I know that if a ray is incident at the medium boundary at critical angle, the ray after refraction moves towards the medium boundary. So, is it possible that if a ray is passed through the medium boundary, it will deviate towards the denser medium? if not then can it be called an exception of principle of reversibility of light rays. Please help me through this paradox.
A lot of textbooks and exam boards claim that light incident at exactly the critical angle is transmitted along the media boundary (i.e. at right-angles to the normal), but this seems to violate the principle of reversibility in classical physics.  How would a photon or ray travelling in the reverse direction "know" when to enter the higher refracting medium?  It can't know, so I conclude that such light is simply reflected?  Is this correct?
One-sided limits of $f'(x)$ at a point vs. one-sided limits of the difference quotient at that point I know that if a function is piecewise, then when differentiating it one needs to deal separately with the points of intersection of the pieces, because the derivative may not be defined there.  Also, if a one-sided limit of $f'(x)$ at these "crucial points" is infinite or doesn't exist, it doesn't preclude the limit of the difference quotient from existing and being finite. For example, take $f(x)=x^2\sin\frac1x, f(0)=0$: $f'(x)= 2x\sin\frac1x-\sin\frac1x $ has no limit as $x\to0$ but $\lim\limits_{h\to0}\frac{f(h)-f(0)}{h}=0=f'(0)$.  On the contrary, looking at $f(x)=\sin x +\lvert \sin x\rvert $ on $[0,2\pi]$ one has $f'(x)=2\cos x$ on $(0,\pi) $ and $f'(x)=0$ on $(\pi,2\pi)$ so $\lim\limits_{h\to0^+}f'(\pi+h)\ne\lim\limits_{h\to0^-}f'(\pi+h)$, but they are both finite. And in this case it's easy to find that they agree with the one-sided limits of the difference quotient. Why? Does the finiteness of the former imply that they agree with the latter? Is this also true if the one-sided limits of $f'$ at the crucial points agree with each other? The MySQL manual at  covers this.  Usually I just dump the database and reimport it with a new name. This is not an option for very big databases. Apparently RENAME {DATABASE | SCHEMA} db_name TO new_db_name; .  This needs to work with , which stores things very differently than . At work today, I came across the volatile keyword in Java. Not being very familiar with it, I found . Given the detail in which that article explains the keyword in question, do you ever use it or could you ever see a case in which you could use this keyword in the correct manner? I am trying to send an email using a Visualforce template and I want that template to include an icon. Is this possible? add transistor model to LTspice I'm trying to add the mmbt3904lt1g model to LTspice. - I've copied its data from the MMBT3904LT1.LIB.txt in the standard.bjt library. - I've added an NPN to my schematic then I've changed its value to mmbt3904lt1g  - through the Edit->spice directive I've pasted the path to this library I thought it was all to use this device but no simulation is achieved.. I get only: Can't find definition of model "MMBT3904LT1g"  where I'm doing wrong ? I am a Chinese citizen married to a US citizen last year. I am going to apply for a UK tourist visa. The name shows on my passport is my maiden name, while any other documents the UK visa needed, such as bank statement, is with my married name. I can provide my marriage license to prove the link between my maiden name and married mane.   Can I apply for this UK visa with my situation? After the latest Ubuntu updates, I have not been able to open virtualbox, on starting it gives me an error saying the kernel driver is either not loaded or set up properly and that I should reinstall virtualbox-dkms package. After removing virtualbox, and trying to reinstall I get the error below: ERROR: Cannot create report: [Errno 17] File exists: '/var/crash/virtualbox-dkms.0.crash' Error! Bad return status for module build on kernel: 5.8.0-34-generic (x86_64) Consult /var/lib/dkms/virtualbox/6.1.10/build/make.log for more information. dpkg: error processing package virtualbox-dkms (--configure):  installed virtualbox-dkms package post-installation script subprocess returned error exit status 10 dpkg: dependency problems prevent configuration of virtualbox:  virtualbox depends on virtualbox-dkms (&gt;= 6.1.10-dfsg-1~ubuntu1.20.04.1) | virtualbox-source (&gt;= 6.1.10-dfsg-1~ubuntu1.20.04.1) | virtualbox-modules; however:   Package virtualbox-dkms is not configured yet.   Package virtualbox-source is not installed.   Package virtualbox-modules is not installed.   Package virtualbox-dkms which provides virtualbox-modules is not configured yet.  dpkg: error processing package virtualbox (--configure):  dependency problems - leaving unconfigured dpkg: dependency problems prevent configuration of virtualbox-qt:  virtualbox-qt depends on virtualbox (= 6.1.10-dfsg-1~ubuntu1.20.04.1); however:   Package virtualbox is not configured yet.  dpkg: error processing package virtualbox-qt (--configure):  dependency problems - leaving unconfigured No apport report written because the error message indicates it's a follow-up error from a previous failure.                             