Computing integral of $x+y+z$ over the unit cube Compute the integral:  $$\iint_\sigma (x+y+z)\,\mathrm dS $$   on the cube $0\le x\le1, 0\le y\le1, 0\le z \le 1$.   I am really confused how should I compute it as I used to just take the projection of $\ z $ on the $\ xy $ plane but here the projection is just a square from $\ 0 $ to $\ 1 $. Trying to compute $\ \int_0^1 \int_0^1 (x + y) dy dx $ doesn't lead me to the right answer.
Surface integral of a scalar over a unit cube. Evaluate the following integral $$\iint_S (x+y+z) \, dS$$  where $S$ is the surface of the cube $[0,1] \times [0,1] \times [0,1]$  Honestly, I don't know what to do. All I know is that you have to evaluate the integral for each of the 6 surfaces of the cube. The normals for them I can do and the equations of the planes, but other than that I couldn't do much.  Can someone break it down for me, please?  PS: No divergence theorem yet for me.
I have heard that two permutations are conjugate if they have the same cyclic structure. Is there an intuitive way to understand why this is? su options - running command as another user I was wondering how to run a command as another user from a script.  I have the script's owner set as root. I also have the following command being run within the script to run the command as the hudson user:  su -c command hudson   Is this the correct syntax? Are fractions with prime numerator and denominator dense? How can one change an entire document to use bold font? How do I decrease the size of the page file? I just upgraded my laptop RAM to 12GB (W510, 4 DIMM slots), and since I am using a SSD, I find hard drive space to be quite precious. However, since I have more than enough RAM for most stuff, I think I can get by with probably around 4GB of page file. however, on C drive, there is a pagefile.sys which I assume to be reserving space for page files in the OS, how would I decrease the size of this file? I noticed that after joining a table to a shapefile the headers of each fields in the table used for the join are changed by QGIS and they are attributed to a different name.  For instance, I had a shapefile with vegetation_codes and a table with vegetation_codes and veg_definition (these were the headers). After joining the table to the shapefile and also saving the shapefile and creating a new one, the vegetation_classes header was changed into vegetati_1.  Is there a way to keep the original headers?  I am using QGIS Dufour on Windows 7 and the table that I join to the shapefile is CSV (tried with dbf as well and the problem is the same). Is the term "errored out" a grammatically correct phrase, or just a colloquial one? Confusion about the definition of function Yesterday I was talking to one of my friends about the definition of function. The formal definition of function is given by Cartesian Products but my friend's question was whether it is possible to define a function without being acquainted with any concept of Cartesian Products.   To answer this question I told him the definition of a function may be given by defining a function $f$ from the set $X$ to $Y$, denoted by $f:X\to Y$, to be some "rule" (the word rule being an undefined concept) which associates, to each member of $X$, exactly one element of $Y$.     But my friend said that the definition using the concept of "rule" isn't rigorous and there are examples that don't obey the definition but still is a function in the Cartesian Product sense.    My questions are,   Why the definition of function using concepts such as "rule" isn't rigorous?  Is there really any example which doesn't obey the definition but still is a function in the Cartesian Product sense? How do I update Android on my device? Why is it “a defeated Napoleon, not “the defeated Napoleon” who rode off the battlefield and into exile? I noticed an infinite article; ‘a’ was used before ‘defeated Napoleon’ in the following sentence of Jeffery Archer’s novel, "Fales Impression" :  “General Sir Harry Wentworth sat at the right hand of the Iron Duke that night, and was commanding his left flank when a defeated Napoleon rode off the battlefield and into exile."  I’m curious to know why it should be “a defeated Napoleon,” not “the defeated Napoleon” or simply “defeated Napoleon.” Isn’t it only one Napoleon who was defeated the battle and exiled (to St. Helena). It looks like as though there were many (defeated) Napoleons who were exiled.  It’s always headache for me to handle articles as we Japanese (I think Chinese too) don’t have article as the basic part of speeches in our language system. I’ve studied it on English grammar text books at school, but have never reached full understanding of the usage of articles, because we don’t have that practice. Is there any special knack to master how to use articles? I have noticed that when I leave comments on a question and someone votes it up, I do not get a notification. If the upvote count would at least be more than 5, I would expect to be notified since that leads to a badge. Lie bracket for Lie algebra of $SO(n,m)$ I need a way to determine the type of an HTML element in JavaScript. It has the ID, but the element itself could be a &lt;div&gt;, a &lt;form&gt; field, a &lt;fieldset&gt;, etc. How can I achieve this? Suppose $f$ is a continuous single-variable function, prove that:  $$2\int_a^b \int_a^x f(x)f(y) \, dy \, dx = \left[ \int_a^b f(x) \, dx \right]^2$$  This question was just on my Calculus III final exam and I couldn't figure it out, could someone provide some insight, please? Blown out blue/red light making photos look out of focus I have 16GB of system RAM. On startup with no applications open except the task-manager Windows is using about 3gb of RAM. I looked in the processes tab, but nothing seems to be out of the ordinary. How can I find out why my Windows is using so much RAM.    all processes from all users      Reading from the poolmon it seems that my wireless broadcom driver is using about 0.4GB of of RAM. Even if I remove  it would still be using  2.6GB on start-up, which is still too much.      After reinstalling the wireless driver associated with the memory leak. I have a new screenshot and would like to confirm that it was indeed a memory leak. I would like to make a debian system where you can only open one GUI program at boot, no other graphical interface, no minimize, or any X apart from that program, is there a way to do it on debian if possible, or any other custom distro? I just want to boot open the program and allow the user to only see and use that program. Why does ('0' ? 'a' : 'b') behave different than ('0' == true ? 'a' : 'b') Proof that the largest eigenvalue of a stochastic matrix is $1$ Proof that $\mathrm{ker}(f^*) = \mathrm{im}(f)^{\perp} $ Where $ f: V \rightarrow V  $ is a linear operator on an inner product space $V$, I'm trying to show that $$ \mathrm{ker}(f^*) = \mathrm{im}(f)^{\perp} $$  i.e. the Kernel of the adjoint is equal to the orthogonal complement of the image of f.  I taking the inner product of $f(v)$, where $v \in V$  and an arbitary element of the kernel, $k \in \mathrm{ker}(f^*) $  $$\langle f(v), k\rangle = \langle v, f^*(k)\rangle = \langle v, 0\rangle = 0$$  And then taking the inner product of $f(v)$ with $w$, where $w \in  \mathrm{im}(f)^{\perp}$  By definition  of the orthogonal complement to the image of f,   $$ \langle f(v), w\rangle = 0 $$  My question is, now that we have   $$ \langle f(v), k\rangle = \langle f(v), w\rangle = 0 $$  What can we conclude? I would like to conclude that because $w$ and $k$ are arbitary elements of their respective subspaces, that these must be equal, i.e. $ \mathrm{ker}(f^*) = \mathrm{im}(f)^{\perp} $  However, I am not sure this is mathematically sound, is that conclusion valid or are there any technical barriers or additional statements that would have to be made to make that watertight?