According to my lecture notes, the normal vector to a surface $f$ is given by $\text{grad}(f) = \underline{\nabla} f$. However, surely the normal vector to the surface $f$ should be perpendicular to the vector $\underline{\nabla}f$?
If we have an implicit surface $g(x, y, z)=...$, the gradient is a normal vector. But why? Do I have to start with tangent vectors and then show the gradient is perpendicular? Not sure how to approach this.
My Realtek NIC in my Windows Home Server (essentially w2k3) box refuses to run at 1000Mb/s even if I change the driver settings from Auto Negotion to 1000Mb/s full duplex. I've installed the latest driver and tried changing the speed setting but I'm having no luck. Not sure what to try next. Any ideas?  I have a Netgear GS605 v2 Switch supports 10/100/1000 D Link DIR 655 router.  All my machines bar one/2 have 1000Mb/s NICs (Old laptop/Xbox being the exceptions)  The reason I noticed the problem was because I was copying files between two machines and maxing one NIC out and only seeing 10% on the other machine. I do have a fairly lengthy cable between the switch and the router (maybe 20m) could that be the issue? Compute: $\lim_{x\to 0}\frac{\sqrt[m]{\cos(\alpha x)}-\sqrt[n]{\cos{(\beta x)}}}{x^2},\;m,n\in\mathbb N,\alpha,\beta\in\mathbb R$ Compute: $$\lim_{x\to 0}\frac{\sqrt[m]{\cos(\alpha  x)}-\sqrt[n]{\cos{(\beta x)}}}{x^2},\;m,n\in\mathbb N,\alpha,\beta\in\mathbb R$$   My attempt:  $$L=\lim_{x\to 0}\frac{\sqrt[m]{\cos(\alpha x)}-\sqrt[n]{\cos{(\beta x)}}}{x^2}=\lim_{x\to 0}\frac{\sqrt[m]{\cos(\alpha x)}-1+1-\sqrt[n]{\cos(\beta x)}}{x^2}\\\displaystyle=\lim_{x\to 0}\frac{\left(1+(\cos(\alpha x)-1)\right)^{\frac{1}{m}}-1-\left(\left(1+(\cos(\beta x)-1)\right)^{\frac{1}{n}}-1\right)}{x^2}$$  Transformed expression: $$-\frac{(1+(\cos(\alpha x)-1))^{\frac{1}{m}}-1}{\cos(\alpha x)-1}\cdot\frac{1-\cos(\alpha x)}{\alpha^2x^2}\alpha^2+\frac{(1+(\cos(\beta x)-1))^{\frac{1}{n}}-1}{\cos(\beta x)-1}\cdot\frac{1-\cos(\beta x)}{\beta^2x^2}\beta^2$$  I applied the standard limits from the table: $$\lim_{x\to 0}\frac{(1+x)^a-1}{x}=a\;\&amp;\;\lim_{x\to 0}\frac{1-\cos x}{x^2}=\frac{1}{2}$$ $$L=-\frac{1}{m}\cdot\frac{1}{2}\alpha^2+\frac{1}{n}\cdot\frac{1}{2}\beta^2=\frac{m\beta^2-n\alpha^2}{2mn}$$ Is this correct? GIS Point or discrete longitude and latitude columns? Conditional Rendering inside Visualforce pages I have a server running with the timezone set to UTC. It seemed like that was generally a good practice (please correct me if I'm wrong).  Anyhow, one of the servers I connect to, in order to scp files, is running on EDT and stores files that I need to copy in the format /path/to/filename/data20120913  I looked at trying to rsync files using something like find's -mtime -1 flag for files modified in the last day, but I didn't have any luck.  I don't mind just using scp to copy the current day's file, but as of right now there is a 4-hour window where running date +%Y%m%d will give a different day on each server and that bugs me a little.  Looking through man date I see that I can have the time output as UTC, but I don't see a way to have it output as another timezone like EDT  I suppose I could also use something like the GNU date extension date -d 20100909 +%s to get the date in seconds from the epoch, apply a manual 4 * 60 * 60 second calculation, and see about rendering that as a date - but then when daylight time kicks in it will still be an hour off.  Is there a simpler way to output the date in a YYYYMMDD format for EDT on a server that is set to UTC ? Two different solutions to integral Orthogonal basis for infinite-dimensional vector spaces Least rank of a matrix Let $A$ be a $7\times 7$ matrix such that it has rank $3$ and $a$ be $7\times 1$ column vector.  Then least possible rank of $A+(a a^T)$ is? ($a^T$ is transpose of the vector)  Intuitively I think it's $2$ as $a a^T$ has rank $1$, and the most it could affect is one row of $A$. Functional equations leading to sine and cosine Convergence of $\sum_{n=1}^\infty{\left(\sqrt[n]{n}-1\right)}$ What does the 'cosmological constant' represent? Newtons theory of gravity involves a gravitational constant $G$, however one does not refer to it directly, we speak instead of gravity or the force of gravity.   Now Einsteins introduced a  in his equation for GR, and ever since it's been referred to as such; however, this usage seems strange to me, given the above parallel; it seems to me we ought to be referring to something physical, which can then be quantified and whose quantification involves the cosmological constant.   Does the cosmological constant then refer to a cosmological force? A force that is, contra gravity, repulsive? And which is even weaker than gravity, kicking in on the scale of galaxies rather than solar systems? What is "angle of view" in photography? I am looking for a package to create bullet gauge graphs, but I couldn't find any.  A bullet graph looks like this:    and an ample description/specification can be found   It is available in Mathematica 9 ()  Is it possible create this kind of graphs in a easy way with TeX (like Matehmatica 9 or similar)? I am using XeLaTeX and memoir to write a longer text with several parts, chapters and sections. When printing the \tableofcontents I get a spacing problem with 2-figure numbers almost overlapping the relevant section name, as per enclosed picture.  I am aware that the memoir manual has a section dedicated to this theme, but the most part of it is doubledutch to me. Is there any easy way to avoid this problem? Evaluate $\sum_{n=1}^\infty nx^{n-1}$ How can you evaluate $\sum_{n=1}^\infty nx^{n-1} = \frac{1}{(1-x)^2}$ without relying on the fact that it's the derivative of $\sum_{n=1}^\infty x^n = \frac{1}{1-x} $? Get two letter country code in Apex method I want to write an Apex method that takes a country name as input and returns the equivalent two letter country code (ISO A2). Is there a way to write it without explicitly hard coding all countries and code in the method? $\Delta U$, $C_p$, $C_v$ for an ideal gas process How can I get multiple stacked plots (ybar stacked, stack plots=y) in one diagram, such that there are multiple stacked bars for the same x-coordinate, each slightly shifted? Is there a way to reset pgfplot's zero levels after an \addplot command, such that the next \addplot[xhshift=...] doesn't add something to the current stack, but rather starts a fresh stack?  FWIW, overlaying two plots with scale only axis won't help; I need this to work with groupplot Backup of clients' directory to Windows Server 2008 We have a machine running Windows Server 2008, and we need to take backup of folder from a desktop on LAN. This folder is not shared.   Is there any tool or software to do this, or can a scheduled task be made which should automatically take the backup of that folder date-wise &amp; make a copy on server? Ie can we do it through Schedule Tasks option ?  Desktop systems have Windows Vista/Windows 7. How to make a domain receiving email behind cloudflare?