Changing variables to obtain derivatives of new variables
Analogue for Maple's dchange - change of variables in differential expressions
If string is stretched just by weight, where does the gravitational potential energy goes if only half is converted to elastic potential energy? If a spring is stretched by a weight of mass m, (so the extension is $\Delta x$) then $ k = \frac W{\Delta x} = \frac {mg}{\Delta x}$. So $ k\Delta x= mg $.  When the Spring is stretched by distance $\Delta x$ (by the weight) then it looses gravitational potential enegry. ($ \Delta GPE= mg\Delta x$)  But, when we calculate the change in elastic potential energy, We get  $ U = \frac 12 k(\Delta x)^2 $. Since $ k\Delta x = mg$, $U=\frac 12 mg\Delta x$  At, equilibrium we have no kinetic energy $E_k = 0$  Doesn't that violate Energy Conservation?  Where does the rest of lost GPE goes?  Here's a illustration Seeking open raw LiDAR or laser scanner data? In the search results of stackexchange.com/search?q=*, allow people to click on a tag and go to that website with the tag page open.   For example,  shows me this:        But clicking on the hovered-on tag takes me to  and the panda.   Instead, it should take me to    The math.SE URL is already there in  /#question-summary-608993 &gt; div.summary &gt; div.started.fr &gt; a #question-summary-608993 &gt; div.hot-question-site-icon &gt; a Exponent and abelian groups This looks like a question asked , but it isn't.  $$T(n)=\begin{cases} T (\sqrt{n}) + 1 \quad &amp; \text{ if } n&gt;1 \\  1 &amp; \text{ if }n=1\end{cases}$$  My professor gave this to me in class yesterday. This is where I'm stuck:   $$T(n) =  T(n^{1/2})+1 =  T(n^{1/4})+2 = T(n^{1/8})+3 = \dots =   T(n^{1/ 2 ^k})+k$$  Now this will continue till  $$n^{1/ 2 ^k}=1$$  How do I proceed ahead? If I take log on both sides, then RHS becomes zero. How do I solve the relation?  I don't want to use the Master theorem, I want to know where and why am I stuck. How do I disable updating a form in Windows Forms? During a complicated update I might prefer to display all the changes at once. I know there is a method that allows me to do this, but what is it? I need to implement the following circuit in LTSpice but I don't know if my schematic is ok. Also, I don't know what  values for I should use for Cinf and R2.  The following constraints are imposed:   VCC= 10V VCM = 1.5V Vgis a sine wave with a frequency of 1 kHz. Its amplitude will be chosen so as to ensure minimal distortion of the output signal (i.e. v0 should also be a sine wave) R1= 15kΩ.    I need to choose a value for R2 and Cinf in order to have a (theoretical)gain value Av=1500.   How can I compute the theoretical expressions of the DC collector currents of bipolar transistors Q1–Q8, considering VBE≈ 0.6V and the theoretical expression of the small-signal voltage gain Av= vo/vg?   The first picture is the schematic and the second is the LTSpice implementation.  What I found is that:   ICq5=ICq6=ICq7=ICq8=(Vcc - Vbe)/R2 ICq1=ICq2 = [ICq7 / 2 ] * [Beta/(Beta+1)] ICq4=iCq2-current of Q5 ICq3=ICq1-current of Q3-current of Q4  Beta in my case is 182.1 because I'm using a BC847A transistor. A=Vo/Vg = Gm_q2 * Beta_q5 * R1Gm_q2=ICq2 / Vt that is approx 40*ICq2   How can I find the value of R2? This is what I have now, an sine like wave for Vg Merging dictionaries in C# What's the best way to merge 2 or more dictionaries (Dictionary&lt;T1,T2&gt;) in C#? (3.0 features like LINQ are fine).  I'm thinking of a method signature along the lines of:  public static Dictionary&lt;TKey,TValue&gt;                  Merge&lt;TKey,TValue&gt;(Dictionary&lt;TKey,TValue&gt;[] dictionaries);   or  public static Dictionary&lt;TKey,TValue&gt;                  Merge&lt;TKey,TValue&gt;(IEnumerable&lt;Dictionary&lt;TKey,TValue&gt;&gt; dictionaries);   EDIT: Got a cool solution from JaredPar and Jon Skeet, but I was thinking of something that handles duplicate keys. In case of collision, it doesn't matter which value is saved to the dict as long as it's consistent. I have this MySQL query:  SELECT DAYOFYEAR(`date`)  AS d, COUNT(*)  FROM  `orders`  WHERE  `hasPaid` &gt; 0 GROUP  BY d ORDER  BY d   Which returns something like this:  d  | COUNT(*) | 20 |  5       | 21 |  7       | 22 | 12       | 23 |  4       |   What I'd really like is another column on the end to show the running total:  d  | COUNT(*) | ??? | 20 |  5       |   5 | 21 |  7       |  12 | 22 | 12       |  24 | 23 |  4       |  28 |   Is this possible? Python Implementation of Track-to/aim constraint Debian 10 systemd, how to disable autostart of graphical environments I installed a graphical environment on a server and I have decided that I no longer want to run that environment at boot. How can I disable it with systemd? This is gnome desktop environment. cat file : prints prior to "user :/" prompt What's the difference between convolution and crosscorrelation? What's the difference between convolution and crosscorrelation? So why do you use '-' for convolution and '+' for crosscorrelation? Why do we need the "time reversal on one of the inputs" when doing convolution? I was shooting the lighting of the city that is visible at the top of the trees if you shoot landscape.  