Why does the Equivalent resistance decrease when we connect resistors in parallel? I know current divides and potential difference is constant but my question is that if Equivalent resistance have same flow of current as indiviual resistance then why parallel equivalent resistance is smaller than smallest indiviual resistance. I dont want derivation of parallel resistance formula.
I understand that for resistors in series we need to add R for each resistor to get the equivalent resistance, where R is resistance. I also understand the mathematical manipulations that show that for resistors in parallel we need to add the 1/R for each resistor to get the equivalent resistance. However I have a couple of questions that have to do with the physical interpretation of the mathematical manipulations involved.   First, I am trying to understand why in the case of parallel resistors we have to add 1/R's instead of adding R's as in the case of resistors in series i.e. could it be thought that the current in the wire is like a sort of water flow in the pipe that has two horizontal parallel pipes positioned one on top of the other and in between connected by a number of vertical pipes that have a tap each that could be adjusted to represent different resistances. Then at the top of each vertical pipe the water pressure P exists and at the bottom the pressure will be reduced by P*R.   For example, suppose we have three vertical pipes, so then the pressure at the bottom of each (after the tap) is P*R for the total of 3*P*R so why is the equivalent resistance not 3*R*P but 1/R = (1/R1 + 1/R2 + 1/R3) and how can I be able to visualize this relation?
According to Wikipedia      extra disks should not be mounted in /mnt/ according to the first link, but doesn't seams so according to the second link.  Both say thought that they shouldn't be in /media.  Question  So where should extra permanent disks be mounted? I'm flying Etihad Airways to Melbourne and I've a 14 hours layover at Abu Dhabi.  Some say that as long as you are inside the airport terminal you don't require a transit visa while some say you need a transit visa if your layover is more than 10 hours.  Can I stay inside the airport without a transit visa? Is convex surface normal in cast iron skillet? Please give me the proof for the following: Let $a_1,\,a_2,\,\dots\,a_n$ be $n$ positive real numbers whose product is equal to $1$. Prove that $$a_1+\cdots+a_n \geq n$$ and that the equality sign holds only if every $a_k$ is  equal to $1$. Use Objects.hash() or own hashCode() implementation? Let $$ f(x) =    \begin{cases}\begin{align}     1\quad&amp;\text{ if }\; 0\leq x \leq 1 \\     2 \quad&amp;\text{ if }\; 1&lt;x \leq 2 \\   \end{align}\end{cases}$$  Then $\int^2_0 f(x)dx=3$ and an anti derivative of f(x) is $$F(x)=\begin{cases}\begin{align}     x\quad&amp;\text{ if }\; 0\leq x \leq 1 \\     2x \quad&amp;\text{ if }\; 1&lt;x \leq 2 \\   \end{align}\end{cases}$$ But, $F(2)-F(0)=4-0=4 \neq 3.$ Why has this happened?  Find an anti-derivative $G(x)$ of $f(x)$ such that $G(2)-G(0)=3$, the correct answer.  My attempt: $F(2)-F(0)=4 \neq 3$ because we need to break it up into two parts like this: $\int^2_1 2x dx + \int^1_0 x dx = x^2|^2_0 + \frac{1}{2} x^2|^1_0 =4-1+\frac{1}{2}-0=3.5$ but this is not right either. :(  For the "Find an anti-derivative $G(x)$..."I'm completely lost but perhaps if we solve the first part then I'll understand what it's asking for there.  Any help would be appreciated. I have 3 accounts that I am deleting - however, several hours after the 24 hour (+ grace period - end about 4-5 hours ago), the countdown completed, but disappeared and the accounts are still there.   I would rather not go through another 24 hours, and would like to have these accounts deleted (and yes, I have thought about it and this decision is not up for discussion).  Currently, the accounts are still there.  Could this please be seen to? Insert HTML markup into Gmail signature Is there any way to add HTML markup signature in Gmail settings? I mean, insert some HTML markup without using rich editor, just inserting HTML? Why commutator subgroup is normal to G? If $G$ is a group and $G'$ is generated by $\{xyx^{-1}y^{-1}|x,y\in G\}$, then $G'\trianglelefteq G$ and  $G/G'$ is Abelian.  At first, I thought this is easy because I thought $g[x,y]g^{-1}=[gxg^{-1},gyg^{-1}]$. But it was a mistake. I could not prove that it is also subgroup of G and $G/G'$ is abelian. I need help. Compute $e^{e^z}$, where z is a complex number I am self-studying the MAT 67 linear algebra course and I got stuck at a problem in Chapter 2:     Compute the real and imaginary parts of $e^{e^z}$ for $z$ $\in$ $\Bbb C$.   I tried to expand it using Euler's formula but it seemed endless. Any help? Can anyone explain reverse bias pn junction using only electrons as charge carriers In forward bias condition p-type semiconductor is connected to positive side and ntype semiconductor to negetive side and the electrons flow from n-type to p-type to cathode of battery.  But imagine the same situation when the situation is reverse biased so now p-type is connected to negetive terminal of battery and n-type to positive terminal of battery.  Now there is excess of electrons in n-region and it is connected to positive terminal of the battery.So electron can flow from n-type region to positive terminal of the battery due to potential difference created by battery whereas in p-type electrons can flow from one covalent bond to another as they do in forward biasing.  