Partial Definition and 1000 Threads

$$\sum_i (\frac{\partial}{\partial q_i}(\frac{\partial T}{\partial q_j}\dot{q}_i)+\frac{\partial}{\partial q_i}(\frac{\partial T}{\partial q_j})\ddot{q}_i)+\frac{\partial}{\partial t}(\frac{\partial T}{\partial \dot{q}_j})$$ They wrote that above equation is equal to...

Intersecting the graph of the surface z=f(x,y) with the yz -plane. This is the option I have chosen, but it's wrong. I don't understand why. x is fixed so I thought the coordinates: y and z are left. I thought this source may be helpful...

Good day I just want to confirm if a function f(x,y) who has directional derivatives has automatically partial derivatives (even though the function itself is not necessarly differentiable)? Can we consider that partial derivatives are special cases of directional derivatives? Thank you in advance!

Hello, I am trying to calculate the partial derivative of a convolution. This is the expression: ##\frac{\partial}{\partial r}(x(t) * y(t, r))## Only y in the convolution depends on r. I know this identity below for taking the derivative of a convolution with both of the functions only...

I started by taking a derivative: $$E = \sum_{i=1}^{\alpha} (E_i^{(p)} n_i) \ \ \ | \cdot \frac{\partial}{\partial n_i}$$ $$\frac{\partial E}{\partial n_i}=\sum_{i=1}^{\alpha} [\frac{\partial E_i^{(p)}}{\partial n_i}n_i + E_i^{(p)} \frac{\partial n_i}{\partial n_i}]$$ $$\frac{\partial...

Problem: If sequence ## (a_n) ## has ##10-10## as partial limits and in addition ##\forall n \in \mathbb{N}.|a_{n+1} − a_{n} |≤ \frac{1}{n} ##, then 0 is a partial limit of ## (a_n) ##. Proof : Suppose that ## 0 ## isn't a partial limit of ## (a_n) ##. Then there exists ## \epsilon_0 > 0 ## and...

Heisenberg equation of motion for operators are given by i\hbar\frac{d\hat{A}}{dt}=i\hbar\frac{\partial \hat{A}}{\partial t}+[\hat{A},\hat{H}]. Almost always ##\frac{\partial \hat{A}}{\partial t}=0##. When that is not the case?

I've attached images showing my progress. I have used Maxwell relations and the definitions of ##\alpha##, ##\kappa## and ##c##, but I don't know how to continue. Can you help me?

It looks very easy at first glance. However, the variable S is a variable in the given expression. I have no clue to relate the partial derivatives to entropy and the number of particles.

Can someone explain me some studies I saw about partial reprogramming and rejuvenation?. In Vivo Amelioration of Age-Associated Hallmarks by Partial Reprogramming - https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5679279/ Multi-omic rejuvenation of human cells by maturation phase transient...

Hi all, I am having some problems expanding an equation with index notation. The equation is the following: $$\frac {\partial{u_i}} {dx_j}\frac {\partial{u_i}} {dx_j} $$ I considering if summation index is done over i=1,2,3 and then over j=1,2,3 or ifit does not apply. Any hint on this would...

Hi all, I was wondering is if the following partial derivative can be computed without a specific ##u(t,x)## $$\partial_tu\big[(t,x-t\kappa V)\big]$$ I was thinking it can't be done, because we could have $$u_a(t,x)=tx \Rightarrow \partial_tu\big[(t,x-t\kappa...

I have read numerous times that equilibrium vapor pressure (EVP) is a function ONLY of temperature. This at least partly makes sense to me (so I think) given energy of molecules and movement associated with such. But apparently this is not true for the partial pressures? I once thought that...

I have tried to do it in standard way by integrating in PDE's but it turned out that ##\psi## is a function of y, so now I have no clue to start this. I know the range of ##\sqrt {g}y## from ##\frac{-\pi}{2}## to ##\frac{\pi}{2}##

ial this is the question.

a) ONLY The common way to solve this problem is minimizing the two-variable equation after using the substitution ##z^2=1/(xy)##. Yet I wondered if it is possible to optimize the distance equation with three varibles. So I wrote the following equations: Distance: $$f(x,y,z)=s^2=x^2+y^2+z^2$$...

