In the derivation for ρgh we consider the weight above a given area to give pressure but because of it pressure should only be in vertical direction why is it found in every direction possible and in every direction it is equal to ρgh, for two opposite nonvertical directions it could have taken any value since at the end Net force is zero in any direction.
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 $d \mathbf{F}=-Pd \mathbf{A}$
Create terrazzo material/pattern with the standard textures is it possible to create this kind of material with the standard textures? The Voronoi texture (set to cells) is coming close but I don’t know if it is possible to a) set the single cells apart from each other and b) to give them seperate colors. Resolve build errors due to circular dependency amongst classes How to maintain a file's 'last modified' date? The function $f(z)=z+\frac{1}{z}$ maps $\Omega$ biholomorphically on the upper plane The question states:     Let $\Omega$ be the upper half plane minus the unit disk. Show that the map $f: z \mapsto z+1/z$ maps $\Omega$ biholomorphically on the upper plane. Describe the preimages of the lines $\mathrm{Im}(z)=\text{a constant}&gt;0$.   So how exactly does one show the mapping is biholomorphic. I'm not sure where I should start, so unfortunately I haven't made any progress Resolving heteroscedasticity in Poisson GLMM Astrophotography picture with too much noise. How do I correct this in postprocessing? This is the Milky Way above Kakadu National Park, Australia, and my first try at astrophotography.    Thirty second exposure, f/4.0 at 17mm with ISO 12800 (!!) which in hindsight probably was way too high and resulted in a lot of noise. I also used Picasa to increase contrast, but I probably went a bit overboard there as well.  , I am looking for input on how to better postprocess this picture to reduce noise. Let $(\mathcal{H}, (\cdot, \cdot))$ be a complex Hilbert space, and $A :  \mathcal{H} \to \mathcal{H}$ a positive, bounded operator ($A$ being positive means $(Ax,x) \ge 0$ for all $x \in \mathcal{H}$). Prove that $A$ is self-adjoint. That is, prove that $(Ax,y) = (x, Ay)$ for all $x,y \in \mathcal{H}$. Here's what I have so far. Because $A$ is positive we have $\mathbb{R} \ni (Ax,x) = \overline{(x,Ax)} = (x,Ax)$, all $x \in \mathcal{H}$. Next, I have seen some hints that tell me to apply the polarization identity: $$(x,y) = \frac{1}{4}((\lVert x+y \rVert^2 + \lVert x-y \rVert^2) - i(\lVert x + iy \rVert^2 - \lVert x - iy \rVert^2)),$$ where of course the norm is defined by $\lVert \cdot \rVert^2 = (\cdot, \cdot)$. So my guess is that I need to start with the expressions: $$(Ax,y) = \frac{1}{4}((\lVert Ax+y \rVert^2 + \lVert Ax-y \rVert^2) - i(\lVert Ax + iy \rVert^2 - \lVert Ax - iy \rVert^2)),$$ $$(x,Ay) = \frac{1}{4}((\lVert x+Ay \rVert^2 + \lVert x-Ay \rVert^2) - i(\lVert x + iAy \rVert^2 - \lVert x - iAy \rVert^2)),$$ and somehow show they are equal. But here is where I have gotten stuck. Hints or solutions are greatly appreciated. Where does the term "fun fact" originate?-- namely, not with the compositional meaning but rather with the idiomatic usage to introduce some sort of unusual, esoteric, absurd or otherwise "controversial" fact? 1.13 Detect if no mobs with a specific tag exist and execute commands My 1.12 solution can be found . When trying to update command blocks to 1.13, I have the following:  scoreboard objectives add invert dummy   (run at start) then:  scoreboard players set FakePlayer invert 1   (impulse unconditional repeat) chained into:  execute at @e[tag=summoned] run scoreboard players reset FakePlayer invert   (chain unconditional repeat) chained into:  execute if score FakePlayer invert = 1 invert run say "All summoned animals are dead!"   