Yes, there are many threads seeking book recommendation on multivariable calculus. Me personally enjoyed "Differential and Integral calculus - Piskunov" and looking something similar in writing on multivariable calculus at any level (intermediate to advance)  Any recommendation will be appreciated.
Due to my ignorance, I find that most of the references for mathematical analysis (real analysis or advanced calculus) I have read do not talk much about the "multivariate calculus". After dealing with the single variable calculus theoretically, it usually directly goes to the topic of measure theory.    After reading the wiki article "", I'd like to find the rigorous proof for this test. The first book comes to my mind is Courant's Introduction to Calculus and Analysis which includes the multivariate case in the second volume.   Motivated by this, I'd like to put the question here:     What are the usual references for the   theoretical treatment for   ?
Clean install of Windows 10 OEM: license key issues? How to set static IP address? I have configured network adapter of guest Ubuntu to work in a bridged mode so I could ssh to it from the host machine.   The problem is that the guest Ubuntu ip address keep changing.   Even though it changes  within a very short range 192.168.0.4-10 it is still takes time to configure Putty and other programs each time.  Is there a way to make the guest Ubuntu ip address to be static?  My guest OS has the full GUI. Are upvotes on competing posts counted per question or per answer for Sportsmanship? The definition of the :     Up voted 100 competing answers   doesn't specify whether each upvote on competing answers to the same question counts towards this criteria, or if it's limited to one "point" per question no matter how many additional competing answers you upvote. Can someone explain how it works? Let $E$ be a finite dimensional vector space over field $\mathbb R$ with $E^*$ as the dual space of $E$.($\dim E=n$)   $\Omega^P(E):=\{ \alpha\colon \overbrace{E\times\cdots\times E}^{p- times}\rightarrow \mathbb R\  \ , \alpha \text{ is alternating multilinear map}\}$ $\Omega^P(E^*):=\{ u\colon \overbrace{E^*\times\cdots\times E^*}^{p- times}\rightarrow \mathbb R\  \ , u \text{ is alternating multilinear map}\}$  I wish to prove that there exist an $\textbf{unique}$ bilinear form $B\colon \Omega^p(E^*)\times\Omega^p(E)\rightarrow\mathbb R$   such that for all $ f_i\in E^* \ , u_j\in E^{**}\simeq E \qquad  i,j=1,\cdots,p$, it has the following rule:  $B(u_1\wedge\cdots\wedge u_p,f_1\wedge\cdots\wedge f_p)=\det[f_i(u_j)]$.  $\wedge$ is wedge product between alternating maps i.e. $\alpha\wedge\beta(v_1,\cdots,v_{p+p})=\dfrac{1}{p! p!}\displaystyle\sum_{\sigma\in S_{p+p}}sign(\sigma)\alpha(v_{\sigma(1)},\cdots,v_{\sigma(p)})\beta(v_{\sigma(p+1)},\cdots,v_{\sigma(p+p)})$  How can I do this? Can I disable full-disk encryption? What is the concept of erasure in generics in Java? What is the concept of erasure in generics in Java? The more I read about this, the more confused I become. Illuminance, as I understand it,  measured in lux, is the perceived brightness of light on a surface. It's what you'd measure with an incident light sensor. Luminance, then, is how much reflected light you'd perceive looking at that surface. This is what the light meter in a camera measures. Am I correct so far?  In both cases, the word "perceived" is important, because the scales are weighted using a  to map wavelengths to . However, in the case of illuminance, there's only actual perception if your eye happens to be the surface in question.  I can basically grasp this, but then I come across charts saying things like "family living room: 50 lux". Wait, hold on! Does that really mean that the typical lights in a house are that bright, or is it just confused and wrong, or am I confused and wrong?  If you're not taking pictures of light sources directly, why would an incident light meter reading ever be useful in photography? Reflected light recorded on film or by a sensor is usually what makes a photograph. So, if I have an incident light meter, how does that reading relate meaningfully to my camera settings?  Since incident light meters are sold and used, this implies that there must be some useful conversion. But this is where my brain explodes. Google tells me that 1 lumen equals 1 candela, so therefore 1 lumen/m² (i.e. 1 lux) must equal 1 candela/m2 (i.e. 1 nit). But clearly something is missing from this. There's something called a "steradian". Cones are involved. I'd never heard of this before; how does it fit in? I can see how it might be determined when calculating the usefulness of LED home lighting, but for a photo, I'm at a loss.  Some part of my (exploded, from the last paragraph) brain is trying to relate this to the difference between flash metering with TTL and with an incident meter. But how can the incident meter work without knowing the reflectance properties of the objects in the scene? Is this what the "C" in the  is all about? Is it just an average-this-probably-will-work value, or is there more to it than that? And if it is just an average, what knowledge is necessary for compensating for off-averages scenes? (As with the , where the photographer simply judges if the scene should be rendered brighter or darker than the average given by the meter.)  So yeah, so much confusion. In short:   What's the difference? Can one meaningfully convert between the two? When and how are illuminance / incident light measurements useful for photography?     