I have no answer or solution to this. So I'm trying to seek a confirmation of whether this is correct or not: ##df = \frac{\partial f}{\partial x}dx + \frac{\partial f}{\partial t}dt ## ##\frac{df}{dt} = \frac{\partial f}{\partial x} \dot x + \frac{\partial f}{\partial t} ## Therefore, ##...

Hi can someone give an hand on understanding how to handle such kind of problems ? can't come up with a valid solution. if i do ##\frac {0,56}{2*14.00}## do i get the molar fraction of nitrogen?

Greeting I'm trying to study the convergence of this serie I started studying the absolute convergence because an≈n^(2/3) we know that Sn will be divergente S=∝ so arcatn (Sn)≤π/2 and the denominator would be a positive number less than π/2, and because an≈n^(2/3) and we know 1/n^(2/3) >...

So I start by isolating v the speed here would be the square root of the partial t derivative divided by the sum of the partial x and y derivatives. the amplitude, phi and the cos portion of the partial derivatives would all cancel out. What I am left with is the sqrt(43.1 / ( 2.5 + 3.7 ) =...

My progress: P(HE) = 0.877*0.75/(0.75+0.5) = 0.5262 atm is this right?

𝝏w/𝝏x=1 and then I wasn't sure about 𝝏x/𝝏s, so I tried implicitly differentiating s: 1=(3x^2)(𝝏x/𝝏s)+y(𝝏x/𝝏s)+x(𝝏y/𝝏s)+(3y^2)(𝝏y/𝝏s) And then I shaved my head in frustration.

zx = 2xy + y2 -3y = 0 and zy = 2xy + x2 - 3x = 0 Subtracting one equation from the other gives y2 - 3y = x2- 3x ⇒ y (y-3) = x (x-3) This leads to the following solutions ( 0 , 0) , (0 ,3) , (3 , 0) but the answer also gives ( 1, 1) as a solution. What have i done wrong to not get this...

Divergence & curl are written as the dot/cross product of a gradient. If we take the dot product or cross product of a gradient, we have to multiply a function by a partial derivative operator. is multiplication by a partial derivative operator allowed? Or is this just an abuse of notation

Can you please help me ? I have tried to do it many times but I end up getting the wrong answer. Thank you in advance.

From the solution of my thermodynamics homework, $$ ({\frac{\partial h_{fg}/T}{\partial T}})_P \\ = ({\frac{\partial h_{g}/T}{\partial T}})_P - ({\frac{\partial h_{f}/T}{\partial T}})_P = \frac{1}{T} ({\frac{\partial h_{g}}{\partial T}})_P - \frac {h_g}{T^2} - \frac{1}{T} ({\frac{\partial...

I am going through some proofs for Damping oscillations in relation to partial differentials. Can someone help on why the numbers are switched around after giving inequality condition? Please see the images for better clarity. The highlighted characters that gets switched around. Thank you in...

I had already found the Mass of the product (C3H3N) produced by this reaction (theoretical mass at 100% yield) in a previous problem. I did this by finding the Limiting Reagent (C3H6) in the reaction , calculating the number of moles of C3H6 and using the Molar Ratios in the balanced reaction...

Why the summation of the following function will be canceled out when we take the partial derivative with respect to the x_i? Notice that x_i is the sub of (i), which is the same lower limit of the summation! Can someone, please explain in details?

Can anyone please help me to write partial derivative of Vxx w.r.t. r in terms of Vxx as shown in the hand written box at the end.

Selective and cross-reactive SARS-CoV-2 T cell epitopes in unexposed humans Science 04 Aug 2020: eabd3871 DOI: 10.1126/science.abd3871 URL: https://science.sciencemag.org/content/early/2020/08/04/science.abd3871 The Coronavirus family of viruses is a cause of multiple human illnesses...

I am not sure how to determine the sign of this derivatives. (a) first we can pass a plane by (1,2) parallel to XZ (y fixed) and see how the curve belongs to the plane will vary with x, but what about the next partial derivative, with respect to y?