No matter what I do, the image is blurry. What have i done:   ISO 100 Manual focus towards infinity Using a tripod Using really sharp prime &amp; good camera   Basically, everything I could. The only example i have is here, but it is not good, because image is underexposed:    The main question: Why can't night photographs be as sharp as daily photographs (with approperiate exposure times of course)? Is GTA Online cross-platform? Grand Theft Auto Online exists on five different platforms: XBox 360, PS3, XBox One, PS4, and PC. Do any of these play on the same servers, allowing people on differing platforms to play together? Output two sided, but only on recto page -- occasional images on verso I am preparing a thesis. In order to reduce the number of physical sheets of paper, I need to print certain figures/tables on the verso side. However the university regs state that, while figs/tables may appear on the verso, the body text must appear on the recto side.  I would like to therefore output the vast bulk of the work on the recto page, but with occasional verso pages. Ideally they would be numbered as a "subpage" of the preceeding recto page (e.g. 131, 131a, 132...) or similar.  The approach which I will tentatively adopt is to use a "twoside" flag, then have "empty" and non-numbered verso pages until a page with content is required.   I'm sure I'm not the first person to have had a similar issue, but am unable to imagine other ways of posing the question. a) There's a radially symmetric charge density $\rho(r)$ centered around the origin. Determine the dipolemoment of that charge density. b) Let $\rho(r)$ be an arbitrary charge density now. Under what circumstances does the dipole moment of the displaced charge density  $\rho '(\vec{r}) = \rho (\vec{r}-\vec{b})$ differ from the one not displaced at all.  Here were my ideas so far: a) Just thinking about the situation it has to be zero, right? I mean, since there's no real dipole. But how do I show that mathematically? I was thinking of just going like this (it may be wrong): Let the charge density be $\rho (r)=kr$, then we can get the charge q by integrating: $$q=4\pi \int_0^R kr\cdot r^2dr=\pi k R^4$$ I'm looking at the charge distribution as a spherical electron cloud with radius $R$. Then, since $p=qd$ and $d$ is zero because there are no two different charges the dipole moment is zero. Is that sufficient as an answer? b) I don't know how to approach this one. My guess is that its dipole moment is also zero because we're only looking at a displacement here. Anyone got any idea? I would appreciate any advice on this. Zooming in to a selection of vertices with the period key does not work as expected I am following through the exercises in BlenderBasics_4thEdition2011.pdf and have gotten to the section on texturing.  The book goes through the creation of a Lighthouse and now I am texturing it.  I want to have a closeup view of the roof of the lighthouse after partitioning it as the lesson describes on page 54.    In order to get a close up view I selected the roof and want to center the screen on the vertices of it so I pressed the Numpad . key.  When I do that the view zooms so that the roof is off the screen.    What am I am doing wrong?    I tried Ray's answer and was able to move the origin but pressing the Numpad . key still did not work.  The centering was still on the bottom of the lighthouse, as if the origin was still there. For multiple OTP keys, why not do this:  Key = Hash[(EC)DH Shared Secret + Counter]  The counter will increase per message, and the output of the hash will be random for each new unique input into it. What are the problems or weaknesses with this design??  EDIT: For clarification: both parties will have the shared secret from an ECDH exchange, random numbers from Pythons os.urandom for generating the ECC key I can't find my bin folder. Where is it located? The other day I was trying to download something for Minecraft and it involved my bin folder. So I was looking for it and couldn't find it. I then thought that I might have accidentally deleted it, but didn't find it in the trash. I have no idea how to get the bin folder back with all the Minecraft binary files in them.