everywhere I look I can only find derivations of the Gauss–Ampère law $\partial_\alpha F^{\alpha\beta}=\mu_0 J^{\beta}$, and this follows quite simply from the variational method with the Lagrangian Density, but I cannot seem to find any way to derive the Gauss-Faraday law $\partial_\alpha\left(\frac{1}{2}\epsilon^{\alpha\beta\gamma\delta}F_{\gamma\delta}\right)=0$ (equivalently as $\partial_\gamma F_{\alpha\beta} + \partial_\alpha F_{\beta\gamma} + \partial_\beta F_{\gamma\alpha}=0$ &quot;Bianchi identity&quot;) also from the Lagrangian Density and Euler-Lagrange Equations. As far as I know the Gauss–Ampère law represents only half of the Maxwell's equations, the other half being represented by the Gauss-Faraday law, does this mean that Faraday's law of induction and Gauss's law for magnetism are not equations of motion of the fields in question?
In Heaviside-Lorentz units the Maxwell's equations are:  $$\nabla \cdot \vec{E} = \rho $$  $$ \nabla \times \vec{B} - \frac{\partial \vec{E}}{\partial t} = \vec{J}$$ $$ \nabla \times \vec{E} + \frac{\partial \vec{B}}{\partial t} = 0 $$ $$ \nabla \cdot \vec{B} = 0$$  From EM Lagrangian density: $$\mathcal{L} = \frac{-1}{4} F^{\mu \nu}F_{\mu\nu} - J^\mu A_\mu$$   I can derive the first two equations from the variation of the action integral: $S[A] = \int \mathcal{L} \, d^4x$. Is it possible to derive the last two equations from it?
I want to create this:  Note that wire lay chaotically, so spiral object not the same thing.  There is a cool way to do this task in 3ds max:  Basically he deform straight spline/curve and then bend it couple iterations (over 9000 degrees angle in max).   Blender can't bend that way with simple deform as I know, only up to 360 degree. Is there any other way to make wire wrapping similar to photo? Show that every large integer has a large prime-power factor  That is, if $P(n)$ designates the largest number $p^a$ which divides $n$, then $\lim_{n\to\infty}P(n)=\infty.$ Pure verbal nouns/deverbal nouns vs. gerunds This is a follow-up to a previous question which I am still trying to understand. I think I'm making progress in my understanding, but I would appreciate feedback to help me refine my thinking.  Here is the sentence:     Good writing requires hard work.   Originally, I thought that "writing" was acting as a gerund in this phrase.  After reading through the replies to that question along with back posts related to gerunds, I've since concluded that "writing" in this sentence is a , also termed a deverbal noun.  My reasoning?  —In the sentence above "writing" is modified by "good" and is thus acting more like a noun than a verb. —It has no direct object.  Is my thinking correct here?  If not, could someone politely help me to refine it?  However, what if I had written the following?     Writing well requires hard work.     or   To write well requires hard work.   In these examples, I'm thinking that "writing" would be a different kind of verbal noun—a non-finite verb acting as a gerund or infinitive, which is one piece of a larger complement noun clause.  The last bit is muddy for me, but I will leave it at that. Thank you in advance for your patience and insight. Prove that a topological space equipped with a delta-complex structure is Hausdorff I was doing a very simple search:  grep -R Milledgeville ~/Documents   And after some time this error appeared:  grep: memory exhausted   How can I avoid this?  I have 10GB of RAM on my system and few applications running, so I am really surprised a simple grep runs out of memory. ~/Documents is about 100GB and contains all kinds of files.  grep -RI might not have this problem, but I want to search in binary files too. How to find correctly the .bib file? UISearchDisplayController - how to preload searchResultTableView When running Internet Explorer from the the modern UI, I can't find out how to access tabs.  I can create tabs using Ctrl + T, but I'm not sure how to switch between tabs, see what tabs I have open, etc.  How do I navigate between tabs when using the modern UI Internet Explorer? What is an adjective referring to “someone who puts people down”? I was just thinking about someone I went to school with who had a penchant for putting people down, and I was trying to think of a word that describes this, but couldn’t find it. It would be a synonym of caustic, abrasive, discouraging, and undermining, but these all have slightly different connotations (to wear away", cause discouragement or to make fall). I want an adjective that has the connotation of a hammer hitting you on the head. The word does not necessarily need to describe someone who does this on purpose, out of evil intent. Is there such a word?    