Is use of negative sign in faraday's law a convention because of using right hand thumb rule? We use right hand thumb rule to relate $dS$ and $dl$ like if $dS$ points in the direction of thumb then direction of $dl$ is taken in the direction of fingers but if we use left hand thumb rule to relate $dS$ and $dl$ then there is no need of negative sign in the equation? $$\oint\vec{E'}\cdot\vec{dl'}=-\frac{d\phi_B}{dt} $$
What does the minus sign in Maxwell's third equation imply? If we write out Maxwell's equations with magnetic charges, we get $$ \begin{align} \nabla \cdot \mathbf{E} &amp;= 4 \pi \rho_e \tag{1}\\ \nabla \cdot \mathbf{B} &amp;= 4 \pi \rho_m \tag{2}\\ -\nabla \times \mathbf{E} &amp;= \frac{\partial \mathbf{B}}{\partial t} + 4 \pi \mathbf{J}_m \tag{3}\label{Eq:Faraday}\\ \nabla \times \mathbf{B} &amp;= \frac{\partial \mathbf{E}}{\partial t} + 4 \pi \mathbf{J}_e \tag{4}\label{Eq:Ampere} \end{align} $$ In particular, Faraday's law \eqref{Eq:Faraday} contains a minus sign that Ampere's law \eqref{Eq:Ampere} does not. This always struck me as odd because it's often said the fields are dual to each other (i.e. you can replace E with B and &quot;get the same result&quot;), but that requires a bit of mental recalibration to accommodate that minus. So I'm curious what the origin of that negative is and what it means. Are there any intuitive explanations for how to think about it?
Are there any places to leave luggage in Brussels, for short and/or long periods of time? I wanted to left align a set of equations. I've tried the solution provided in  . It worked. But, I wanted to have a single equation number for the entire flalign in order to refer to it later. Like in case of equation+split. While using equation+align, I am able to get it. But, if I use equation+flalign the equations are still being centered. Is there a way to do it?  Example using equation+align,  \begin{equation}      \begin{align}       \mbox{Minimize } &amp; x_1+x_2+x_3 \\       \mbox{Subject to} &amp; \\        &amp; x_1+x_2 \leq 10 \\        &amp; x_2+x_3 \leq 8 \\        &amp; x_1+x_3 \leq 5       \end{align} \end{equation}   I would like to do this while the equations are left aligned.  It will be even better if the equations can be spaced a little (for example, 1 cm) from the left margin instead  of starting from the left margin.  Thanks in advance Prove that $\frac{R/ \ker \phi}{(\ker \phi + J)/ \ker \phi} \cong \frac{S}{\phi(J)}$ Let $\phi: R \to S$ be a surjective homomorphism. Prove that $\frac{R/ \ker \phi}{(\ker \phi + J)/ \ker \phi} \cong \frac{S}{\phi(J)}$ for an ideal $J$ of $R.$  Obviously, $S \cong R/ \ker \phi$(first isomorphism theorem) and $\phi(J)$ is an ideal of $S$, since $\phi$ is surjective( $\forall s \in \phi(J), s = \phi(j)$ for some $j \in J, \forall s' \in S, s' = \phi(r)$ for some $r \in R$, so $ss' = \phi(j)\phi(r) = \phi(jr) \in \phi(J)$, and $ss' = \phi(r)\phi(j) = \phi(rj) \in \phi(J)$ ).  However, I'm having trouble with going on. If it's important, I mean a unitary ring by a ring. I'm currently working my way through Harvard's online abstract algebra lectures (if you're interested, you can find them ).  The lectures come complete with notes and homework problems.  Of course, since I don't actually go to Harvard, I can't hand in the homework assignments to find out if I'm doing them correctly.  So I've decided to try to crowd source the grading of my solutions.  I'll post individual questions as I finish them and wait for comments.  I would like to get critiques of not just my reasoning but the style of my write ups as well.  Also, any alternative approaches to the problem would be welcome.  I've looked for forums dedicated to these kinds of OCW courses but have not been able to find any.  This surprises me.  It seems like, with the advent of these free online educational resources, an online meeting place for those who take advantage of them would be a natural offshoot.  So this might be a bit of an experiment.  Unless I'm the 873rd person to post something like this here.  If that's the case, sorry for being so long winded.  Anyway, on with the question.  This one is assigned in the 4th lecture.     Let $V$ denote the Klein 4-group.  Show that $\operatorname{Aut}(V)$ is isomorphic to $S_3$.   (I've moved my solution below to keep this question from showing up in the unanswered list.)  Thanks... Finding languages such that $L_{1} \subseteq L_{2} \subseteq L_{3}$ where $L_{1}, L_{3} \notin \mathbb{R}$, $L_{2} \in \mathbb{R}$ I am struggling to find such languages $L_{1}$, $L_{2}$, and $L_{3}$ such that  $$ L_{1} \subseteq L_{2} \subseteq L_{3} $$ where $L_{1}, L_{3} \notin  \mathbb{R}$ and $L_{2} \in  \mathbb{R}$.  I know they exist, I need help finding them. Please allow using nested expressions when filtering with tags Please support the use of nested expressions like  and (or ) when searching.  Looking here, on meta, at questions tagged   and at questions found with +search +tag +boolean I didn't see any information on this topic. Prove the following $A \setminus (B\setminus C) = (A\setminus B)\cap (A \cup C)$ I want to prove the following so I decide to rewrite the RHS  $$A \setminus (B\setminus C) = (A\setminus B)\cap (A \cup C)$$ $$ A\setminus(B\setminus C) = A\setminus (B \cap C^c) = A\cap (B\cap C^c)^c = A\cap(B^c\cup C) = (A\cap B^c)\cup(A\cap C) $$ there is way that $(A\cap B^c)\cup(A\cap C)\rightarrow (A\setminus B)\cup(A\cap C) = (A\setminus B)\cap (A \cup C)$?  thanks. EDIT  LHS: $$x\in A \vee x\in A\cap C \rightarrow x\in A \wedge x\in C$$ RHS: $$x\in A \wedge x\in A\cup C \rightarrow x\in A \vee x\in C$$  how it can help me? Is time continuous or discrete? I was coding a physics simulation, and noticed that I was using discrete time. That is, there was an update mechanism advancing the simulation for a fixed amount of time repeatedly, emulating a changing system.  I though that was interesting, and now believe the real world must behave just like my program does. It is actually advancing forward in tiny but discrete time intervals? Adjectives order: opinion or size? speaking, opinion goes before size, i.e. opinion, size, age, shape, color, origin, material, purpose. However, I saw this example in Practical English Usage, third edition, page 12:      a big beautiful garden   Or you have probably heard the known expression  (BBW). What am I missing? What is a terminal and how do I open and use it? What is a terminal? How do I open a terminal window? How do I use the terminal? Vector Applications in Euclidean Geometry At work we were discussing this as my boss has never heard of normalization. In Linear Algebra, Normalization seems to refer to the dividing of a vector by its length. And in statistics, Standardization seems to refer to the subtraction of a mean then dividing by its SD. But they seem interchangeable with other possibilities as well.   When creating some kind of universal score, that makes up $2$ different metrics, which have different means and different SD's, would you Normalize, Standardize, or something else? One person told me it's just a matter of taking each metric and dividing them by their SD, individually. Then summing the two. And that will result in a universal score that can be used to judge both metrics.  For instance, say you had the number of people who take the subway to work (in NYC) and the number of people who drove to work (in NYC).   $$\text{Train} \longrightarrow x$$ $$\text{Car} \longrightarrow y$$  If you wanted to create a universal score to quickly report traffic fluctuations, you can't just add $\text{mean}(x)$ and $\text{mean}(y)$ because there will be a LOT more people who ride the train. There's 8 million people living in NYC, plus tourists. That's millions of people taking the train everyday verse hundreds of thousands of people in cars. So they need to be transformed to a similar scale in order to be compared.  If $\text{mean}(x) = 8,000,000$  and $\text{mean}(y) = 800,000$  Would you normalize $x$ &amp; $y$ then sum? Would you standardize $x$ &amp; $y$ then sum? Or would you divide each by their respective SD then sum?  In order to get to a number that when fluctuates, represents total traffic fluctuations.  Any article or chapters of books for reference would be much appreciated. THANKS!  Also here's another example of what I'm trying to do.   Imagine you're a college dean, and you're discussing admission requirements. You may want students with at least a certain GPA and a certain test score. It'd be nice if they were both on the same scale because then you could just add the two together and say, "anyone with at least a 7.0 can get admitted." That way, if a prospective student has a 4.0 GPA, they could get as low as a 3.0 test score and still get admitted.  