Which textbook of differerntial geometry will have these formulas about conformal transformation? $$\tilde g_{ij} = e^{2\varphi}g_{ij}$$  $$\tilde \Gamma^k{}_{ij} = \Gamma^k{}_{ij}+ \delta^k_i\partial_j\varphi + \delta^k_j\partial_i\varphi-g_{ij}\nabla^k\varphi $$ $$\tilde R_{ijkl} = e^{2\varphi}\left( R_{ijkl} - \left[ g {~\wedge\!\!\!\!\!\!\bigcirc~} \left( \nabla\partial\varphi - \partial\varphi\partial\varphi + \frac{1}{2}\|\nabla\varphi\|^2g    \right)\right]_{ijkl}  \right)$$ $$\tilde R = e^{-2\varphi}\left[R + \frac{4(n-1)}{(n-2)}e^{-(n-2)\varphi/2}\triangle\left( e^{(n-2)\varphi/2} \right) \right] $$  I've read many textbooks about differential geometry, such as Do Carmo, Kobayshi, Novikov and so on. But I never found these formulas. Who can give me a reference about these formulas.
I have a background in QFT, GR and differential geometry at the level of a master student in theoretical physics. I would like to touch the area of CFT. I know the textbook of Philippe Di Francesco. It may be too big for the beginner like me. Are there some good introductory lectures or textbooks adapted to the needs of beginners?
How do you choose Spells Known when creating a new character above 1st level? Does more reflective aluminum foil make a room cooler compared to less reflective foil? Aluminum foil is said to be not absorbing light at all. It reflects light. So, does it mean that a more shiny aluminum foil will reflect more light and thus make the room more cooler as compared to less shiny foil?  If yes, then where does unreflected light go in case of less shiny aluminum foil?  Does less shiny and more shiny actually make a difference? Why?  In order to keep whole room cool will it help if I stick the foil to the outside of wooden door (exposed to the sunlight) too? What are valid values for "data defined" labeling settings? Is there a converter that will convert LaTeX to HTML? I have tried several services but the LaTeX diagrams and some of the maths came out weird.  I actually converted the Latex to PDF without a problem. Then I used  to convert from PDF to HTML. The square root signs did not appear properly and some of the Cartesian coordinate plane diagrams had nothing on them except the grid.   \documentclass[12pt,a4paper]{report} \usepackage{amssymb} \usepackage{amsthm} \usepackage{amsmath} \usepackage{pstricks,pst-plot} \usepackage{pst-eucl} \addtolength{\hoffset}{0.2cm} \addtolength{\textwidth}{1.1cm} \addtolength{\textheight}{1.0cm} \addtolength{\voffset}{-0.7cm} \linespread{1.3} \begin{document}  If $x$ is non-positive then $x \leq -\sqrt{\frac{1}{2}}$     $\hspace{30 mm} $     (6)\\  (4) and (6) tell us that\\  (4)  $-1 \leq x \leq 0$  \\  (6)  If $x$ is non-positive then $x \leq -\sqrt{\frac{1}{2}}$\\  Therefore $-1 \leq x \leq -\sqrt{\frac{1}{2}}$\\  \vspace{50 mm}  \begin{align*} \psgrid[unit=2cm](0,0)(-2,-2)(2,2) \pnode(-1.8,.87){b}\uput[50](b){b} \pnode(-1.8,1.8){a}\uput[50](a){a} \psset{linecolor=green} \psline[linewidth=1pt](-1.8, 0)(-1.8,4) \psset{linecolor=green} \psline[linewidth=1pt](-1.8, 0)(-1.8,-4) \psset{linecolor=blue} \psellipticarc(0,0)(2,2){0}{180} \psset{linecolor=red} \psline[linewidth=1pt,linestyle=dashed,dash=2pt](-2,2)(-1.41,1.41) \psset{linecolor=red} \psline[linewidth=1pt](-1.41,1.41) \psset{linecolor=red} \psline[linewidth=1pt]{-&gt;}(4,-4) \psset{linecolor=red} \psline[linewidth=1pt]{-&gt;}(-2,2)(-4,4) \end{align*}  \end{document}   PDF output:    PDF->HTML output: Why did Palpatine wait so long before giving Order 66? C# Unsafe/Fixed Code Proof of $\pi$ not being a quadratic irrational number. Does someone know a proof (books, articles) that $\pi$ is not a quadratic irrational?   The proof should not use that $\pi$ is transcendental.  Any hints would be appreciated. A particle experiences an acceleration described by  $$ a=kx^{-2} $$ where x is the displacement from the origin and k is an arbitrary constant.  