Partial derivatives in Lagrangian formalism Suppose I have a function $f = xy$. A partial derivative of $f$ with respect to $x$ implies holding $y$ constant:   $$ \frac{\partial f}{\partial x} = y $$  Does this mean that in order to evaluate this derivative, $y$ cannot depend on $x$? For example, if $y = x$ then   $$ \frac{\partial f}{\partial x} = \frac{\partial (x^2)}{\partial x} = 2x $$   Which is inconsistent with the first calculation of $\frac{\partial f}{\partial x}$.  If $y$ indeed cannot depend on $x$, then how does the Lagrangian formalism of classical mechanics make sense? The Lagrangian is a function of $q$ and $\dot{q}$, and when we evaluate $\frac{\partial L}{\partial \dot{q}}$ we 'ignore' all of the $q$ dependence, even though $\dot{q}$ is a function of $q$.
Calculus of variations -- how does it make sense to vary the position and the velocity independently? In the calculus of variations, particularly Lagrangian mechanics, people often say we vary the position and the velocity independently. But velocity is the derivative of position, so how can you treat them as independent variables?
How should I act when I see tiny "Have you tried this..?" type of answers? Subdividing Causes Strange Edges / Issues on Shape Access a glyph on fontspec using a GID (Glyph ID) I'm trying to access a glyph, but it's only mentioned in the "Basic Latin and Latin 1" set for a font. The glyph has no name or unicode ID, but it does have a glyph ID or GID. fontspec doesn't mention possibilities for using this glyph. Neither does Adobe in the adobe feature file syntax:   I'd still like to be able to access a glyph through its ID, since it's the only way I can access it (I think).  My specific case requires to access the glyph ID 554 from Stevens Titling Pro Sable Brush. I will only post this font if it's strictly necessary, since I don't like to distribute these fonts to the wide world.  I posted an MWE containing another (free) font. This is not the font I intend to use, but perhaps a starting point to try and access glyphs manually.   \documentclass{article}  \usepackage{fontspec} \setmainfont{EB Garamond}  \parindent=0pt \begin{document}  Some text int EB Garamond. How can I access glyph 123 now?  \end{document} Solving an equation over the reals: $ x^3 + 1 = 2\sqrt[3]{{2x - 1}}$ Solve the following equation over the reals:$$ x^3  + 1 = 2\sqrt[3]{{2x - 1}} $$ I noticed that 1 is a trivial solution, then I tried raising the equation to the 3rd, then dividing the polynomial by $(x-1)$.. But I can't see the solution, how do I go from here? How do I install iTunes? I have an iPhone and run Ubuntu on all of my personal computers. Since I did not want to keep a separate partition with Windows on it for the sole purpose of running iTunes, I attempted to install It using Wine.   I installed Wine 1.4 from the Software Center and installed iTunes 10.6.3. When I tried to run it I got a slew of error messages. I hopped over to google where it was suggested that I install it through PlayOnLinux. I did so with the same result. Further googling revealed that iTunes 10.6.x is confirmed to work with Wine 1.5.1 and up.   I installed Wine 1.5.1 following the instructions I found and was unable to get it to open. I did the same with 1.5.9 with the same results. I opened the Package Manager and installed the Wine 1.5.9 packages through it, and it appears to have installed properly.   When trying to install iTunes I got he error "This iTunes installer requires Windows Vista 64 bit or later". Realizing that Wine uses XP as a default I ran winecfg and changed it to Windows 7. This changed nothing and I tried changing it through winetricks to no avail. I even changed it to Vista with the same results.   Does anyone know what is going wrong here and how to fix it? Thanks Find the equation of a plane containing a point and passing through a line Find the general form equation of a plane that contains the point $(2, -2, 1)$ and passes through a line $(x, y, z) = (1, 2, -3) + t(2, -3, 2)$.  I managed to use the fact that it contains a given point to write the equation of the plane as $A(x-2) + B(y+2) + C(z-1) = 0$ but got stuck there. Is it possible to access a static resource of a managed package in your local environment?  Thanks Obtaining client IP address in WCF 3.0 Is there any reference where Captain America's shield is made from stolen Wakanda vibranium? I am going to Santa Monica for 4 weeks to polish my English in July and would like to volunteer at the Planetary Society located in Pasadena.  I am a Swiss citizen and have a biometric passport. However, I don't have any special visa, as there is no visa needed for the school I am attending and using the Visa Waiver Program, I am allowed to stay up to 90 days in the US.  Am I allowed to do volunteer work during my stay? Preventing brute force attacks against ssh? What tool or technique do you use to prevent brute force attacks against your ssh port. I noticed in my Security logs, that I have millions of attempts to login as various users through ssh.  