Simple, possibly stupid question, but here it is.   So I know $$\nabla \cdot\vec E = \frac{\rho}{\epsilon_0},$$ but how do I use a given charge density to calculate an electric field?   I guess the root of my question is about the mathematical procedure of "undoing" the div operator.
This question is a sequel of sorts to my  about .  In the paper, the authors performed molecular dynamics (MD) simulations of parallel-plate supercapacitors, in which liquid resides between the parallel-plate electrodes.  The system has a "slab" geometry, so the authors are only interested in variations of the liquid structure along the $z$ direction.  In my previous question, I asked about how particle number density is computed.  In this question, I would like to ask about how the electric potential is computed, given the charge density distribution.  Recall that in CGS (Gaussian) units, the Poisson equation is   $$\nabla^2 \Phi = -4\pi \rho$$  where $\Phi$ is the electric potential and $\rho$ is the charge density.  So the charge density $\rho$ is proportional to the Laplacian of the potential.  Now suppose I want to find the potential $\Phi(z)$ along $z$, by integrating the Poisson equation.  How can I do this?  In the paper, on page 254, the authors write down the average charge density $\bar{\rho}_{\alpha}(z)$ at $z$:  $$\bar{\rho}_{\alpha}(z) = A_0^{-1} \int_{-x_0}^{x_0} \int_{-y_0}^{y_0} dx^{\prime} \; dy^{\prime} \; \rho_{\alpha}(x^{\prime}, y^{\prime}, z)$$  where $\rho_{\alpha}(x, y, z)$ is the local charge density arising from the atomic charge distribution of ionic species $\alpha$, $\bar{\rho}_{\alpha}(z)$ is the average charge density at $z$ obtained by averaging $\rho_{\alpha}(x, y, z)$ over $x$ and $y$, and $\sum_{\alpha}$ denotes sum over ionic species.  The authors then integrate the Poisson equation to obtain $\Phi(z)$:  $$\Phi(z) = -4\pi \sum_{\alpha} \int_{-z_0}^z (z - z^{\prime}) \bar{\rho}_{\alpha}(z^{\prime}) \; dz^{\prime} \; \; \; \; \textbf{(eq. 2)}$$  My question is, how do I "integrate the Poisson equation" to obtain equation (2)?  How do I go from  $\nabla^2 \Phi = -4\pi \rho$ to equation (2)?  In paricular, where does the $(z - z^{\prime})$ factor come from?   Thanks for your time.
For any natural number $n\ge 1$, given pairs $(a_1,b_1),(a_2,b_2),...,(a_n,b_n)$ of integer numbers, there exist integer number $c$ and $d$ such that  $$\prod_{i=1}^{n}(a_i^2+b_i^2) = c^2+d^2$$    My initial approach is  Base Case: $(a_1^2+b_1^2) = a_1^2+b_1^2$ which is true. (Although it is trivial)  Prove the statement is true when $n=2$: We have $$(a^2+b^2)(c^2+d^2) = (ac-bd)^2+(ad+bc)^2$$ (Thanks André Nicolas for pointing it out)   So if $a,b,c,d$ are integers, $ac,bd,ad,bc$ are all integers and integers are closed under addition and subtraction. Hence $(ac-bd),(ad+bc)$ are integers.   Inductive Hypothesis: $\prod_{i=1}^{n}(a_i^2+b_i^2) = c^2+d^2$ is true  Inductive Step: $$\prod_{i=1}^{n+1}(a_i^2+b_i^2) = \prod_{i=1}^{n}(a_i^2+b_i^2)\cdot (a_{n+1}^2+b_{n+1}^2) = (c^2+d^2)\cdot (a_{n+1}^2+b_{n+1}^2)$$  Where $c$ and $d$ are integers.   But when we apply $n=2$, we have  $(c^2+d^2)\cdot (a_{n+1}^2+b_{n+1}^2) = (e^2 + f^2)$ where $e$ and $f$ are integers.  Hence, by the principle of induction, the statement we needed to prove is true. Is it "re-offend" or "reoffend"? I want to know whether there is a hyphen in the word re-offend, or if it is spelt reoffend. I looked in Oxford English dictionary and the word "reoffend" appears, but then I checked Merriam-Webster and the word is not in there.  So is reoffend a word or is it re-offend?  