Proof of the existence of the energy-momentum tensor I have a problem providing or finding a general proof for this statement i found in , section $10.3.2$:  Due to the locality of the theory there exists a local field $T_{\mu\nu}(x)$, called the stress-energy tensor, defined by the variation of the local action $S[\varphi]$ under the infinitesimal transformation $x^{\,\mu} \rightarrow x^{' \mu} = x^{\,\mu} + \epsilon^{\,\mu}(x)$  $$ \delta S \, =\, \frac{1}{(2 \pi)^{D-1}} \int d^Dx \, T_{\mu\nu}(x)\, \partial^{\,\mu} \epsilon^{\,\nu} $$  where it says that the factor $(2 \pi)^{D-1}$ is there for later convenience   Now I've seen the proof for a real scalar field which is clearly a field without local interactions.  What is the proof in the general case? Is there a place (book or article) where I can find the proof?
Energy momentum tensor from Noether's theorem in the book Quantum Field Theory by Itzykson and Zuber the following derivation for the stress-energy tensor is proposed (p.22):  Assume a Lagrangian density depending on the spacetime coordinates $x$ only through fields and their gradients. Under a translation we have $$\mathcal L (x+a)=\mathcal L[\phi_i(x+a),\partial_\mu\phi_i(x+a)].$$  Consider an infinitesimal $x$-dependent transformation $$\delta\phi_i=\delta a^\mu(x)\partial_\mu\phi_i(x),$$ $$\delta\partial_u\phi_i(x)=\delta a^\nu \partial_\nu\partial_\mu\phi_i(x)+\partial_\mu[\delta a^\nu (x)]\partial_\nu \phi_i(x).$$  The proof then proceeds with variation of the action and integration by parts. But why do we consider an $x$-dependent local transformation instead of a global transformation?
What are Extension Methods? What are extension methods in .NET?    EDIT: I have posted a follow up question at I understand that people explain (in layman's terms at least) that the presence of mass "warps" space-time geometry, and this causes gravity. I have also of course heard the analogy of a blanket or trampoline bending under an object, which causes other objects to come together, but I always thought this was a hopelessly circular explanation because the blanket only bends because of "real" gravity pulling the object down and then pulling the other objects down the sloped blanket. In other words, to me, it seems that curved space wouldn't have any actual effect on objects unless there's already another force present.   So how is curved space-time itself actually capable of exerting a force (without some source of a fourth-dimensional force)?  I apologize for my ignorance in advance, and a purely mathematical explanation will probably go over my head, but if it's required I'll do my best to understand. Do not give the impression that all votes are unanimously in favor of the winning close-reason. I closed for "belongs on superuser," yet SO claims I closed for a completely different reason.  Source: Windows Active Directory naming best practices? This is a  about Active Directory domain naming.   After experimenting with Windows domains and domain controllers in a virtual environment, I've realized that having an active directory domain named identically to a DNS domain is bad idea (Meaning that having example.com as an Active Directory name is no good when we have the example.com domain name registered for use as our website).   , but I'm still not sure about what other rules there are around naming Active Directory domains.  Are there any best practices on what an Active Directory name should or shouldn't be? Why do soap operas have the soap opera effect? The soap opera effect occurs when frame interpolation is used. For example, if a movie is shot at 24 frames per second and then shown on a TV screen which is capable of displaying 60 frames per second then it is possible to use a computer to do motion interpolation. In other words the computer can average (interpolate) two frames and create a synthetic third frame and stick it between the two real frames.  This interpolation makes the picture look unnatural and plastic, somewhat like soap operas look on TV, hence the name.  My understanding of why soap operas have the soap opera effect has something to do with the fact that many of them are shot using camcorders instead of film, but I am not completely sure.  Why do soap operas have the soap opera effect? Who built the fortress of Dol Goldur, and when was it built? And was it always a place for evil, or Sauron just occupied it?  