This is on a FreeBSD box, but I imagine it would be applicable anywhere. To quantize a scalar field theory with the action:  $$S=\int \mathrm d^Dx\mathscr{L}(\phi,\partial_\mu\phi)=\int \mathrm dx^0L(\phi,\partial_0\phi)$$  we promote $\phi(\vec{x})$ and $\pi=\frac{\delta L}{\delta(\partial_0\phi)}=\pi(\vec{x})$ to be field operator at fixed time $x^0$ (with the canonical commutation relation $[\phi(\vec{x}),\pi(\vec{y})]=\delta^{D-1}(\vec{x}-\vec{y})$), then we use the classical equation of motion to find $\phi$ and $\pi$ at all time.  What principles lie behind this? Why can we assume that the field operators satisfy the classical equation of motion?    By example if we have a classical theory with $S=\int \,\mathrm dtL(\phi,\partial_t\phi)$, momentum $\pi=\frac{\delta L}{\delta(\partial_t\phi)}$ hamiltonian $H=\pi\phi-L$, the classical equation of motion is:  $$\frac{\mathrm d\phi}{\mathrm dt}=\frac{\delta L}{\delta(\partial_t\phi)}$$  When we do the canonical quantization, we promote $[\phi,\pi]=\delta(\vec{x}-\vec{y})$ and the equation of motion for $\phi$ is $$\frac{\mathrm d\phi}{\mathrm dt}=i[H,\phi]$$  Why should $i[H,\phi]$ has the same form as $\frac{\delta L}{\delta(\partial_t\phi)}$? This fact comes from where? Calculate work done by a hovering helicopter over time This is likely to be very simple, but...  How does one calculate work done by a hovering unmoving aircraft over time?  As in work in Joules.  In this scenario, to remain hovering the aircraft has to exert a force that counters gravity (which would be its mass times G, pointing in opposite direction of gravitational pull).  However, work is defined as newton per meter, and the aircraft does not move.  Meanwhile the battery charge or fuel is going to be expended, so the energy is spent.  The other question here:   is dealing with conservation energy and not calculating drain/work performed by hovering. How to change Facebook username? Do I have to re-register a new Facebook account to change my facebook.com/USERNAME?  Or are there any ways to change it? "Improve" treated as improper response to proper response to audit When a Battle Master fighter gets a critical hit and uses a maneuver that adds a superiority die to the damage, is that damage die doubled? The Battle Master fighter has access to a number of maneuvers that add a superiority die to the damage roll. For instance, the Trip Attack maneuver says (PHB, p. 74):     When you hit a creature with a weapon attack, you can expend one   superiority die to attempt to knock the target down. You add the   superiority die to the attack’s damage roll, and if the target is   Large or smaller, it must make a Strength saving throw. On a failed   save, you knock the target prone.   The Critical Hits rule states (PHB, p. 196):     When you score a critical hit, you get to roll extra dice for the   attack’s damage against the target. Roll all of the attack’s damage   dice twice and add them together. Then add any relevant modifiers as   normal. [...] If the attack involves other damage dice, such as from   the rogue’s Sneak Attack feature, you roll those dice twice as well.   During our game, the fighter PC rolled a critical hit. He then wanted to use a superiority die to use the Trip Attack maneuver to trip the enemy. Based on the above rules, is the damage from the superiority die also doubled?  I'm assuming the answer is yes based on the last statement in the Critical Hits section, but I wanted to confirm. Allow rejected editors to be @replied to Transforming positively skewed data with positive and negative values My data (see image below) is positively skewed and contains both positive and negative values. I'd like to transform it to achieve normality so I can apply a repeated measures ANOVA, but can't find a technique that works.  I've considered the following options:   Square-rooting: doesn't keep the sign Square-rooting then making the original negative values negative: makes my data bimodal Natural log transform: can't do this with negative values Add a constant then log transform: this arbitrarily leaves my lowest value as an outlier '' transform (Cox, 2005; also Whittaker et al. 2005): makes my data bimodal Outlier removal (2.5 SD): doesn't fix normality, but an ANOVA on this data has similar results to an ANOVA on the raw data (for better or worse). Perform the non-parametric equivalent (Friedman test) and compare the results: will do this today   My supervisor suggests I should stick with the outlier removal though the data is still visibly skewed, and 'see what the reviewers suggest' but I just wanted to check here in case there was anything I was missing.  Original data:    With outliers (2.5 SD) replaced with the value at that criterion. I later decided to just exclude the outliers, so imagine the highlight data points in each condition removed. New database server hardware and software inserts slower than old setup For the following block of code:  For I = 0 To listOfStrings.Count - 1     If myString.Contains(lstOfStrings.Item(I)) Then         Return True     End If Next Return False   The output is:  Case 1:  myString: C:\Files\myfile.doc listOfString: C:\Files\, C:\Files2\ Result: True   Case 2:  myString: C:\Files3\myfile.doc listOfString: C:\Files\, C:\Files2\ Result: False   The list (listOfStrings) may contain several items (minimum 20) and it has to be checked against a thousands of strings (like myString).  Is there a better (more efficient) way to write this code?