P^{t}$ and that $P^{t}$$P$=$I$, $P$$P^{t}$=$I$, where $I$ is the identity matrix. Any suggests are gracefully appreciated. Feel free to edit my question for clarity. Thank you missing $ inserted error for equation I get the missing $ inserted error for the following code:  \begin{equation}  C = $ \frac{2}{\omega^2N^2\mu\pia} $  \end{equation}   I have tried out different changes based on Googling for this error, but nothing seems to help. Any suggestions would be appreciated. Why do objects move when you apply a force that's equal in magnitude to the friction? If the friction on an object is 200 N and you apply 200 N, it seems, the forces should immediately cancel each other out since the friction and your applied force go in opposite directions, leaving you with an object that is motionless. Yet, from what I've read, that's not the case. I feel like I'm misunderstanding something. Could it be I'm wrong about needing to apply 200 N initially? Might the amount of force I actually need be greater, but instantaneously decrease to 200 N? How To Change The Fonts In Windows 8 In Windows 7, there is an option to change the fonts for things such as the active title bar, menu, etc. It was found by opening the Window Color and Appearance settings in the Control Panel. I just installed Windows 8 and would like to change the fonts, but I can't find anything to change the actual fonts (there is however a limited drop-down list to change a few font sizes in the appearance settings). Does anyone how to change the fonts used in Windows 8? How to align two images side by side if one label has more lines I have two same sized images and want them side by side with a caption under them.  My code looks like this:   \begin{figure}[H] \centering \begin{minipage}{.5\textwidth}   \centering     \includegraphics[width=\textwidth]{3}     \caption{label1}   \label{fig:test1} \end{minipage}% \begin{minipage}{.5\textwidth}   \centering     \includegraphics[width=\textwidth]{4}     \caption{label2}   \label{fig:test2} \end{minipage} \end{figure}   which works fine until one of the labels has 2 lines, than it looks like this:    and I woud like to prevent it to look something like this:    How can it be done?  Thanks How can I create my own Action using Photoshop? Can anyone guide me with a simple example? What's the difference between executing a script like this:  ./test.sh  and executing a script like this:  . test.sh?  I tried a simple, two-line script to see if I could find if there was a difference:  #!/bin/bash ls   But both . test.sh and ./test.sh returned the same information. How do you deserialize json properties that are reserved words in apex? Show that $n$ lines separate the plane into  $\frac{n^2+n+2}{2}$ regions if no two of these lines are parallel and no three pass through a common point.   I know we start with the base case, where, if we call the above equation P(n), P(0), for 0 lines would be 0.  But I really have no idea how to begin the inductive step.  How do we know what k+1 we're supposed to arrive at?  Thanks! Running junit tests in parallel in a Maven build? Difference between spin and polarization of a photon I understand how one associates the spin of a quantum particle, e.g. of a photon, with intrinsic angular momentum. And in electromagnetism I have always understood the polarization of an EM wave as the oscillations of the E and M field, not necessarily being  aligned with the direction of propagation of the wave.     Questions:   But when one talks about the polarization of a photon in Quantum Mechanics, how does it really differ from its spin?  Are they somehow related and what's the physical idea behind photon polarization in contrast to photon-spin? Feel free to use mathematical reasonings as well if you see it fit! How can they see Harry's invisibility cloak when they are not wearing it? Debian root partition resizing 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 Flag dialogue popup is stacked behind realtime-post-deleted-notification banners There is a little bug that becomes visible when a page contains realtime-post-deleted-notification banners (that's the CSS classname of these red banners appearing on answers which just got deleted, see screenshot below) and you try to open the flag dialogue above them:    Obviously, the dialogue appears below the banners, which it clearly should not do. It also has no input focus, trying to interact with the dialogue fails and either closes it (clicking outside of the red banners) or reloads the page (clicking on the red banners).  The cause for this is simply the z-index ordering in the site's CSS. Relevant CSS snippets:  .realtime-post-deleted-notification {     position: absolute;     text-align: center;     z-index: 100; }  .popup {     display: none;     padding: 15px;     position: absolute;     z-index: 10; }   So the fix for this minor bug would probably be to raise the z-index of the popup class to something greater than 100. So, we know that there is no "natural" isomorphism between a finite dimensional vector space $X$ and its dual $X^{\vee}$. However, given a basis for $X$ $(e_1, \dots, e_n)$ we can always construct a (dual) basis $(e^1, \dots, e^n)$ for $X^{\vee}$ that satisfies $e^i(e_j) = \delta^i_j$. It follows that $V \approx V^{\vee}$ simply because they have the same finite dimension $n$. Now, it seems to me that we can construct an isomorphism by assigning $$ e_i \mapsto e^i $$ and extending by linearity. That is, for an arbitrary vector $v \in X$ given by $v = \sum v^ie_i$ we can specify a linear map   $$\phi:V \rightarrow V^{\vee}$$  by  $$ \phi(\sum v^ie_i) := \sum v^ie^i $$ It is a quick exercise to check that $\phi$ is an isomorphism. Now, although this construction seems correct, there is something that doesn't quite sit right with me about it. For one thing, there is no way to really make sense of the summation convention as the right side of the function will always contain two superscripts thus requiring an explicit summation sign. The other thing that feels rather off is actually taking a lower index and moving it to an upper index, i.e., $e_i \mapsto e^i$ (which, of course, is the reason that the summation convention doesn't work).  So, with this background my questions are   Is there any sense in which the above isomorphism is "favored" or "canonical"? Is there another way to construct an isomorphism between a vector space and its dual, that would "conserve indexes", for lack of a better way to state it. My guess here is that the answer is no but becomes possible if one assumes $V$ has an inner product. Multiplicative nature of the separability degree In what follows, let $E / F$ be an algebraic extension, $h(x),f(x)\in F[x]$ polynomials, $h(x)$ irreducible.   Definitions.   We say $h(x)$ is separable if it has not repeated factors. We say $f(x)$ is separable if all its irreducible factors are separable. We say $\alpha\in E$ is separable over $F$, if its minimal polynomial over $F$ is. The extension $E/F$ is separable if all the elements of $E$ are separable over $F$. We define the field $$\newcommand\sclo{\operatorname{Scl}}\sclo_F E= \{\alpha\in E : \text{ $\alpha$ is separable over $F$ }\}$$ as the separable closure of $F$ in $E$. The separability degree of the extension $E/F$ is defined as $$[E:F]_s = [\sclo_F E:F].$$   Now, suppose the extension $E/F$ is finite.  My question is: how can one proof that with these definitions, if $B$ is an intermediate field of the extension $E / F$, then $$[E:F]_s = [E:B]_s[B:F]_s?$$  I know that the purpose of the separability degree is to count in how many ways we can extend certain injective homomorphism to an automorphism. If one starts defining separability degree this way, as done in the book by Lang, the result follows without much difficulty.  I wonder how to achieve it in this way. Why I cannot ping 192.168.0.108 (my computer IP?) How can I stop being prompted to unlock the 'default' keyring on boot? Whenever Ubuntu boots up, a dialogue pops up asking me to unlock my default keyring. Is there some way this can unlock automatically through PAM or some other magical way?