I'm in Australia where we observe UK English if that matters. Prove that the point of intersection of the diagonals of the trapezium lies on the line passing through the mid-points of the parallel sides Prove,by vector method,that the point of intersection of the diagonals of the trapezium lies on the line passing through the mid-points of the parallel sides.    My Attempt: Let the trapezium be $OABC$ and that the O is a origin and the position vectors of $A,B,C$ be $\vec{a},\vec{b},\vec{c}$.Then the equation of $OB$ diagonal is $\vec{r}=\vec{0}+\lambda \vec{b}................(1)$ And equation of $AC$ diagonal is $\vec{r}=\vec{a}+\mu(\vec{c}-\vec{a}).......(2)$ And the equation of the line joining the mid points of $OA$ i.e.$\frac{\vec{a}}{2}$ and $BC$ i.e. $\frac{\vec{b}+\vec{c}}{2}$ is $\vec{r}=\frac{\vec{a}}{2}+t(\frac{\vec{b}+\vec{c}}{2}-\frac{\vec{a}}{2}).........(3)$. Here $\lambda,\mu,t$ are scalars. I do not know how to solve $(1)$ and $(2)$ and put into $(3)$ to prove the desired result. Please help me.Thanks. Occasionally I see an old Meta post that has no status tag (like ), but from the answers it looks like the question is resolved.  Example:  should be  since the answers say HTTP compression actually is enabled.  Is it worth flagging these questions, or is that a waste of time? Suppose we have a sample of size n= 9, where the sum equals 27, and the sum of the squaredvalues equals 113. Sample mean is 27/9=3 and mean of squared values is 113/9 = 12.56 Can we then say that the sample variance $S^2 = E(\overline X^2) - E(\overline X)^2 = 12.55 - 9 = 3.56$ ? Proving $|P(A\cap B)-P(A)P(B)|\leq \frac{1}4$ Let $A$ and $B$ be two events of a probability space. Prove that  $\displaystyle|P(A\cap B)-P(A)P(B)|\leq \frac{1}4$   I think it's a very challenging problem, and I've made no progress so far ...  Can someone give me a hint ? Help + Noun + Gerund or Infinitive Help my sister peel oranges. Help my sister to peel oranges. Help my sister peeling oranges.  Help my sister with peeling oranges.   Which of the above is/are correct, and why are the others incorrect? What is the maximum possible length of a .NET string? What is the longest string that can be created in .NET? The docs for the String class are silent on this question as far as I can see, so an authoritative answer might require some knowledge of internals. Would the maximum change on a 64-bit system?  [This is asked more for curiosity than for practical use - I don't intend to create any code that uses gigantic strings!] I plan on building a computer with  which has no GPU:     Graphics Specifications      Processor Graphics:   None   on a  motherboard which has an onboard integrated graphics processor:     Integrated Graphics Processor:         1 x D-Sub port, supporting a maximum resolution of 1920x1200@60Hz   1 x DVI-D port, supporting a maximum resolution of 1920x1200@60Hz   1 x HDMI port, supporting a maximum resolution of 4096x2160@24Hz or 2560x1600@60Hz   Support for up to 3 displays at the same time   Maximum shared memory of 512 MB      I have asked  and know that even if the CPU has no GPU, I can still get video as long as the motherboard has onboard graphics. However, when I went to buy the parts at a computer center yesterday, the technician said motherboards nowadays use CPU's GPU to display, including this Gigabyte GA-Z97X-UD5H motherboard (which he meant this board has no real onboard graphics).  So for Xeon E3 1231v3 + Gigabyte GA-Z97X-UD5H to display, do I need a video card or not?  (I don't really care about the graphics quality. I use the computer mostly to write programs, web browsing, and Photoshop, and seldom play games or watch videos.) Is there a (basic) way to understand why a limit cannot be brought inside of a sum, in certain situations? I have seen that the method of "dominated convergence" implies that sometimes a limit cannot be brought inside of a sum, but is there a more basic way to understand this, preferably without using integrals?   