Why direction of Ampere's Law follows right hand rule?
I am currently wondering about the famous .  Where does it come from mathematically that when you point with your thumb in the direction of the current, your curved fingers will point in the direction of the $B$-field? In other words, which mathematical operation does it come from? Probably it has to do with some vector product or Stokes theorem application, but I am not quite sure about it.
Are there infinitely-many numbers that are both square and triangular? I just started to read "Friendly Introduction to Number Theory" but I am getting stuck in the first exercises.     1.1. The first two numbers that are both squares and triangles are 1 and 36. Find the next one and, if possible, the one after that. Can   you figure out an efficient way to find triangular–square numbers? Do   you think that there are infinitely many?     I found how to find out the number which is both square and triangle. (don't know if this is effective way)    However how can I answer "Do you think that there are infinitely many?"? I think I need to find the formula but no idea yet. Can you please help me? Does a four-divergence extra term in a Lagrangian density matter to the field equations? Greiner in his book "Field Quantization" page 173, eq.(7.11) did this calculation:     ${\mathcal L}^\prime=-\frac{1}{2}\partial_\mu A_\nu\partial^\mu A^\nu+\frac{1}{2}\partial_\mu A_\nu\partial^\nu A^\mu-\frac{1}{2}\partial_\mu A^\mu\partial_\nu A^\nu $   $\space\space\space\space=-\frac{1}{2}\partial_\mu A_\nu\partial^\mu A^\nu+\frac{1}{2}\partial_\mu[A_\nu(\partial^\nu A^\mu)-(\partial_\nu A^\nu) A^\mu]$        The last term is a four-divergence which has no influence on the field equations. Thus the dynamics of the electromagnetic field (in the Lorentz gauge) can be described by the simple Lagrangian        ${\mathcal L}^{\prime\prime}=-\frac{1}{2}\partial_\mu A_\nu\partial^\mu A^\nu$     Yes, if it is a four-divergence of a vector whose 0-component doesn't contain time derivatives of the field, indeed according to the variational principle this four-divergence will not influence the field equation.  And actually I calculated the time-derivative dependence of 0-component of $[A_\nu(\partial^\nu A^\mu)-(\partial_\nu A^\nu) A^\mu]$, in which only $[A_0(\partial^0 A^0)-(\partial_0 A^0) A^0]$ could possibly contain time-derivative, which vanishes fortunately, so whatever the general case it doesn't matter in this present case.  But how can he seem to claim that it holds for a general four-divergence term,The last term is a four-divergence which has no influence on the field equations?  EDIT: I only assumed the boundary condition to be $A^\mu=0$ at spatial infinity, not at time infinity. And the variation of the action $S = \int_{t_1}^{t_2}L \, dt$ is due to the variation of fields which vanish at time, $\delta A^\mu(\mathbf x,t_1)=\delta A^\mu(\mathbf x,t_2)=0$, not having the knowledge of $\delta \dot A^\mu(\mathbf x,t_1)$ and $\delta \dot A^\mu(\mathbf x,t_2)$, which don't vanish generally, so the four-divergence term will in general contribute to the action, $$\delta S_j=\delta \int_{t_1}^{t_2}dt\int d^3\mathbf x \partial_\mu j(A(x),\nabla A(x),\dot A(x))^\mu =\delta \int_{t_1}^{t_2}dt\int d^3\mathbf x \dot j^0 =\int d^3\mathbf x [\delta j(\mathbf x, t_2)^0-\delta j(\mathbf x, t_1)^0]$$ which does not vanish in general! How to sandbox applications? I want to run small untrusted programs, but restrict them from accessing any files outside of their folder, network access, and everything else that they don't really need. What is the simplest way to achieve this? product distribution of two uniform distribution, what about 3 or more $G$ is non abelian simple group of order $<100$ then $G\cong A_5$ How many 5-cards poker hands are there containing at least 3 of the 4 suits?  I don't how know how to properly handle this question.  There 13 ranks of each card, and 4 suits  My Attempt:  Begin with the first 3: C(13,3) X 4 ways to choose suits  Then pick last 3 cards of anything:  C(49,2)  My solution: 1,345,344  Thanks, Rerender funnel highcharts with new data Editing LaTeX generated PDF Is there an application (in Windows, Unix, or Mac) that allows you to edit the LaTeX generated PDF. I know there is pdfsync, but it only gives you the line, and then you have to search where in the line you want to make the change. This is OK sometimes, for a couple of changes, but when there are lots of changes it becomes really tedious.  