The word should give you a mental picture of someone barraging you with negative comments, but it not necessarily affecting you. Think of someone who so often has something negative to “contribute” that many can just either ignore it or roll their eyes at it, and that is the extent of its effect. I keep getting the mental picture of the word buffet as in, how waves buffet the shore. A constant hitting, without the hitting necessarily being devastating in its destruction. Can anyone please tell me whether the following function $d:\mathbb{R}^2 \times \mathbb{R}^2 \to  \mathbb{R}$ is really a metric or not on $\mathbb{R}^2$.  $$ d((x_1,x_2),(y_1,y_2))= \begin{cases} |x_1-y_1|, &amp;\text{ if } x_2=y_2\\ |x_1|+|y_1|+|x_2-y_2|, &amp;\text{ if } x_2 \neq y_2. \end{cases} $$  If it really is indeed a metric, please give the proof of triangle inequality.(I'm OK with the other two conditions). Does the alarm work when my phone is turned off? Will the alarm still go off if I turn off the phone before?  Example: when I have low battery, I have to go to sleep and wake up at X o'clock, and there is no charger around. Should I turn off the phone?   Note: If it makes any difference my phone is Galaxy S running froyo Using lasso for feature selection, followed by a non-regularized regression I use Lasso logistic regression in order to identify a smaller subset of important variables. I start with N=51 (28/23) and 32 predictors.   So far it looks pretty promising, because I can identify four important predictors in my optimal model.   Now I would like to take those four predictors and examine them along with some control variables in a standard logistic regression.   My question is, does that analysis strategy make sense? Is there a better way to include controls or other variables that might be interesting?  For a better understanding:   Identify important variables via Lasso logistic regression Do further analysis including identified predictors and other control variables using standard logistic regression (using AIC to check model fit) date command not resolving properly in bash alias How to download a whole FTP site using command line? I need to download an FTP site (a web site source code actually) containing a very high quantity of relatively small files (downloading it with FileZilla took more than a day, but I believe it could be downloaded much faster if many files were downloaded at once). Unfortunately there is no SSH access to the site and no way to archive the files on the server side.  So at least the question is how to download a whole FTP site into a folder using command line tools. A better it'd be if the download could be parallelized by downloading many files simultaneously instead of one by one. Linear Discriminant Analysis for $p=1$ Supersymmetry in Quantum Field Theory I have an idea of supersymmetry in quantum mechanics, can you suggest a book on "supersymmetry in quantum field theory", which has sufficient mathematical rigour like "Peskin and Schroeder" So I've been thinking through some test cases. If $a_n = n$ then $\sum_n \frac{a_{n+1} - a_n}{a_n}$ is the harmonic series which diverges. And if $a_n = \sum_{k=1}^n 1/k$ then $\sum_n \frac{a_{n+1} - a_n}{a_n}$ diverges like $\sum_n 1/(n \log n)$. So that got me thinking, if $a_n$ is a strictly increasing unbounded sequence, does $\sum_n \frac{a_{n+1} - a_n}{a_n}$ necessarily diverge? Why does a proof of $\exists f: X\to Y$ injection $\iff \exists g: Y \to X$ surjection requires the axiom of choice? Why does a proof of $\exists f: X\to Y$ injection $\iff \exists g: Y \to X$ surjection requires the axiom of choice?  This question is answered here:   I'm not interested in a proof itself of this theorem, I want to know why this requires the Axiom of Choice. How to determine the class of a generic type? The question is based on the following exercise in real analysis:     Assume that $A,B\subset{\Bbb R}$ are both bounded and $x&gt;0$ for all $x\in A\cup B$. Show that $$ \sup(AB)=\sup A\sup B $$   where    $$ AB:=\{ab\in{\Bbb R}:a\in A, b\in B\}. $$   Since $0&lt;a\leq\sup A$ and $0&lt;b\leq\sup B$ for all $a\in A$ and $b\in B$, we have $$ ab\leq\sup A\sup B $$ for all $ab\in AB$ which implies that $\sup AB\leq\sup A\sup B$. I have trouble with another direction: $$ \sup AB\geq\sup A\sup B $$ I was trying to show that for every $\epsilon &gt;0$, $\sup AB-\epsilon \geq \sup A\sup B$. If one uses the definition of supremum, one has the estimates that for every $\epsilon&gt;0$, $$ \sup A-\epsilon\leq a, \quad \sup B-\epsilon\leq b $$ for some $a\in A,\ b\in B$. It follows that $$ \sup A\sup B\leq (a+\epsilon)(b+\epsilon)=ab+\epsilon(a+b)+\epsilon^2\leq \sup AB+\epsilon (a+b)+\epsilon^2 $$ which seems quite close to what I want. How can I go on?