From Wiki and from physics fundamentals lections I received info that  is scalar value. But in definition you have relation between projections of two vector values to normal axe-force and surface area with normal orientation. Are you interesting only in absolute values of this two vectors? (if so then the pressure is always positive.)   Can pressure be negative?  I don't want to discuss the averaged model of the pressure compare to tension elastic model.  At current moment from the post answers and comments I realized that pressure scalar field can be defined so:    $f$ is support force to support equilibrium in $ds$ surface under other molecules attack. pressure scalar value in some circumstances can be negative. If smth. was wrong here please let me know!
I have a final exam tomorrow for fluid mechanics and I was just looking over the practice exam questions. They do not provide solutions. But pretty much I have to define pressure at a point and also say why pressure is scalar instead of a vector.  I am thinking pressure at a point is $P=\lim_{\delta A \to 0} \frac{\delta F}{\delta A}$. Please let me know if I am wrong.   But I do not know at all why pressure is a scalar instead of a vector. I know it has something to do with $d \mathbf{F}=-Pd \mathbf{A}$
Is the collection of all functions between two sets a set? Are S.P.E.C.I.A.L. skills capped at 10? We know that the maximum level a dweller can train to on any SPECIAL skill is 10, however this does not consider bonuses given from their outfit.    Is there any additional benefit for going over 10? For example, a dweller with an Intelligence skill of 10 wearing a lab coat that boosts Intelligence by 3?   If it is a hard cap, would it be more useful to use outfits that raise a secondary characteristic instead? Is it possible to recover Classical Mechanics from Schrödinger's equation? Let me explain in details. Let $\Psi=\Psi(x,t)$ be the wave function of a particle moving in a unidimensional space. Is there a way of writing $\Psi(x,t)$ so that $|\Psi(x,t)|^2$ represents the probability density of finding a particle in classical mechanics (using a Dirac delta function, perhaps)? How do I install VMware player on a Mac os X? I went to their site, and downloaded a .bundle which automagically turned into a .txt file after downloading.   Long story short, I can't run the txt file because I guess Mountain Lion Mac os X isn't a strict Linux box and is missing some core components to do that type of installation.  Does anyone know how to install VMware? Does $\int_a^bf(x) = 0$ imply $f(x) = 0$ for a Riemann Integral function? Let $a &lt; b$ and $f:[a, b] \rightarrow \mathbb R$ be a Riemann integral function. Suppose that $f(x) \geq 0$ for all $x \in [a, b]$. I want to show whether $\int_a^bf(x) = 0$ implies $f(x) = 0$ or not, but I'm not sure where to start. I don't know much about Riemann integral functions in general. I only know that $$\underline \int_a^b f= \overline \int_a^b f.$$ Also, I know the fact that if $f$ is continuous, then $f(x) = 0$ for all $x\in [a, b]$. Can anyone provide any hints for me, because I'm really confused about this whole thing. What does first authorship really mean? In my field (theoretical computer science), authors of any paper are always listed alphabetically; our papers .  (Well... hardly ever.)  In most other disciplines, at least within science and engineering, the ordering of authors is a signal about their relative contributions to the paper, with the first author indicating the most significant contributor.  Hiring and promotion committees do give extra weight to "first-author papers" (and sometimes have to be reminded that not all areas have them).  As an outsider, I find this practice confusing.  What does first authorship actually mean in your discipline? I understand vaguely that the first author is supposed to be the one who "", but what counts as "work" in this comparison?  Does "most" mean "more than all the other coauthors together" or just "more than any other coauthor"?  What happens when the comparison is unclear?   How often is "did the most work" the actual truth, versus a cover story for a more complex political decision?  I realize that the precise answer is different for every paper.  I'm looking for general guidelines for how an outsider (like me) should interpret first authorship in your field.  Pointers to guidelines from journals or professional societies would be especially helpful.  Please give only one answer per discipline. I want to leave my Mac awake overnight so it syncs 4GB of data from Dropbox. I need to prevent the machine from going to sleep because that pauses the Dropbox sync  I've been using  to keep the computer awake, but it also keeps the computer unlocked which is insecure.  I need a method to keep my Mac awake (so that Dropbox can sync overnight) but also locked (so no one can read my email). It'd be great if this method also turned off the screen (to preserve power), but honestly that's not even 100% necessary. Duplicate questions in feed, and questions incorrectly marked as unanswered I see duplicate questions in the feed (Why is the book still wanted if Aida has memorized it?) and the question is marked as "unanswered" even though it has an upvoted answer.  I would expect unique questions in the feed, as well as questions with answers to not be marked as unanswered.   