Inversely, if someone had a 3.0 GPA, they could still get admitted with a 4.0 test score.  But it's not like that. The ACT is on a 36 point scale and most GPA's are on 4.0 (some are 4.3, yes annoying). Since I can't just add an ACT and GPA to get some kind of universal score, how can I transform them so they can be added, thus creating a universal admission score.  And then as a Dean, I could just automatically accept anyone with a score above a certain threshold.  Or even automatically accept everyone whose score is within the top 95%.... those sorts of things.  Would that be normalization? standardization? or just dividing each by their SD then summing? How do I deal with a compromised server? Are there any rules for adjudicating a ranged attack fired at a target engaged in melee combat with an ally? While running a session recently, one of the players chose to "assist" his companion who was in melee with a goblin, by firing an arrow at the pair of them. I couldn't find any particular rules for this scenario so just had him roll at a disadvantage, counting a miss as a hit on his companion (damage to be ascertained by a second roll).   This seemed to work okay, and became a running joke for the night as the arrow bounced off the companions back, but are there specific rules for this scenario? Can I install the 64 bit version? I have to prove by induction that $$\sum_{r=0}^{n}\begin{pmatrix}n \\ r \end{pmatrix}=2^k$$, using the identity $\begin{pmatrix}n \\ r-1 \end{pmatrix}+\begin{pmatrix}n \\ r \end{pmatrix}=\begin{pmatrix}n+1 \\ r \end{pmatrix}$. I'm fine with everything except the inductive step, where I only managed to get as far as $$\sum_{r=0}^{k+1}\begin{pmatrix}n \\ r \end{pmatrix}= \sum_{r=0}^{k}\begin{pmatrix}n \\ r \end{pmatrix}+\begin{pmatrix}n \\ k+1 \end{pmatrix}$$ I tried to then use the identity and substitute $\begin{pmatrix}n \\ r \end{pmatrix}=\begin{pmatrix}n+1 \\ r \end{pmatrix}-\begin{pmatrix}n \\ r-1 \end{pmatrix}$ but all this did was just take me around in a circle. I also couldn't substitute $2^k$ for $\sum_{r=0}^{k}\begin{pmatrix}n \\ r \end{pmatrix}$ since this didn't properly simplify to give $2^{k+1}$. However, I think that I have to obtain the expression $\sum_{r=0}^{k}\begin{pmatrix}n \\ r \end{pmatrix}$ since it simplifies to $2^{k+1}$. But I've not been able to obtain this in anyway so I'm wondering if anyone can guide in what I need to or what direction I need to take to complete the inductive step. Remove particularly tricky items from context menu I've found various sources of info about removing these annoying things which I never use... but I haven't succeeded in getting rid of the following.    (Avira-related:) - Scan selected files with Avira  (I believe all this is M$ crapware... I presume it's all been lovingly crafted so they're all impossible to remove. You never know, though: some super-uber-uber-geek might just have found a way) - Restore previous versions  - Cut - Copy - Delete  - Rename $H_1 ,H_2 \unlhd \, G$ with $H_1 \cap H_2 = \{1_G\} $. Prove every two elements in $H_1, H_2$ commute Will hardware changes affect system? When a spaceship is experiencing a constant acceleration of $10m/s^2$, the astronauts will be moving at nearly the speed of light after about a year in the earth's reference frame. This means the spaceship's energy will start to diverge as a function of the speed $v$ so there will be a huge amount of energy necessary to increase the speed of the ship any further. This way, the speed of light can never be crossed.  All of this is clear to me, but all of this is also formulated in earth's reference frame. But from the astronaut's reference frame: the spaceship is simply accelerating at $10m/s^2$ and so the force on the spaceship is constant. Then why would we need huge amounts of energy to accelerate the spaceship?  For example, I read somewhere that the amount of energy that would be needed to accelerate a large spaceship to half the speed of light is more than 2000 times the current world annual energy consumption. How does this make sense in the astronaut's (non-inertial) frame?