Despite the fact that Mordor is much better fortified; artificially by the old fortifications,  and geographically, as it is surrounded by the mountains. Also considering the presence of mount doom that seem to enhance Sauron's power, Sauron chose to start his campaign from Dol Goldur, despite Mordor being ready for quite some time.  Why did he choose this enigmatic fortress, and not go straight for Mordor or even go earlier? word choice - realize notice find out figure out Does the alternating sum of prime reciprocals converge? $$\sum_{n = 1}^\infty \frac{(-1)^n}{p_n}$$ where $p_n$ is the $n$th prime.  I have computed this to 10000 rather than infinity. My results suggest that convergence does happen but it's very slow. But I can't even be sure about the first few digits: $-0.26959$?  I have looked at the "questions that may already have your answer" and some of the "similar questions," but they involve somewhat different formulas. How do I add a .sty file to my MacTeX/TeXShop installation? $(this) inside of AJAX success not working Construct a field of 25 elements. We need to construct a field of 25 elements.  By using a result : For a prime p and a monic irreducible polynomial $p(x)$ in $\Bbb F_p [x]$ of degree $n$ , then the ring $(\Bbb F_p [x] / &lt;p(x)&gt;)$ is a field of order $p^{n}$.  So , for the question problem above , we need a irreducible polynomial of degree $2$ in $\Bbb Z_5 [x]$.  Solution says , $x^{2} + x +1$ does our job , since it's irreducible in $\Bbb Z_5 [x]$.  My question : Can't we take $ x^{2} + 2$ , it has no zeroes in $\Bbb Z_5$ , and is irreducible. In every solution I've gone through , the polynomial $x^{2} + x+1$ is taken. Can't we consider $ x^{2} + 2$ ? How can I safely create a nested directory in Python? Help with footer always to bottom How to select an index value from a String Array in a JOptionPane I've just set up a new machine with Ubuntu Oneiric 11.10 and then run  apt-get update apt-get upgrade apt-get install git   Now if I run git --version it tells me I have git version 1.7.5.4 but on my local machine I have the much newer git version 1.7.9.2  I know I can install from source to get the newest version, but I thought that it was a good idea to use the package manager as much as possible to keep everything standardized.  So is it possible to use apt-get to get a newer version of git, and what is the right way to do it? Flowchart using Tikz I'm trying to create a fine flowchart that is a bit complicated. Therefore I need a block for a predefined process and a connector block as well as a data block. Can someone help me? In comment is what I'd like to see in my code to keep my code nicely arranged. The rest of the code is a simple example for the Tikz site.  \documentclass{article}  \usepackage[latin1]{inputenc} \usepackage{tikz} \usetikzlibrary{shapes,arrows} \begin{document} \pagestyle{empty}   % Define block styles \tikzstyle{decision} = [diamond, draw, fill=blue!20,      text width=4.5em, text badly centered, node distance=3cm, inner sep=0pt] \tikzstyle{block} = [rectangle, draw, fill=blue!20,      text width=5em, text centered, rounded corners, minimum height=4em] \tikzstyle{line} = [draw, -latex'] \tikzstyle{cloud} = [draw, ellipse,fill=red!20, node distance=3cm,     minimum height=2em]  %\tikzstyle{subroutine} = [draw,rectangle split, rectangle split         horizontal,rectangle split parts=3,minimum height=1cm]  %\tikzstyle{connector} = [draw,circle,...] %\tikzstyle{data} = [draw, trapezium, ...]  \begin{tikzpicture}[node distance = 2cm, auto]     % Place nodes     \node [block] (init) {initialize model};     \node [cloud, left of=init] (expert) {expert};     \node [cloud, right of=init] (system) {system};     \node [block, below of=init] (identify) {identify candidate models};     \node [block, below of=identify] (evaluate) {evaluate candidate models};     \node [block, left of=evaluate, node distance=3cm] (update) {update     model};     \node [decision, below of=evaluate] (decide) {is best candidate     better?