To what value does the velocity v of the particle converge to as x approaches infinity if the particle starts at some point x0?  If I approach this problem with energy, then $$ W = \int F \mathrm{d}x $$ $$  = \int_{x_0}^\infty mkx^{-2} \mathrm{d}x $$ $$  = mk(-\infty^{-1}+{x_0}^{-1}) $$ $$ W = K = mk{x_0}^{-1} $$ $$ \frac{1}{2}mv^2 = mk{x_0}^{-1} $$ $$ v = \sqrt{2k{x_0}^{-1}} $$  How would I solve this problem with pure kinematics? (there appears to be some sort of cyclical dependency where acceleration affects velocity, velocity affects displacement, and displacement affects acceleration)  Likewise, two particles experience accelerations described by $$ a_1=k_1x^{-2} $$ $$and $$ $$ a_2=-k_2x^{-2} $$ where x is the distance between the two particles  What two velocities do the particles reach as x approaches infinity if the two particles are initially separated by some x0? Show $\,p\mid 9^q -1 \implies q\mid p-1$ for odd primes $p$ and $q$ If $p$ and $q$ are odd primes show that $$p|(9^q -1) \implies q|(p-1).$$ ....  I am having difficulty to prove this.  Yes that was the question I have misunderstood between 9 and q!!! how do you make commands not show up on the chat I am making a server, and I put a comparitor clock down to constantly have clear weather. How do I silence the commands? What is this type of flexible contact strip called? The LCD below is from a cheap digital clock which was (obviously) damaged (broken glass).    The electrical connection between the PCB and the display was realized with this pink strip.    Its flexible...    ...and consists of alternating rows of conductive and insulating material between the pink boarders.    [above image was altered in contrast/brightness to highlight the alternating strips]  What is this kind of connector called? Is it only used for displays?   I am pretty impressed that the connection is reliable just from the very slight pressure between the strip and the PCB. Also, its easily removable from the display. There seems to be some conductive layer on the glass, from where the signal goes into the displays internals. This connection seems to be damaged for the three rightmost signals, where the glass is broken. A childish question (literally) -    My 8 year old asked me this morning: "Dad, what does 'physical' mean?" - and I found myself at loss for an ordinary language answer.   Every answer I could come up with ended up sounding like a definition of empiricism: "physical things are those things that can be observed", "physical means something that can be measured", "whatever can be verified and tested", etc....   Claiming that things that are physical are those things which are material or things that are described by physics seems circular.   Google:     phys·i·cal   ˈfizik(ə)l/   adjective   1.   relating to the body as opposed to the mind.   "a whole range of physical and mental challenges"   synonyms:   bodily, corporeal, corporal, somatic; More   2.   relating to things perceived through the senses as opposed to the mind; tangible or concrete.   "pleasant physical environments"   synonyms:   material, concrete, tangible, palpable, solid, substantial, real, actual, visible   "the physical universe"   And various dictionaries (here Merriam-Webster):      Definition of physical   1   a :  of or relating to natural science   b (1) :  of or relating to physics (2) :  characterized or produced by the forces and operations of physics   2   a :  having material existence :  perceptible especially through the senses and subject to the laws of nature   everything physical is measurable by weight, motion, and resistance — Thomas De Quincey   b :  of or relating to material things     lead to similar definitions.   