Let $M$ be a smooth manifold. Given affine connection $\nabla:\Gamma(TM)\times\Gamma(TM)\to\Gamma(TM)$, I want to check that curvature tensor field $R(X,Y)Z=\nabla_X\nabla_YZ-\nabla_Y\nabla_XZ-\nabla_{[X,Y]}Z$ is $Z$  $C^\infty(M)$ linear.   So I want to check $R(X,Y)fZ=fR(X,Y)Z$. I can surely check this in local coordinates.   Q: Is there any trick I can use to simplify my checking?
For an affine connection $\nabla$, prove the curvature R  $R(X,Y,Z,\alpha)=\alpha(\nabla_X \nabla_Y Z - \nabla_Y \nabla_X Z -\nabla_{[X,Y]}Z)$  with $X,Y,Z$ vector fields and $\alpha$ a co-vector, is a tensor.  So I realise the aim is probably to show that each of the three terms in the expression for $R$ are tensors themselves as then the result will obviously follow. However, I'm not too sure how to show each of these terms are tensors. Do I need to expand out each covariant derivate to get a bunch of connection coefficients and then see some cancellations? Still though, I'm not too sure what I'll want to see after all of this so that I can say "…therefore $R$ is a tensor".  Also, very much related to the question, I'm a bit confused with the difference between  $\nabla_Y Z$ $\space \space $ and $\space \space $  $\nabla_{\mu}\omega$.  The second one I know as the covariant derivative of a 1-form, but what is meant by the covariant derivate of a vector $\textit{field}$? As in, what's the difference between the $\nabla_Y$ and $\nabla_{\mu}$?  Edit: In answering this last question, I have realised that $\nabla_YZ=Y^{\mu}\nabla_{\mu}Z^{\nu}$ and $\nabla_{\mu}\omega=\nabla_{\mu}\omega_{\nu}$.
Find a closed form of the series $\sum_{n=0}^{\infty} n^2x^n$ The question I've been given is this:  Using both sides of this equation:  $$\frac{1}{1-x} = \sum_{n=0}^{\infty}x^n$$  Find an expression for $$\sum_{n=0}^{\infty} n^2x^n$$ Then use that to find an expression for $$\sum_{n=0}^{\infty}\frac{n^2}{2^n}$$  This is as close as I've gotten:  \begin{align*} \frac{1}{1-x} &amp; = \sum_{n=0}^{\infty} x^n \\ \frac{-2}{(x-1)^3} &amp; = \frac{d^2}{dx^2} \sum_{n=0}^{\infty} x^n \\ \frac{-2}{(x-1)^3} &amp; = \sum_{n=2}^{\infty} n(n-1)x^{n-2} \\ \frac{-2x(x+1)}{(x-1)^3} &amp; = \sum_{n=0}^{\infty} n(n-1)\frac{x^n}{x}(x+1) \\ \frac{-2x(x+1)}{(x-1)^3} &amp; = \sum_{n=0}^{\infty} (n^2x + n^2 - nx - n)\frac{x^n}{x} \\ \frac{-2x(x+1)}{(x-1)^3} &amp; = \sum_{n=0}^{\infty} n^2x^n + n^2\frac{x^n}{x} - nx^n - n\frac{x^n}{x} \\ \end{align*}  Any help is appreciated, thanks :) Omission of "the" in "elected him president" and "made captain" Why is there no the before president and captain?     They elected him president.       She was made captain of the team. For a CDF that is not strictly increasing, i.e. its inverse is not defined, define the quantile function   $$F^{-1} (u) =\inf \{x: F(x) \geq u \},\quad 0&lt;u&lt;1. $$  Where U has a uniform $(0,1)$ distribution. Prove that the random variable $F^{-1} (u)$ has cdf $F(x)$.  In case of a strictly increasing CDF the proof is quite easy because the inverse is defined. Define $X=F^{-1} (u)$  $$ P\left[X&lt;x \right]=  P \left [F^{-1} (U) \leq x \right]= P \left[U \leq F(x) \right] =F(x)  $$  But how do I accomondate the nondecreasing CDF whose inverse is given by the quantile function? I am a begginer so any help is welcome. Thank you. Is there a way to unfold a mesh? Is there a way to take a mesh, cut it with seams or sharp edges and flatten it, so that the faces stay the same shape and size?  Making paper models is one aspect, where you could use that technique.  No, I'm afraid SZ0Return is not going to do it properly.  And misusing the UV-unwrap is only a last option.    Note, while it's an option to revert to Blender 2.49 which had a script to do this, I'd like a solution that I can use inside the current workspace. Wrapfigure in an enumerate environment. I want to use wrapfigure in the enumerate environment. For example,   \documentclass[11pt, a4paper]{article} \usepackage{graphicx} \usepackage{float} \usepackage{wrapfig}  \begin{document} \normalsize {\bf Multiple Choice: } \begin{enumerate}     \item \begin{wrapfigure}{r}{4.5cm}           \includegraphics[scale=0.5]{Exam1_Fig1.jpg}           \end{wrapfigure}           As shown in the figure. This rock is phaneritic and contains quartz, K-feldspar, and plagioclase in nearly equal amounts. 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I tried using System Preferences -> Shortcuts and Gestures:    but I cannot set "meta" alone as being a shortcut.  I use Kubuntu 14.10 Plasma 4 and Kubuntu 14.04 LTS x64.   (On Windows 7 SP1 x64 Ultimate I can achieve that using the program .) What's the action of banging two beers together called? Proving that $7^n(3n+1)-1$ is divisible by 9 I'm trying to prove the above result for all $n\geq1$ but after substituting in the inductive hypothesis, I end up with a result that is not quite obviously divisible by 9.   Usually with these divisibility induction problems, it falls apart nicely and we can easily factorise say a 9 if the question required us to prove that the expression is divisible by 9. However in this case, I do not end up with such a thing.  My work so far below:  Inductive Hypothesis: $7^k(3k+1)-1=9N$ where $N\in\mathbb{N}$  Inductive Step:  $7^{k+1}(3k+4)-1 \\ =7\times 7^k(3k+1+3)-1 \\ =7\times \left [ 7^k(3k+1)+3\times 7^k \right ] -1 \\ = 7 \times \left [ 9N+1 + 3 \times 7^k \right ] -1 \\ = 63N+21\times 7^k+6 \\ = 3 \left [ 21N+7^{k+1}+2 \right ]$   So now I need to somehow prove that $21N+7^{k+1}+2$ is divisible by 3, but I'm not quite sure how to proceed from here... How to get the day, year, hours, min Individually from date format "yyyy-MM-dd'T'HH:mm:ss.SSSZ"? Does "FN" stand for or mean anything? Simulating group_concat MySQL function in Microsoft SQL Server 2005? Why is the electric field zero inside a hollow conducting sphere? How To Get Notified When Hard Drives Fail In Dell PowerEdge Servers I have a Dell PowerEdge R415 and R515 and wondering how I can be alerted remotely when a hard drive fails in either server? Both are running RAID, so I need to be notified, so I can swap out the drive. 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Why do some sites need less reputation to be a trusted user? Array of zero length