};     \node [block, below of=decide, node distance=3cm] (stop) {stop};     % Draw edges     \path [line] (init) -- (identify);     \path [line] (identify) -- (evaluate);     \path [line] (evaluate) -- (decide);     \path [line] (decide) -| node [near start] {yes} (update);     \path [line] (update) |- (identify);     \path [line] (decide) -- node {no}(stop);     \path [line,dashed] (expert) -- (init);     \path [line,dashed] (system) -- (init);     \path [line,dashed] (system) |- (evaluate); \end{tikzpicture}   \end{document} If I enter the US by land, not needing ESTA, would I then need ESTA to enter by air 3 days later? I am a UK citizen, and therefore qualify for the USA's  (VWP). If I enter by  I need to apply for (and pay for) ESTA beforehand. If I enter by , I don't need ESTA. But I do get a 90 day visa waiver which allows me to go to Canada and return to the USA, on the same VWP.   My question is: If I first enter the USA on foot, from Canada (at Niagara Falls), return to Canada on foot, and then 3 days later fly from Canada into the USA, do I need ESTA? Or as my existing visa waiver is still in place, is that good enough? Cross referencing between chapters of a thesis Hi I am new to the forum and fairly new to Tex, if I miss anything out please let me know and I will be glad to post it. I am writing up my PhD thesis using Texmaker on a windows system. I have my document organised with a main file and a sub-directory containing chapter 1 to 5. I want to be able to use the labels I created in chapter two for example in the appendix and vice versa. Is there an easy way to do this?   I have tried following the advice of various posts of including the commands \usepackage{xr} and \externaldocument{Chapters/Chapter_2} in the individual chapters but this leads to an error when compiling.   My Preamble is as follows:  \documentclass[12pt,hidelinks,twoside]{report} \usepackage[utf8]{inputenc} \usepackage[T1]{fontenc} \usepackage[greek,spanish,english]{babel} \usepackage{textgreek} \usepackage[version=3]{mhchem} \usepackage[a4paper,width=150mm,top=25mm,bottom=25mm,bindingoffset=6mm]{geometry} \usepackage{amsmath, amsfonts, pifont, float, color, url} \usepackage[pdftex]{graphicx} \usepackage[style=numeric,backend=biber,sorting=none]{biblatex} \addbibresource{references.bib} \usepackage[pdftex, plainpages=false, pdfpagelabels, bookmarks=true]{hyperref} \usepackage{fancyhdr} \usepackage[center,font={it,small}]{caption}   I then have a section such as this in the main document:  \begin{document} %%... Title page ... \chapter{Theory and Methods} \label{chap:Theory and Methods} \include{Chapters/chapter_2} %%... Other chapters .... \end{document}   In Chapter 2 I have for example a label for a diagram which I may want to reference in other chapters:  \begin{figure}[H] \centering \includegraphics[height=10cm,width=\textwidth]{RF_colour_hi_res.pdf} \caption{Random Forest} \label{fig:2.1 Random Forest} \end{figure}   How can I have this label show up in other chapters after I type \ref ?  Many thanks in advance.  If any other information is required please let me know.   This has been answered in the comments although the references do not appear in the auto complete if they are copied and pasted in they still work. I've installed Xcode and MacPorts on a brand new mid-2013 MacBook Air, but MacPorts doesn't work, saying that it cannot find anything.  mba:~ cnst$ sudo port install wget Warning: The Command Line Tools for Xcode don't appear to be installed; most ports will likely fail to build. Warning: See http://guide.macports.org/chunked/installing.xcode.html for more information. ---&gt;  Computing dependencies for wgetError: Unable to execute port: can't read "build.cmd": Failed to locate 'make' in path: '/opt/local/bin:/opt/local/sbin:/bin:/sbin:/usr/bin:/usr/sbin' or at its MacPorts configuration time location, did you move it? mba:~ cnst$   And, indeed, there is no make or clang in the path, but it is definitely installed, though:  mba:~ cnst$ find /Applications -name "clang" -or -name "make" /Applications/Xcode.app/Contents/Developer/Toolchains/XcodeDefault.xctoolchain/usr/bin/clang /Applications/Xcode.app/Contents/Developer/Toolchains/XcodeDefault.xctoolchain/usr/lib/clang /Applications/Xcode.app/Contents/Developer/usr/bin/make mba:~ cnst$   Why is there no make or clang in the path?  Why MacPorts doesn't work?  I notice that  mentions that you have to install something from ADC, Apple Command Line Developer Tools, but ADC doesn't work since last Thursday or Friday due to their security breach, so, the developer tools are nowhere to be found.  What's the proper solution here? Evaluating string "3*(4+2)" yield int 18