No apport report written because the error message indicates it's a follow-up error from a previous failure.                                                         Processing triggers for menu (2.1.47ubuntu4) ... Processing triggers for bamfdaemon (0.5.3+18.04.20180207.2-0ubuntu2) ... Rebuilding /usr/share/applications/bamf-2.index... Processing triggers for desktop-file-utils (0.24-1ubuntu3) ... Processing triggers for mime-support (3.64ubuntu1) ... Processing triggers for hicolor-icon-theme (0.17-2) ... Processing triggers for gnome-menus (3.36.0-1ubuntu1) ... Processing triggers for systemd (245.4-4ubuntu3.3) ... Processing triggers for man-db (2.9.1-1) ... Processing triggers for shared-mime-info (1.15-1) ... Errors were encountered while processing:  virtualbox-dkms  virtualbox  virtualbox-qt E: Sub-process /usr/bin/dpkg returned an error code (1)  This happens even when I boot from the previous kernel using the grub menu, any help would be very appreciated. I have also tried to use aptitude to remove any conflicting versions of virtualbox from other repositories but still the same issue. How to write startup script for Systemd? I have 2 graphics cards on my laptop. One is IGP and another discrete.   I've written a shell script to to turn off the discrete graphics card.  How can I convert it to systemd script to run it at start-up? iPhone Storage Space Inaccuracy (No Storage Space) How to create an iso from current installation I installed Ubuntu 14.04.1 on one of my laptops and now I want distribute that exact installation to a few of my friends. I tried making an iso with Gparted on a live session but it makes a 500 GB file (I've only used around 8 GB) .   Thanks in Advance ! Suppose we have two functions, $f:X\rightarrow Y$ and $g:Y\rightarrow Z$. If both of these functions are onto, how can we show that $g\circ f:X\rightarrow Z$ is also onto? Dealing with rust on a steel frame Evaluating the limit of a sequence given by recurrence relation $a_1=\sqrt2$, $a_{n+1}=\sqrt{2+a_n}$. Is my solution correct? Problem  The sequence $(a_n)_{n=1}^\infty$ is given by recurrence relation:   $a_1=\sqrt2$, $a_{n+1}=\sqrt{2+a_n}$.   Evaluate the limit $\lim_{n\to\infty} a_n$.  Solution   Show that the sequence $(a_n)_{n=1}^\infty$ is monotonic. The statement $$V(n): a_n &lt; a_{n+1}$$ holds for $n = 1$, that is $\sqrt2 &lt; \sqrt{2+\sqrt2}$. Let us assume the statement holds for $n$ and show that $V(n) \implies V(n+1)$. We have that $$a_n &lt; a_{n+1}.$$ Adding 2 to both sides and taking square roots, we have that $$\sqrt{2+a_n} &lt; \sqrt{2+a_{n+1}},$$ that is $a_{n+1} &lt; a_{n+2}$ by definition. Find bounds for $a_n$. The statement $$W(n): 0 &lt; a_n &lt; 2$$ holds for $n=1$, that is $0 &lt; \sqrt2 &lt; 2$. Let us assume the statement holds for $n$ and show that $W(n) \implies W(n+1)$. We have that $$0 &lt; a_n &lt; 2.$$ Adding two and taking square roots, we have that $$0 &lt; \sqrt2 &lt; \sqrt{2+a_n} &lt; \sqrt4 = 2.$$ The limit $\lim_{n\to\infty} a_n$ exists, because $(a_n)_{n=1}^\infty$ is a bounded monotonic sequence. Let $A = \lim_{n\to\infty} a_n$. Therefore the limit $\lim_{n \to\infty} a_{n+1}$ exists as well and $\lim_{n \to\infty} a_{n+1} = A$. (For $(n_k)_{k=1}^\infty = (2,3,4, \dots)$, we have that $(a_{n_k})_{k=1}^\infty$ is a subsequence of $(a_n)_{n=1}^\infty$, from which the statement follows.) We have that $a_{n+1} = f(a_n)$. That means that $A = \lim_{n\to\infty} a_n = \lim_{n \to\infty} {f(a_n)} = f(\lim_{n \to\infty} a_n) = f(A) = \sqrt{2 + A}$. Solving the equation $A = \sqrt{2 + A}$, we get $A = -1 \lor A = 2$. Putting it all together, we get that $A = 2$, because the terms of the sequence are increasing and $a_1 &gt; 0$.   Is my solution correct? How is a pitch supporting fast bowlers or spin bowlers? Under Windows, most applications and application data are stored in a special directory known as C:\Program Files (and occasionally C:\Program Files (x86)). What is the Ubuntu/Linux equivalent to this path? Is there even one? What is the concept of erasure in generics in Java? What is the concept of erasure in generics in Java? List AppDomains in Process What Canon lens would one use for Sartorialist-like street photography? You probably know Scott Schuman and his blog (). What kind of lens (that is not expensive) would you recommend for Canon 550D to achieve similar effect (in terms of DoF etc.)? I'm a newbie so I'm not sure how to properly interpret those photos' EXIF info. My guess is, that a similar effect can be achieved using .   PS:   Update: Thank you guys for all the answers, it's much more clear for me right now. I think I'll try the Canon 50mm f/1.4 lens. Split a commit into smaller commits with Git? Can I change a user's keyboard input?