Let $\omega \in \Omega^k(M)$ and $X_i \in \Gamma(TM)$. In Spivak Volume I page 213 the exterior derivative is given invariantly by the formula \begin{align*}d \omega(X_0, \ldots, X_k) = &amp; \sum_i (-1)^i X_i . (\omega(X_0, \ldots, \hat{X_i}, \ldots, X_k)) \\ &amp;+ \sum_{i&lt;j} (-1)^{i+j} \omega([X_i, X_j], X_0, \ldots, \hat{X_i}, \ldots, \hat{X_j}, \ldots, X_k) \\ \end{align*} whereas in Kobayashi and Nomizu Volume I page 26 the exterior derivative is given by almost the same formula, but with a factor of $\frac{1}{k+1}$ in front of the sums: \begin{align*}d \omega(X_0, \ldots, X_k) = &amp; \frac{1}{k+1}\sum_i (-1)^i X_i . (\omega(X_0, \ldots, \hat{X_i}, \ldots, X_k)) \\ &amp; + \frac{1}{k+1} \sum_{i&lt;j} (-1)^{i+j} \omega([X_i, X_j], X_0, \ldots, \hat{X_i}, \ldots, \hat{X_j}, \ldots, X_k) \\ \end{align*} I'm confused about where this extra factor is coming from. Even if Spivak and Kobayashi and Nomizu use different conventions for the wedge product, shouldn't the invariant formula for the wedge product be the same?
In Helgason, "Differential Geometry, Lie Groups and Symmetric Spaces"  we have the formula for the exterior derivative  $(p+1)d\omega(X_{1}, \ldots X_{p+1})=\Sigma_{i=1}^{p+1} (-1)^{i+1} X_{i} \cdot \omega(X_{1}, \ldots , \hat{X_{i}}, \ldots X_{p+1})+\Sigma_{i&lt;j} (-1)^{i+j}\omega([X_{i}, X_{j}], X_{1}, \ldots , \hat{X_{i}}, \ldots \hat{X_{j}}, \ldots , X_{p+1})$  but in other sources on-line they have the same formula without the factor of $p+1$ on the left-hand side.  Are there two different definitions of the exterior derivative, or is there an error in Helgason?
What are the light blue lines on my imported objects One of my columns is called from. I can't change the name because I didn't make it. Am I allowed to do something like SELECT from FROM TableName or is there a special syntax to avoid the SQL Server being confused? Say my opponent plays a planeswalker during their first main phase.  I have a  or some other effect that will kill it after it is already in play, but before they use any of its abilities.  I could respond to them using a +1 ability, but I want to kill it before that effect goes on the stack.  In most casual situations, it would basically come down to whether I say "Hero's Downfall" first, or they say "+1" first.  But I can't imagine that's how it's supposed to work.  Is there a time between when the planeswalker hits the battlefield and when they activate its abilities that they must let me respond? It seems to me that situations like this arise frequently, and they are resolved simply by who speaks up first, which is horrible for such a rules-oriented game. How to show loading spinner in jQuery? In Prototype I can show a "loading..." image with this code:  var myAjax = new Ajax.Request( url, {method: 'get', parameters: pars,  onLoading: showLoad, onComplete: showResponse} );  function showLoad () {     ... }   In jQuery, I can load a server page into an element with this:  $('#message').load('index.php?pg=ajaxFlashcard');   but how do I attach a loading spinner to this command as I did in Prototype? How do I pipe colored diff output to less? Mental estimate for tangent of an angle (from $0$ to $90$ degrees) Does anyone know of a way to estimate the tangent of an angle in their head? Accuracy is not critically important, but within $5%$ percent would probably be good, 10% may be acceptable.  I can estimate sines and cosines quite well, but I consider division of/by arbitrary values to be too complex for this task. Multiplication of a few values is generally acceptable, and addition and subtraction are fine.  My angles are in degrees, and I prefer not have to mentally convert to radians, though I can if necessary. Also, all angles I'm concerned with are in the range of [0, 90 degrees].  I am also interested in estimating arc tangent under the same conditions, to within about 5-degrees would be good.  Backstory  I'm working on estimating the path of the sun across the sky. I can estimate the declination pretty easily, but now I want to estimate the amount of daylight on any given day and latitude. I've got it down to the arc cosine of the product of two tangents, but resolving the two tangents is now my sticking point. I also want to calculate the altitude of the sun for any time of day, day of the year, and latitude, which I have down to just an arc tangent. What is "it" in the following sentence: It is clear that Bob likes doughnuts A great Python IDE for Ubuntu Replace Line Breaks in a String C# Is there a streaming API for JSON? Is DOM the only way to parse JSON? If $R$ is an infinite ring, then $R$ has either infinitely many zero divisors, or no zero divisors Please help me to prove that if $R$ is an infinite ring, then $R$ has either an infinite number of zero divisors, or it has no zero divisors. Let $t\in \mathbb R$. Prove that $\displaystyle{\sum_{n=1}^{\infty}n^{3}\sin\left(\frac{1}{n}\right)-n^{2}+t}$ converges iff $t=\frac{1}{6}$.  