Update: I appreciate Stan's answer, which covers the third point of when and how pretty well. And I think I've basically got the first point figured out, as described above. But I'd appreciate some answers covering the issue of conversion as well, both in the mathematical abstract and as the practical for photography. And I wouldn't mind more why and how, either. Why would layout_width="wrap_content" take up the entire width of the screen? How do I split a string with multiple separators in JavaScript? I'm trying to split on both commas and spaces, but AFAIK JavaScript's split() function only supports one separator. How to get a list of all the labels in a LaTeX document in WinEdt I am writing a large document using LaTeX and WinEdt.  I have labeled each chapter, section and so on.  I would like to print out a list of just the labels (so that I don't duplicate one later on, and so that I have it handy in order to write \refs).    Is there a way to do this? Why does my hard drive LED light blink every second? The light blinks faintly about once a second.   It is slowly driving me insane.  How do I find out which program is causing this, so I can disable it?  Acer 5274 laptop/core i3 Windows 7 Home Premium Hitachi HTS545050B9A300 hard drive Repairing corrupt MP4 I took a video from my Android phone (Sony Xperia P, if you need to know), and before I could stop recording it, the phone memory got full and the phone got hanged too. I rebooted the phone, the video was there, but I couldn't play it from any media player on Android. It's in mp4 format.    So I tried to open it on PC and Mac with different video player, like VLC, quick Time, Windows Media Player etc., but none of them can play it. I know that the video is corrupted, but I believe it's repairable, since it got more than a GB in size. Is there anything that I can try  to repair the video file? The video is really important. Gosper's unusual formula connecting $e$ and $\pi$ I have a Lubuntu (now 14.04) installation. Recently, I logged in to find that the Wi-Fi icon had disappeared, and there was some sort of a keyboard icon in it's place. I have restarted multiple times, but it doesn't fix it. I can't find the same applet in the panel settings, and resetting the panel didn't help. When I try to manually start nm-applet, it starts, but prints a bunch of warnings to the console:  nm-applet-Message: using fallback from indicator to GtkStatusIcon   It then starts, using ugly icons for the Wi-Fi connectivity symbol.  What do I need to do to make it start correctly? Once it starts, why isn't it using the right icons? Prove by Double Counting Method $\sum\limits_{k = 0}^m \binom{m}{k}\binom{n}{r + k} = \binom{m + n}{m + r}$ Show by combinatorial arguments that:   $$\sum_{k = 0}^m \binom{m}{k}\binom{n}{r + k} = \binom{m + n}{m + r}$$  Hello ,  I want to prove this argument by double counting method, i kinda have an idea of proving by comb args. but not by double counting... Can someone give any clues or ideas? thanks :) Confidence bands for QQ line This question doesn't specifically pertain to R, but I   chose to use R to illustrate it.  Consider the code for producing confidence bands round a (normal) qq-line:  library(car) library(MASS) b0&lt;-lm(deaths~.,data=road) qqPlot(b0$resid,pch=16,line="robust")   I'm looking for an explanation of (or alternative a link to a paper/online document explaining) how these confidence bands are constructed (I have  seen a reference to Fox 2002 in the help files of R  but sadly I don't have this book handy).   My question will be made more precise with an example.  Here's how R computes these particular CI's (I've  shortened/simplified the code used in car::qqPlot)  x&lt;-b0$resid good&lt;-!is.na(x) ord&lt;-order(x[good]) ord.x&lt;-x[good][ord] n&lt;-length(ord.x) P&lt;-ppoints(n) z&lt;-qnorm(P) plot(z,ord.x,type="n") coef&lt;-coef(rlm(ord.x~z)) a&lt;-coef[1] b&lt;-coef[2] abline(a,b,col="red",lwd=2) conf&lt;-0.95 zz&lt;-qnorm(1-(1-conf)/2) SE&lt;-(b/dnorm(z))*sqrt(P*(1-P)/n)     #[WHY?] fit.value&lt;-a+b*z upper&lt;-fit.value+zz*SE lower&lt;-fit.value-zz*SE lines(z,upper,lty=2,lwd=2,col="red") lines(z,lower,lty=2,lwd=2,col="red")   The question is: what is the justification for  the formula used to compute these SE (e.g. the line SE&lt;-(b/dnorm(z))*sqrt(P*(1-P)/n)).   FWIW this formula is very different from the formula of the usual How to build Blender as a Python module? I have seen this question now many times and every question here and on other forum points to  experimental tutorial by . Now sadly a lot of answers are invalid right now because this central link is down. I couldn't find anything by Googling so if anybody has a fork of the tutorial or a second source that would be great? The link can't be gone for too long, I saw an answer with it 4 months ago.  The only other thing I have found is , but it also references the now missing tutorial. Any ideas how to find it? Is the space $\mathbb{N}$ is a baire spaces ? yes/No Is  the  space $\mathbb{N}$  is a  baire  spaces ?  My attempts :   No ,  It  will not form a  baires  spaces ,  let $C_n= \mathbb{N}\setminus\{n\}$. This is open (since $\{n\}$ is closed in $\mathbb{R}$, hence in the induced topology of $\mathbb{N}$) and dense, since every open ball with center in $\mathbb{n}$ intersects ${C_n}$. But  $$\bigcap_{n\in\mathbb{N}} {C}_n = \emptyset$$  So $\mathbb{N}$ is not a baires  spaces because for  Baire Space the intersection of a countable family of open dense sets must be dense.   Is  Its correct or not ? A photon emitted from a receding source (Doppler redshift) has less energy when detected at an observer's location. Please explain the energy loss from the perspective of energy conservation. I have two integers, x and y so that x &lt; y. How many prime numbers are there between x and y (exclusive). Is there a formula or algorithm to compute?