I would really like to accelerate this process of correcting mistakes (mostly grammar, missing commas etc.). in my document, without going through all the markup. How to use the Bevel node in 2.79b In the Harry Potter world, there's the term "muggle" for people with no magical abilities, "muggle-born" for those with magical abilities who's parents are muggles and "squib" for those with no magical abilities who's parents are magical.  But is there any collective term for all magical people (as opposed to wizards for males and witches for females)?  I don't recall seeing any word that refers to all men and women and children with magical abilities, and it's the only group left without noun or collective noun to describe them. I have an unlocked North American (Canada) iPhone 5s, where can I buy a data only SIM card in Japan ? I'm not interested in renting nor in a voice plan, I'll be staying for 4 weeks.  I'll be landing in Tokyo, so preferably around there. Can I get them from the airport somewhere ? I'm assuming I won't have problems with the network/frequencies, am I correct ? Tristate Checkboxes in Java Can Hooke's law be derived? Can we derive  from the theory of elasticity? I know it is not a fundamental law and therefore can be derived from more basic considerations. Google Chrome (Windows) not displaying text/encoding correctly on certain websites I experience a problem similar to    Here's a screenshot on the website     I've tried:   Removing fonts I have installed Setting the encoding to Western, UTF-8, Greek, Automatic Changing the fonts used in Google Chrome Removing Google Chrome, including all application data/user profiles Running CCleaner Reinstalling Google Chrome, twice Installing Chromium (fonts display fine in Chromium) Turning off Windows ClearType Changing Windows/Chrome languages from English (Australia) to US, UK Restarting Windows several times after making changes   All of which did not work. I do not connect over VPN/proxy.   Can anyone give me any tips?  Windows 7 x64 Ultimate Is there a set of lines in $\mathbb{R}^3$ such that their union is equal to the set of points with not all coordinates rational? Does there exist a set L of pairwise disjoint lines, each line in an axis direction, in $\mathbb{R}^3$ such that their union is equal equal to $\{ x : x \text{ has at least one irrational coordinate}\}$? A specified Include path is not valid. The EntityType does not declare a navigation property with the name * Do different databases use different name quote? For example, mysql quote table name using   SELECT * FROM `table_name`;   notice the `   Does other database ever use different char to quote their table name Electron shells in atoms: What causes them to exist as they do? I have seen similar posts, but I haven't seen what seems to be a clear and direct answer.  Why do only a certain number of electrons occupy each shell? Why are the shells arranged in certain distances from the nucleus? Why don't electrons just collapse into the nucleus or fly away?  It seems there are lots of equations and theories that describe HOW electrons behave (pauli exclusion principle), predictions about WHERE they may be located (Schrödinger equation, uncertainty principle), etc. But hard to find the WHY and/or causality behind these descriptive properties.  What is it about the nucleus and the electrons that causes them to attract/repel in the form of these shells at regular intervals and numbers of electrons per shell?  Please be patient with me, new to this forum and just an amateur fan of physics. Is "chutzpah" used by non-Jewish English speakers? I often see tutorials online that connect various commands with different symbols. For example:  command1 |  command2 command1 &amp;  command2 command1 || command2     command1 &amp;&amp; command2   Others seem to be connecting commands to files:  command1  &gt; file1 command1  &gt;&gt; file1   What are these things? What are they called? What do they do? Are there more of them?    .