While the $(\Rightarrow)$ direction was fairly easy (as $a_n$ should converge to $0$, forcing $t=\frac{1}{6}$), i'm facing some hard time trying to prove the second direction, that is $\displaystyle{\sum_{n=1}^{\infty}(n^{3}\sin\left(\frac{1}{n}\right)-n^{2}+\frac{1}{6})}$ converges. I did notice that $a_n=a_{-n}$ and tried to write this series as a Leibnitz series, but had no luck since the sequence doesn't decrease. I also tried to prove absolute convergence but then again, no luck. Thanks. I cannot see WiFi option in my Ubuntu 15.04. What should I do? Revert asap  lspci -nnk | grep 0280 -A2 output:-  03:00.0 Network controller [0280]: Qualcomm Atheros Device [168c:0042] (rev 30) Subsystem: Foxconn International, Inc. Device [105b:e09a]   rfkill list output:-  0: hci0: Bluetooth Soft blocked: no Hard blocked: no 1: acer-wireless: Wireless LAN Soft blocked: no Hard blocked: no      dmesg | grep ath [    9.545199] ath10k_pci 0000:03:00.0: pci irq msi-x interrupts 8 irq_mode 0 reset_mode 0 [   10.029684] ath10k_pci 0000:03:00.0: Direct firmware load for ath10k/cal-pci-0000:03:00.0.bin failed with error -2 [   10.029701] ath10k_pci 0000:03:00.0: Direct firmware load for ath10k/QCA9377/hw1.0/firmware-5.bin failed with error -2 [   10.029704] ath10k_pci 0000:03:00.0: could not fetch firmware file 'ath10k/QCA9377/hw1.0/firmware-5.bin': -2 [   10.029711] ath10k_pci 0000:03:00.0: Direct firmware load for ath10k/QCA9377/hw1.0/firmware-4.bin failed with error -2 [   10.029713] ath10k_pci 0000:03:00.0: could not fetch firmware file 'ath10k/QCA9377/hw1.0/firmware-4.bin': -2 [   10.029719] ath10k_pci 0000:03:00.0: Direct firmware load for ath10k/QCA9377/hw1.0/firmware-3.bin failed with error -2 [   10.029721] ath10k_pci 0000:03:00.0: could not fetch firmware file 'ath10k/QCA9377/hw1.0/firmware-3.bin': -2 [   10.029726] ath10k_pci 0000:03:00.0: Direct firmware load for ath10k/QCA9377/hw1.0/firmware-2.bin failed with error -2 [   10.029728] ath10k_pci 0000:03:00.0: could not fetch firmware file 'ath10k/QCA9377/hw1.0/firmware-2.bin': -2 [   10.029733] ath10k_pci 0000:03:00.0: Direct firmware load for ath10k/QCA9377/hw1.0/firmware.bin failed with error -2 [   10.029735] ath10k_pci 0000:03:00.0: could not fetch firmware (-2) [   10.029737] ath10k_pci 0000:03:00.0: could not fetch firmware files (-2) [   10.029739] ath10k_pci 0000:03:00.0: could not probe fw (-2) Can we say that $\frac{0}{0}$ is every number? Installing windows to dual-boot I'm looking to install 32-bit windows with IIS to dual-boot with my 64-bit edition of windows on a laptop. I have a ten gig partition on my harddrive - is this large enough for a simple Windows install and a few programs?   I've never dual booted before, so I'm not quite sure how easy it is to mess up the process. Can you recommend any tutorials on how to set up a dual-boot configuration on Windows? I just started in Griffith's Introduction to electrodynamics and I stumbled upon the divergence of $\frac{ \hat r}{r^2} \equiv \frac{{\bf r}}{r^3}$, now from the book, Griffiths says:  Now what is the paradox, exactly? Ignoring any physical intuition behind this (point charge at the origin) how are we supposed to believe that the source of $\vec v$ is concentrated at the origin mathematically? Or are we forced to believe that because there was a contradiction with the divergence theorem? Also how would the situation differ if $\vec v$ was the same vector function but not for a point charge? Or is it impossible? This is my code:  import datetime today = datetime.date.today() print(today)   This prints: 2008-11-22 which is exactly what I want.  But, I have a list I'm appending this to and then suddenly everything goes "wonky". Here is the code:  import datetime mylist = [] today = datetime.date.today() mylist.append(today) print(mylist)   This prints the following:   [datetime.date(2008, 11, 22)]   How can I get just a simple date like 2008-11-22? {5,15,25,35} is a group under multiplication mod 40 Show that the set $\{5,15,25,35\}$ is a group under multiplication modulo $40$. What is the identity element of this group. Can you see any relationship between this group and $\mathbb U(8)$?   I am very stuck on this question and I think my knowledge of abstract algebra is my liability at the moment. Right now I'm most confused by the fact that I learned: An element of a group of this nature must be relatively prime with the working mod.  Obviously here this is not the case, so I don't really know where to begin.  Do I need a Cayley Table?  Thanks. Does a multiple-entry Schengen visa allow visiting other Schengen countries later, without going via the issuing country? My wife was just issued a multiple entry visa by France, which is good for one year.  Our first trip will be to France, the issuing country.  My question is: after we leave the EU and return, must future trips also start in France? Or may we go directly to (say) Greece from the US on a later visit? I continue to find it amazing that something as “bulky” and macroscopic as a static magnetic or electric field is actually a manifestation of virtual photons.  So putting on your QFT spectacles, look closely at the space near the pole of a powerful magnet – virtual photons! Now look between the plates of a charged capacitor – virtual photons again!   But if it’s all virtual photons, how do we get the difference between a magnetic and electric field?  [This question needs the tag "magnetostatics" as well].