How can one define "physical" in ordinary language terms? Is there a way of defining "physical" that doesn't presuppose empiricism? What is the rationalist definition of 'physical'?   Is there a definition that doesn't commit to either dualism or empiricism or to any epistemology? Upgrade to the specific php 7.1/7.2 from php 7.0 in Ubuntu 16.04 How can I upgrade to php 7.1 from php 7.0 in ubuntu 16.04 ?  If I do a php -v I get this message:  PHP 7.0.8-0ubuntu0.16.04.3 (cli) ( NTS ) Copyright (c) 1997-2016 The PHP Group Zend Engine v3.0.0, Copyright (c) 1998-2016 Zend Technologies     with Zend OPcache v7.0.8-0ubuntu0.16.04.3, Copyright (c) 1999-2016, by Zend Technologies Every time I go to install a new app, I get the message "Insufficient storage", and an unable to install anything!  I am using Samsung GT19060 my internal memory is 8GB. I have not downloaded anything but anytime I try to download something it will tell me low memory space. "In here", "from here", and "at here" I just read the discussed topic "look here vs. look at here":  It's got me wondering.  What is the reason for not using the preposition at before here?  Is it because here is an adverb and it is wrong to use a preposition before an adverb?  What if we use here with its nominal meaning, as in "get away from here", or "It's really hot in here", and then say "Look at here" meaning "Look at this place"?    If it is wrong and there is some other reason why at here is incorrect but in here and from here are correct, what is it? So, I have proven already that every open set in $\mathbb{R}$ can be written as a countable union of disjoint open intervals; i.e., that $\mathcal{O} = \bigcup_{i=1}^\infty (a_{x_i}, b_{x_i})$.  Now, consider the closed interval $\left [ a + \frac{1}{n}, b - \frac{1}{n}\right ] \subset (a, b)$.  Since the midpoint of $(a,b)$ is $\frac{b-a}{2}$, we need $\displaystyle n &gt; \frac{2}{b-a}$.  Otherwise, $\displaystyle a + \frac{b-a}{2} = a + \frac{1}{ \frac{2}{b-a}} = b - \frac{1}{ \frac{2}{b-a}} = b - \frac{b-a}{2}$, and our interval will be degenerate.  Note that $\forall (a,b) \in \mathcal{O}$,   $\displaystyle \bigcup_{n = \frac{b-a}{2}+1}^{\infty} \left[ a + \frac{1}{n}, b - \frac{1}{n} \right] = (a,b)$.  So, we have $\displaystyle \mathcal{O} = \bigcup_{i=1}^\infty \left ( \bigcup_{n = \frac{b-a}{2} + 1}^\infty \left[ a_{x_i}+\frac{1}{n}, b_{x_i}-\frac{1}{n} \right] \right)$.  But, I don't like this. It looks like I'm still just taking a union of open sets.  Please help me get this to look like the union of closed sets.  (For unbounded intervals, I know what to do - it should follow from this relatively easily). Show $S = f^{-1}(f(S))$ for all subsets $S$ iff $f$ is injective Let $f: A \rightarrow B$ be a function. How can we show that for all subsets $S$ of $A$, $S \subseteq f^{-1}(f(S))$? I think this is a pretty simple problem but I'm new to this so I'm confused.  Also, how can we show that $S = f^{-1}(f(S))$ for all subsets $S$ iff $f$ is injective? I accidentally installed some USB drivers to my Windows 7 system, they were the wrong ones and now my keyboard and mouse won't work. A system restore would get rid of them, but obviously I can't control my computer to make that happen.  Keyboard and mouse still work before windows has loaded, in the BIOS and boot system, so can I do anything from there? I've already tried booting into another OS, accessing my windows filesystem and deleting the Program Files entry, but it didnt work so they must go deeper than that. Or could i launch it from a remote machine? I've got some VNC server insalled so if i could get that running that could work. Consider the question :    The large number of down-votes displays as 199 rather than a score of -199.  This is clearly unintentional (and unfortunate for questions that possess such voting!), but can this be&nbsp;corrected? Python dateutils print recurrence rule according to iCalendar format (see RFC 5545)