This should reset FakePlayers score continuously until all entities with tag "summoned" are dead, then it should run the final condition. This does not happen, and I can't see why. "New Item" in Finder's right-click menu In Windows when you're in File Explorer, you can right-click anywhere and create a new item of various types (there's then a submenu option to select which type: text file, ZIP file, etc., just to name a few). In Finder there seems to be no such option. I imagine the expected workflow on Mac would be to open an Application and create a new file from within the application, then save it. But I sometimes prefer this alternative workflow of creating a blank file first, and then opening it as the entrypoint into the application.  Is there some way to configure Finder to add a right-click option to create new, empty files? Return matched elements' attribute ar array using jQuery Can a virtualized machine have the CPU and RAM resources of multiple underlying physical machines? We are studying implementing some virtualized servers here, but we don't know what will be better suitable for us. Some folks are saying better have two huge servers, and others are saying have like a ten middle-end servers.  We have a legacy Visual Foxpro application, which nowadays run on Dual Xeon E5405 @ 2GHz and 16Gb of RAM. The currently server its getting too slow due the number of active users and process running on it. Virtualizing this server will give us the benefit of an faster disaster recovery.  So the question is, having like ten physical servers running at 1.7GHz and 4Gb of RAM, we could virtualize one server into 4 machines, and have one virtualized server running at 6.8GHz and 16Gb of memory? If yes, there is some how if one machine stops, automatically manage this virtual machine to another one, and execute the appropriate maintenance on it, and later back to it again? What identifies an action-reaction pair of forces? This is a question on a worksheet that I'm giving to a student:  A man of weight $W$ steps into the lift. The lift then moves downwards with a constant speed. If $R$ represents the force acting on the man by the lift floor, which of the following statements is/are correct? (3) $R$ and $W$ are an action-reaction pair according to Newton's third law  Why isn't this statement (3) correct? As we know, whenever school ends, Harry is sent back to live with the Dursleys, where he suffers from abuse. He would much prefer to live with the Weasleys, and often visits there before returning to school. It stands to reason that they probably don't live that far from each other, since:   Both Harry and the Weasleys use Platform 9&frac34; to get to school Ron was able to take Harry to the Burrow in a flying car within a reasonable amount of time   It's heavily hinted at that the Weasleys are poor. However, it's not like Harry would live there permanently. Three months isn't that bad, not to mention he's pretty well-off from his parents' money. There would be little to no extra expense on the Weasleys.  The only explanation is that Dumbledore goes out of his way to ensure Harry lives with the Dursleys. If so, why is this? Quickly calculate date differences Flag pop-up hidden beneath answer box (and it's still transparent!) The flag window on (a comment to) the last question is hidden beneath the answer box, as shown in the following screenshot:    Linked: , which is currently  - this is not fixed for flags on comments! I accidentally closed the window for the verfication code I wanted to create an account for Ubuntu One, but I got impatient when the email with the verification code didn't arrive... I thought I might have typed the wrong address, closed the window and tried again. it told me the address already existed and I just got the code. But since I closed the window I missed it. Is there any way to recover the code? How to prevent non-alphanumeric input in javascript? Let $P$ be a probability measure on $(\Omega, \mathscr{F})$, If $a,b,\in\mathscr{F}$ and $A\subset B$, then $P(B)-P(A)=P(B\setminus A)\ge 0$.  So far I have:  $A\subset B \implies B\setminus A = B\cap A^c = \emptyset$ as $A$ and $B^c$ are disjoint. Since $A\cup (B\setminus A)=A\cup (B\cap A^c)=(A\cup B)\cap (A\cup A^c)=B\cap\Omega = B$  Then for $A,B\in \mathscr{F}\implies P(A\cup (B\setminus A))=P(A)+P(B\setminus A) = P(B)$ by countable additivity.  Rearranging, $P(B)-P(A)=P(B\setminus A) \ge 0$. Use a execute command once