Why does the virtual image appear right on top of the object and not a little to the right, for example? Is it explained by some formula or just symmetry of the geometry?
What is the formula for Real vs Apparent Depth, when not looking perpendicular to the interface?   I know the formula $$\frac{\text{real depth}}{\text{apparent depth}}=n$$   but I'm not looking for this.
What are independent parameters in Hellmann–Feynman theorem? A typical example in textbooks about the application of Hellmann–Feynman theorem is calculating $\left\langle\frac{1}{r^2}\right\rangle$ in hydrogen-like atoms.  has a nice demonstration of this. At some point in the Wikipedia derivation is used that $$\frac{\partial n}{\partial \ell}~=~1. \tag{1}$$   But why is eq. (1) true? I know that  $$n=n_r+\ell+1,$$   but $n_r$ is just another variable with different physical meaning, so why is $n_r$ independent from $\ell$, whereas $n$ is not? The Wikipedia proof for Hellmann–Feynman theorem does not address the problem of independence of different parameters. What variables are kept fixed during the differentiation $(1)$ and why?  The Wikipedia page seems to have only a vague notion of $\frac{\partial\hat{H}}{\partial\lambda}$ and $\frac{\partial E}{\partial\lambda}$, unlike in, e.g., thermodynamics, where all partial derivatives are typically written like  $$\left( \frac{\partial U}{\partial V} \right)_S \qquad ,\qquad  \left( \frac{\partial U}{\partial V} \right)_p \qquad ,\qquad  \ldots $$ so that it is clear which variables are kept fixed during the differentiations. How to space columns evenly in a table with multicolumns? Getting Started with Unit Testing Unit testing is, roughly speaking, testing bits of your code in isolation with test code. The immediate advantages that come to mind are:  Running the tests becomes automate-able and repeatable You can test at a much more granular level than point-and-click testing via a GUI    My question is, what are the current &quot;best practices&quot; in terms of tools as well as when and where to use unit testing as part of your daily coding? Lets try to be somewhat language agnostic and cover all the bases. how does systemd determine service is stopped? If I left food out of the refrigerator for some period of time, is it still safe? If I left it out too long, can I salvage it by cooking it more? Application of Complex Numbers I was just wondering how complex numbers can be applied in electrical engineering and why we use complex numbers over regular, real numbers for this application (e.g what capabilities does the complex number have that real numbers do not in electrical engineering)?  I have done some research concerning impedances and understand how they are written in complex form, however I am still confused why complex numbers are necessary in this field over regular numbers. Difference between Hashing a Password and Encrypting it Is there a built-in checksum/hash utility on Windows 7? Gimp: how to remove background noise/artifacts and enhance handwritten text I have a lots of scanned handwritten lecture notes (PDFormat) and I want to improve their readability (I have uploaded an example). In particular I'd like to remove the "squares" on the back. I would appreciate any advice you may have. Thanks in advance. Can Quantum Entanglement and Quantum Superposition be considered the same phenomenon? Quantum entanglement is known to be the exchange of quantum information between two particles at a distance, while quantum superposition is known to be the uncertainty of a particle (or particles) being in several states at once (which could also involve the exchange of quantum information for a particle that is known to be in several locations simultaneously).  I was wondering if all of this was nothing more but the exchange of quantum information between different masses, and if this could clear up all the confusion in terms of how quantum systems connect in this field of science.  A clear explanation for how both of these quantum phenomena work, and if they really are connected (the exchange of quantum information?) would be much appreciated. Use of 'prototype' vs. 'this' in JavaScript? Please add the ability to fold blocks of code in questions and answers It would be great to have the option to fold code snippets, entire questions and answer posts on a user basis and allow defining some generic foldable regions within the question/answer editor.  Non generic folding info could be stored in cookies perhaps to move the storage/processing requirement to the client side.  Definable folding can take any free and safe HTML tag or other multi character trick (&lt;>Fold this&lt;>) Are there LaTeX performance profiling tools? Evaluating the limit of a sequence given by recurrence relation $a_1=\sqrt2$, $a_{n+1}=\sqrt{2+a_n}$. Is my solution correct? I'm currently reading the TeXbook, and I use \show a lot to see how Knuth defined plain-TeX macros in terms of TeX primitives. For instance, to have access to the definition of \centerline, I would write \show\centerline in my input file. When I compile  that file with pdftex in interactive mode from the command line (a Unix terminal, in my case), TeX generates an error message that shows me the definition of \centerline in the terminal and offers me a menu of options (as described in Chapter 6: Running TeX in the TeXbook).  I understand that design choice by Knuth of having TeX generate an error message for each instance of \show: he probably wanted to leave the user some time to read what \show... prints in the terminal before letting TeX resume execution, instead of having to open the log file only after the end of the compilation.  However, instances of \show... in the input file have the undesirable side-effect of raising "errors" (false alarms, really) when pdftex is called from within an IDE such as TeXmaker with option -interaction=nonstopmode. For example,  \show\centerline \bye   when compiled with pdftex -interaction=nonstopmode causes TeXmaker to report     Obviously, those false alarms prove really annoying when debugging. I always have to ask myself: Wait... are those errors due to instances of \show in my input file, or are they bona fide errors? On the other hand, my understanding is that you have to specify  -interaction=nonstopmode   if you want to call pdftex from within an IDE. Therefore, my question is:  Is there any way of preventing TeXmaker from reporting those "false alarms"?  Note: the problem I describe may be present in IDEs other than TeXmaker as well. Nullable type issue with ?: Conditional Operator Creating additional Virtual Host Ubuntu Server Lost passport with Schengen visa How can I detect when an Exception's been thrown globally in Java? How can I detect when an Exception has been thrown anywhere in my application?  I'm try to auto-magically send myself an email whenever an exception is thrown anywhere in my Java Desktop Application.  I figure this way I can be more proactive.  I know I could just explicitly log and notify myself whenever an exception occurs, but I'd have to do it everywhere and I might(more likely will) miss a couple.  Any suggestions? I have noticed for a few months that my mouse will randomly jerk across the screen. I'm super paranoid about security. But this is just silly!  Has anyone else experienced this, and found out what causes it? It happens to me every couple of days. The cursor will jump across the screen about 1000+ pixels. I thought it was my mousepad - changed it, and that's not it.  I imagine it's just a hardware issue.  This is on a PC with a Razer Naga, wired mouse. The mouse worked for about a year before I started noticing these issues. They have persisted even after a reformat / reinstall.