App Version: 1.6.3.4 Device: iPhone 6 OS Version: Version 10.2 (Build 14C92) How many times to roll a die before getting two consecutive sixes? Basically, on average, how many times do you have to roll a fair six-sided die before getting two consecutive sixes? How to remove title entry in the bibliography contents? I use plain bibliography style for my book. When I compile it, by default, I get the details including the title in my reference. like this:     H. J. C. Berendsen, J. P. M. Postma, W. F. van Gunsteren, a. Dinola, and J. R. Haak. Molecular dynamics with coupling to an external bath. J. Chem. Phys., 81:3685-3690,1984.   This is too long and I have more than 100 such references which makes the document too long. How can I force it to remove the title from this reference? something like this:     H. J. C. Berendsen, J. P. M. Postma, W. F. van Gunsteren, a. Dinola, and J. R. Haak.  J. Chem. Phys., 81:3685-3690,1984.   Your answer will be greatly appreciated.  By the way, I still would like to use the plain bibliography style. When we define the S-matrix, what are "in" and "out" states? I have seen the scattering matrix defined using initial ("in") and final ("out") eigenstates of the free hamiltonian, with  $$\left| \vec{p}_1 \cdots \vec{p}_n \; \text{out} \right\rangle = S^{-1} \left| \vec{p}_1 \cdots \vec{p}_n \; \text{in} \right\rangle$$  so that  $$\left\langle \vec{p}_1 \cdots \vec{p}_n \; \text{out} \mid \vec{q}_1 \cdots \vec{q}_m \; \text{in} \right\rangle = \left\langle \vec{p}_1 \cdots \vec{p}_n \; \text{in} \mid S \mid \vec{q}_1 \cdots \vec{q}_m \; \text{in} \right\rangle.$$  1) What are "in" and "out" states?   2) Are they Fock states?   3) In Schrödinger or Heisenberg or interaction representation?   4) How are they related? (I believe that I see what they handwavily represent physically, but not formally.)  My main issue is that, if "in" and "out" states are one-particle eigenstates of the free hamiltonian, i.e. if $\left| \vec{p}_1 \text{out} \right\rangle$ describes a free particle with momentum $\vec{p}_1$, and $\left| \vec{p}_1 \text{in} \right\rangle$ also describes a free particle with momentum $\vec{p}_1$, then $\left| \vec{p}_1 \text{out} \right\rangle = \left| \vec{p}_1 \text{in} \right\rangle$ ... which is false. Still, books (some at least) describe these "in" and "out" states like that.  Moreover, I have seen (e.g. in Wikipedia, but also ) that the scattering matrix is a map between two different Fock states, and I don't understand that.   5) Do states of the interacting system live in the same Fock space that asymptotic free states?   6) And if not, where do they live?  Understandable references would be appreciated. try+gerund/infinitive (Difference in meaning) I was very inspired by Michio Kaku's  on the possibilities of quantum computing and also listening to Talia Gershon's  on it.   As I come from a business &amp; analytics background, what materials can I begin exploring to prepare to learn more about the field of quantum computing? Eg. what would I need to study to understand Peter Shor's algorithm,  on physics, etc. Privilege Notifications Determine the Galois group of $\mathbb{Q}(\sqrt{a+b\sqrt{d}})$ external variable in awk The binomial expansion:  $$(x+y)^{n} = \sum_{k=0}^{n} \binom{n}{k} x^k y^{n-k}$$  The General Leibniz rule (used as a generalization of the product rule for derivatives):  $$(fg)^{(n)} = \sum_{k=0}^{n} \binom{n}{k} f^{(k)} g^{(n-k)}$$  Both formulas can be obtained simply by induction; Newton's binomial also has a combinatorial proof . It's striking how these formulas are similar; is there a possible connection between them? I was thinking that maybe the General Leibniz rule could be obtained using a combinatorial argument as well (hence the binomial coefficients)... Allow cross-site queries in data explorer Square roots not on same height This part C\frac{1}{\sqrt{G}}\sqrt{\rho} doesn't seem to be on the same height as this part \sqrt{b}\sqrt{l} when compiled. What am I doing wrong?  \begin{flalign*} T=C\frac{1}{\sqrt{G}}\sqrt{\rho}\sqrt{b}\sqrt{l} &amp;&amp; \end{flalign*} Question: Are there infinitely many primes of the form $4n+3$ and $4n+1$?  My attempt: Suppose the contrary that there exist finitely many primes of the form $4n+3$, say $k+1$ of them: $3,p_1,p_2,....,p_k$  Consider $N = 4p_1p_2p_3...p_k+3$, $N$ cannot be a prime of this form. So suppose that $N$=$q_1...q_r$, where $q_i∈P$  Claim: At least one of the $q_i$'s is of the form $4n+3$:  Proof for my claim: $N$ is odd $\Rightarrow q_1,...,q_r$ are odd $\Rightarrow q_i \equiv 1\ (\text{mod }4)$ or $q_i ≡ 3\ (\text{mod }4)$  If all $q_1,...q_r$ are of the form $4n+1$, then $(4n+1)(4m+1)=16nm+4n+4m+1 = 4(\cdots) +1$  Therefore, $N=q_1...q_r = 4m+1$. But $N=4p_1..p_k+3$, i.e. $N≡3\ (\text{mod }4)$, $N$ is congruent to $1\ \text{mod }4$ which is a contradiction.  Therefore, at least one of $q_i \equiv 3\ (\text{mod }4)$. Suppose $q_j\equiv 3\ (\text{mod }4)$  $\Rightarrow$ $q_j=p_i$ for some $1\leq i \leq k$ or $q_j =3$  If $q_j=p_i≠3$ then $q_j$ | $N = 4p_1...p_k + 3 \Rightarrow q_j=3$ Contradiction!  If $q_j=3$ ($\neq p_i$, $1\leq i \leq k$) then $q_j | N = 4 p_1...p_k + 3 \Rightarrow q_j=p_t$ for some $1 \leq i \leq k$ Contradiction!  In fact, there must be also infinitely many primes of the form $4n+1$ (according to my search), but the above method does not work for its proof. I could not understand why it does not work. Could you please show me?  Regards