Only difference is,here electron move in opposite direction (from one covalent bond to another) i.e. from n-region to depeletion region whereas in forward bias they move from depeletion to n-region.Which shouldn't be a problem   Now in this manner current must flow regardless of biasing so what's the correction? Why doesn't a bike/bicycle fall if going with a high speed? Convergence of sequence of real numbers and (sub)subsequence Let $(x_n)_{n\in \mathbb{N}}$ be a sequence of real numbers. A well known result is:     The sequence $(x_n)_{n\in \mathbb{N}}$ converges to $a\in \mathbb{R}$ if and only if every subsequence $(x_{n_k})_{k\in \mathbb{N}}$ converges to $a\in \mathbb{R}$.   Now, I'm working on an exercise which essentialy asks to prove the following:     The sequence $(x_n)_{n\in \mathbb{N}}$ converges to $a\in \mathbb{R}$ if and only if for every subsequence $(x_{n_k})_{k\in \mathbb{N}}$ there is a (sub)subsequence $(x_{n_{k_j}})_{j\in \mathbb{N}}$ which converges to $a$.   Now, one direction is trivial. If in fact $(x_n)_{n\in \mathbb{N}}$ converges to $a$, then by the result I've mentioned, every subsequence of $(x_{n})_{n\in \mathbb{N}}$ converges to $a$. Let then $(x_{n_k})_{k\in \mathbb{N}}$ be an arbitrary subsequence. In that case, we know that it is a subsequence of itself, and hence we've found a (sub)subsequence which converges to $a$.  The other way around seems quite complicated for me. The reason is as follows: I must assume that every subsequence $(x_{n_k})_{k\in \mathbb{N}}$ has a (sub)subsequence which converges to $a$ and the proof that the original one converges to $a$.  My problem is exactly that "a". If it had all (sub)subsequences converging to $a$, then by the result i've mentionted, the subsequence $(x_{n_k})_{k\in \mathbb{N}}$ would converge to $a$. In that case, we would have every subsequence converging to $a$ and the result would follow.  My idea to prove the claim was then:  First remember that a subsequence is the restriction of the original one $x: \mathbb{N}\to \mathbb{R}$ to an infinite subset $\mathbb{N}'\subset \mathbb{N}$. In that case a (sub)subsequence is the restriction of $\tilde{x} = x|\mathbb{N}'$ to an infinite subset $\mathbb{N}''\subset \mathbb{N}$. But since $\tilde{x}$ is the restriction of $x$ to $\mathbb{N}'$ and since $\mathbb{N}''\subset \mathbb{N}'$, the restriction of $\tilde{x}$ to $\mathbb{N}''$ is itself the restriction of $x$ to $\mathbb{N}''$.  In other words, every (sub)subsequence can be viewed as a subsequence of the original one. Considering this, by hypothesis every subsequence has a (sub)subsequence which converges to $a$. But by this argument, this means that we can replace every subsequence with a convergent one. And then we can use the result I stated.  Is my idea correct? Is this really the way to prove the claim? My only doubt is that I need every subsequence converging to $a$ to use the result. In this case I can't see how I can argue that this implies every subsequence converges to $a$.  How can I finish this proof? how to make a regex string reject a blank field Render just part of a final image from Cycles? I'm using an older Nvidia chip that hates running viewport renders, and I need to check if changes to the materials on a few objects not far from the center of my scene are (I'm rendering a floorplan with an orthographic camera from above, so I'm looking down on objects and want to check if the displacement materials are working correctly).  Is there any way to select a bit of the image and have it render first or without having to render the rest of the image? I'm aware that you can tell the image to render from an edge, but my stuff's about equidistant from either side of the scene. Add parenthesis as an accepted Markdown ordered list delimiter The Markdown editor knows that an empty line, followed by a number, followed by a period, followed by a space, is a list item.  This works great It's almost magic  The problem is that I almost always use a close parenthesis for my lists. I think it's a pretty common practice, but the problem is that the editor / Markdown parser doesn't accept parenthesis as an acceptable list delimiter:  This fails horribly This is on the next line I swear Are there some good overviews of basic facts about images and inverse images of sets under functions? How do I translate strings in Twig templates? I am working on a (my first) Drupal 8 project witch is a multilingual site. Drupal 8 really delivers in terms of native support for content translation.  However, I have a few strings set directly in some if the theme .Twig template files printed out like ex.   {{ 'Read more'|trans }}   I am really stuck on where to handle the translation for strings like these within the Drupal 8 installation.  What steps do I need to take to be able to translate these strings into the language currently selected for on the site? Do I need any additional modules other than the ones included in Drupal 8 core?  Steps I have taken:   Activated the Content Translation &amp; Interface Translation modules Added Twig strings like {{ 'Read more'|trans }} Flushed cache and refreshed pages containing the strings Checked admin/config/regional/translate for the strings with no result I wanted to align a set of equations to left. I found on the Internet that you can do it easily with align* environment.   I found it here:   So I did it:  \begin{align*}  f(b)-f(a)=&amp;f((b_1,b_2,\dots,b_k))-f((a_1,a_2,\dots,a_k))=\ &amp;=\sum_{i=1}^{k}[f((a_1,\dots,a_{i-1},b_i,\dots, b_k))-f((a_1,\dots,a_i,b_{i+1},\dots, b_k))  \end{align*}   Unfortunately, I got bunch of red errors saying that there are missing $s. Why? Doesn't that align* environment provide a math environment already? Full Name Requirement of Commit Phase The Commitment form on Area 51 requires your full name to be provided. Is this publicly viewable? Why not just use the existing usernames?