In most textbooks of physics I've found this demonstration of work-kinetic energy theorem:  $$\begin{align} W &amp;= \int_{x_{1}}^{x_{2}} F(x)\ dx \tag{1}\\ &amp;= \int_{x_{1}}^{x_{2}} m\cdot a\ dx \tag{2}\\ &amp;= m\int_{x_{1}}^{x_{2}}  \frac{dv}{dt}\ dx \tag{3}\\ &amp;=  m\int_{x_{1}}^{x_{2}}  \frac{dv}{dx} \cdot \frac{dx}{dt}\ dx \tag{4}\\ &amp;= m\int_{x_{1}}^{x_{2}} \frac{dv}{dx} \cdot v \ dx \tag{5}\\ &amp;= m\int_{v_{1}}^{v_{2}} v \ dv \tag{6}\\ &amp;= \frac{1}{2}mv_{2}^{2} - \frac{1}{2}mv_{1}^{2} \tag{7} \end{align}$$  I don't fully understand how they go from (5) to (6). It seems that they cancel the dx as if they were algebraic elements. I know from calculus that $\int f(g(x))\cdot g'(x)\ dx = F(g(x))$, where $f(x)=F'(x)$ because we are applying the chain rule reversed. And in order to do so, Leibniz's notation can help us. If we define $u=g(x)$ and the "differential" $du=g'(x)dx$ the former integral turns in  $\int f(u)\ du = F(u)$. But du and dx doesn't exist, they are just a notation we use to realize more easily that we can find the antiderivatives applying the chain rule reversed. So, what are the true mathematical operation they are doing between (5) and (6)?
Referring to unidimensional motion, it is obvious that it doesn't always make sense to write the speed as a function of position. Seems to me that this is a necessary condition to derive formulas like:  $$v^2=v_0 ^2 +2\int_{x_0}^{x}a\cdot dx$$  In fact, in the first step of the demonstration (the one I saw, but I think that this step is crucial) it's required to write $a=dv/dt=(dv/dx)(dx/dt)$, that doesn't make sense if $v$ isn't a function of $x$.   When can one rigorously write $v=v(x)$?
I've heard you can start with pikachu in pokemon go instead of the 3 standard starters. How would one do this? Is this even possible? Comma after introductory words, phrases, clauses: unacceptable, obligatory or optional? Read echo'ed output from another PHP file How could I write my name, for example, to look like the \LaTeX logo? How could I write my name, for example, in a \LaTeX style? It is just interesting, as in Google all other things pop up when I try to search for an answer! Can not start Chrome via command line I have a script that needs to run on multiple machines. The problem is, if I open a command line and type in chrome or chrome.exe, I get the same error message that it is not recognized as an internal or external command... I have already added it to my paths (as per screen shot below, see bottom entry in the paths) . It works this way on other computers.  The command prompt has been restarted (as has the computer) since updating the paths. Why does it not open Chrome as it does on my other machine(s)? Whilst entering C:\Program Files (x86)\Google\Chrome\Application\chrome.exe directly into the command line and pressing enter does make Chrome launch, I don't see why the path is not working. I have an Asus N76VB notebook with Ubuntu 14.04 and TLP for power managment. However, the fan is always on. It's not really loud, but it's annoying. My CPU usage is around 1-2%. It's clocked at 1.20 GHz.  I've added acpi=force to my boot parameters but it doesn't change anything.  /proc/acpi/fan: No such file or directory /proc/acpi/thermal_zone: No such file or directory   I've loaded all Asus ACPI modules.  lsmod | grep asus:  asus_nb_wmi            16990  0  asus_wmi               24191  1 asus_nb_wmi sparse_keymap          13948  1 asus_wmi video                  19476  2 i915,asus_wmi wmi                    19177  2 mxm_wmi,asus_wmi   Somehow, it can read the temperatures.  sensors:  acpitz-virtual-0 Adapter: Virtual device temp1:        +43.0°C  (crit = +108.0°C)  coretemp-isa-0000 Adapter: ISA adapter Physical id 0:  +44.0°C  (high = +87.0°C, crit = +105.0°C) Core 0:         +44.0°C  (high = +87.0°C, crit = +105.0°C) Core 1:         +42.0°C  (high = +87.0°C, crit = +105.0°C) Core 2:         +39.0°C  (high = +87.0°C, crit = +105.0°C) Core 3:         +39.0°C  (high = +87.0°C, crit = +105.0°C)  asus-isa-0000 Adapter: ISA adapter temp1:        +43.0°C     I really hope someone knows how to solve this. Almost all information I could find was about saving power to create less heat.    I got Intel and NVIDIA (GeForce 740M) graphics but I'm almost anytime on Intel.  $ sudo lshw -c display   *-display                       description: VGA compatible controller        product: 3rd Gen Core processor Graphics Controller        vendor: Intel Corporation        physical id: 2        bus info: pci@0000:00:02.0        version: 09        width: 64 bits        clock: 33MHz        capabilities: msi pm vga_controller bus_master cap_list rom        configuration: driver=i915 latency=0        resources: irq:43 memory:f7400000-f77fffff memory:d0000000-dfffffff ioport:f000(size=64)     fwts fan: Is the integral $\int_0^\infty \frac{\mathrm{d} x}{(1+x^2)(1+x^a)}$ equal for all $a \neq 0$? Let $a$ be a non-zero real number. Is it true that the value of $$\int\limits_0^\infty \frac{\mathrm{d} x}{(1+x^2)(1+x^a)}$$  is independent on $a$? One Object won't Show up in Final Render even though it shows up in the Preview Render Writing string at the same position using Console.Write in C# 2.0 I have a console application project in C# 2.0 that needs to write something to the screen in a while loop. I do not want the screen to scroll because using Console.Write or Console.Writeline method will keep displaying text on console screen incremently and thus it starts scrolling.  I want to have the string written at the same position. How can i do this?  Thanks Smoothing/generalizing polygon in QGIS? Is it possible to smooth or generalize a polygon in QGIS, perhaps by using a plugin like the generalizer-plugin for lines?  I could convert my polygons to lines, smooth them and convert them back to polygons but this is not working for my polygons because of several reasons. Java: Avoid inserting duplicate in arraylist How do you search for backdoors from the previous IT person? We all know it happens. A bitter old IT guy leaves a  into the system and network in order to have fun with the new guys and show the company how bad things are without him.  I've never personally experienced this. The most I've experienced is somebody who broke and stole stuff right before leaving. I'm sure this happens, though.  So, when taking over a network that can't quite be trusted, what steps should be taken to ensure everything is safe and secure? mailto link with HTML body Is there an English phrase for an inability to actually *leave* already? There is a Hungarian expression, küszöbgörcs, which literally means "threshold-cramp", and is used to describe that long conversation you have in the entryway, with all the guests awkwardly holding their coats and purses, and every so often somebody says something like "Ok, we really have to leave now, but just one more thing...", and then the conversation is good for another half an hour at least.  I've searched in vain for an equivalent idiom in English, yet I feel sure it must exist: I know English speakers are equally susceptible to this malady, it's not a uniquely Hungarian thing. I will sometimes use literal translations, like the aforementioned threshold-cramp (or doorway-spasm), but then I need to explain it, and that's less than satisfactory. (Plus it leads to more threshold cramping.) Upon trying to upgrade from 10.10 to 11.04 all seemed to go well until the restart. This error message is what comes up:  Kernel Panic - not syncing: VFS: Unable to mount root fs on unknown-block(0,0)   How do we fix that? Definitionally, we have that  $$ \int_\gamma f\ \overline{dz} = \overline{\int_\gamma \overline{f}\ dz} $$  Now let $\int_\gamma f\ dz = w = x +yi$.  Question 1: Is it not the case that $\int_\gamma \overline{f}\ dz = \bar{w} = x - iy$?  Question 2:  Assuming that $\int_\gamma \overline{f}\ dz = \bar{w} = x - iy$, then is it not the case that  $$ \overline{\int_\gamma \overline{f}\ dz} = \overline{x - iy} = x+iy = \int_\gamma f\ dz? $$  Basically, I'm failing to see how $\overline{\int_\gamma \overline{f}\ dz}$ isn't just $\int_\gamma f\ dz$. Good 1st PDE book for self study What is a good PDE book suitable for self study?   I'm looking for a book that doesn't require much prerequisite knowledge beyond undergraduate-level analysis.  My goal is to understand basic solutions techniques as well as some basic theory. GSC: How to move a site from a domain to sub domain, sub folder or new domain with a sub folder? Why is the method executed immediately when I use setTimeout? How do I add a custom keybind to a button on any website?