A conceptual or simple answer would be helpful. Thank you! Plot an elliptic curve in Latex A simple stupid question that I should already know the answer to:  Is this true: $\left|z^5\right| = \left|z\right|^5$ where z is a complex number with a non-zero imaginary component.  Thanks in advance. Number Theory: Find all solutions of $\phi(n)=16$ and $\phi(n)=24$ I have this homework problem assigned and I'm a little confused in solving it:  Find all solutions of $\phi(n)=16$ and $\phi(n)=24$ (where $\phi(n)$ is the Euler phi-function).  The hint that's provided says:  If $n=p_1^{k_1}p_2^{k_2}\cdots p_r^{k_r}$ satisfies $\phi(n)=k$, then $n=\frac{k}{\prod (p_i-1)}\prod p_i$. Thus the integers $d_i=p_i-1$ can be determined by the conditions (1) $d_i\mid k$, (2) $d_i+1$ is prime, and (3) $\frac{k}{\prod d_i}$ contains no prime factor not in $\prod p_i$.  I'm ok with proving the statements in the hint and I know how to get them, but I'm not sure how to apply them to solve the problem. Fourier series identity I need to prove that $\dfrac{a \sin(bx)}{1 - 2a \cos(bx) + a^2} = \sum_{n=1}^\infty a^n \sin(nbx)$ where $|a| &lt; 1$.  It seems that this can be proved by using Euler's formula identities for $\cos(bx)$ and $\sin(bx)$ and substituting $z = e^{ibx}$. From that, I get $$ \dfrac{a \sin(bx)}{1 - 2a \cos(bx) + a^2} = \dfrac{a (z - 1/z)}{2i(1 - a(z + 1/z) + a^2)}$$ $$= \dfrac{a (z^2 -1)}{2i(z - a(z^2 +1) + za^2)}$$ $$= \dfrac{a (z^2 -1)}{-2ai(z^2 - z(1-a^2)/a + 1)}$$  I don't understand what I am supposed to do next. "I wish I didn't do" vs. "I wish I hadn't done" When I define amsthm theorem environments with shared counters, autoref messes up the names of the references. For example, in the output below, we should have "Definition 2" instead of "Theorem 2".  \documentclass{article}  \usepackage{amsthm} \usepackage{hyperref}  \theoremstyle{theorem} \newtheorem{theorem}{Theorem} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition}  \begin{document}  \begin{theorem} \label{wonderful-theorem} This is a wonderful theorem. \end{theorem}  \begin{definition} \label{awesome-definition} This is an awesome definition. \end{definition}  Look at the wonderful \autoref{wonderful-theorem} and the awesome \autoref{awesome-definition}.  \end{document} I have a file containing lists on Solaris:  List A hi hello hw r u  List B Hi Yes  List C Hello   I need to transpose the lists as shown below:  List A    List B    List C hi        Hi        Hello hello     Yes hw r u   How can I do this on Solaris? Why can ECC key sizes be smaller than RSA keys for similar security? Outcast boy meets girl underwater Here's another one I can't find, but somebody will probably recognize right away. A young man has been thrown out of respectable society. To make sure he stays away from people (and they stay away from him), those who have condemned him alter his body so that he always smells bad. Of course, no one can smell underwater - and that's where he meets a girl, while they're both swimming underwater in a swimming pool. I don't remember a lot of details about this story - but I remember the main character was not very nice - he was often mean to other people. Whether he'd always been like that, or whether his nastiness was because everyone had rejected him, I can't remember. I would have read this story in the mid-seventies, but it was probably written earlier. So far, I understand that I have to look at each Jordan block. How do I prove that for a Jordan block with a value of x, $J_n(x)$, the Jordan decomposition of $(J_n(x))^2$ is $J_n(x^2)$ ? I have a feeling this will lead me to the solution. Thanks for your time.