I completely forgot how to solve these so here's my attempt: $$z = au + bv$$ $$z = a(x^2 + y^2) + be ^{-x^2/2}$$ $$z'_x = 2ax - bxe ^{-x^2/2}$$ $$z'_y = 2ay$$ Put that into the original equation and you get $$y * (2ax - bxe ^{-x^2/2}) -x * (2ay) = $$ $$-ybe^{-x^2/2} = xyz$$ $$z =...

Ref. 'Core Principles of Special and General Relativity' by Luscombe. Apologies in advance for the super-long question, but it's necessary to show my thought process. Let ##\gamma:I\to M## be a smooth curve from an open interval ##I\subset\mathbb{R}## to a manifold ##M##, and let...

The pressure of oxygen at sea level = ##\frac{20.9}{100} ~\text{x} ~(21.2 ~\text{x} ~ 10^3) = 4430.8~ \text{Pa}## Then I do not know how to calculate the pressure at altitude 7000 m. I tried using P = ρgh (taking ρ as density of air = 1.3 kg/m3) then subtract the result from 4430.8 Pa but got...

Hi all, I hope this is the correct place to post this. Below is a section of a pipe. The pipe has a radius of 0.848 m. For this example, assume the pipe is buried below ground but a section of it remains exposed. The centre of the pipe is buried 0.590 mbelow the ground. If we assume the pipe...

Hello there, I have stumbled across further examples to derivatives of multivariable functions that confuse me. Similar to my other thread: https://www.physicsforums.com/threads/partial-derivative-of-composition.985371/#post-6309196 Suppose we have two functions, ## f: R^2 \rightarrow R...

It's a detail, but annoying to me: ##{\partial u\over \partial x} = {\partial \phi \over \partial x} \;+ ...## $${\partial u\over \partial x} = {\partial \phi \over \partial x} \;+ ...$$ How do I move up ##\partial u## a little bit so it aligns with ##\partial \phi## ?

I research about coordinate systems and I found the following informations about transformation. Now, if I replace arctan (x/y) (according to the picture above) to φ, I think I can solve. But if I can do this, then what will be replaced to ψ? I mean, I know just taking partial derative about...

1) If we have two functions C(y, r) and I(y, r) can we write: ∂C/∂I×∂I/∂r=∂C/∂r ? Can we also write ∂I/∂C=1/(∂C/∂I) ?

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I've first calculated the partial pressures of each gas: ##N_2: 0.4\times 7.4\times 10^4=3.0\times 10^4 Nm^{-2}\\## ##O_2: 0.35\times 7.4\times 10^4=2.6\times 10^4 Nm^{-2}\\## ##CO_2: 0.25\times 7.4\times 10^4=1.9\times 10^4 Nm^{-2}\\## From here, I do not know how to continue. Could someone...

1. Is it because the initial formula start the series from ##n = 2##? 2. If the initial formula is used, can I find ##S##, which $$S=\lim_{n\to\infty} \frac{2}{n^2-1}=\frac{2}{\infty}=0$$? Why that answer is different if the formula is changed.

I am trying to reproduce the results of a thesis that is 22 years old and I'm a bit stuck at solving the differential equations. Let's say you have the following equation $$\frac{\partial{\phi}}{\partial{t}}=f(\phi(r))\frac{{\nabla_x}^2{\nabla_y}^2}{{\nabla}^2}g(\phi(r))$$ where ##\phi,g,f## are...

While working at home during the COVID-19 pandemic I've taken to seeing if I can still do math from undergrad (something I do once in a while to at least pretend my life isn't dominated by excel). So to that I've been reviewing partial derivatives (which I haven't really thought about in a good...

I'm in a first-year grad course on statistical mechanics and something about multivariable functions that has confused me since undergrad keeps popping up, mostly in the context of thermodynamics. Any insight would be much appreciated! This is a general question, but as an example imagine...

So in my lecture notes on Differential Equations, it states that a first order ODE is exact if A(x,y)dx + B(x,y)dy = 0 and ∂A/∂y = ∂B/∂x. Okay I accept this definition. Then, there is a sentence like this: Our goal is to find the function V(x,y) satisfying Adx + Bdy = dV = ∂V/∂x(dx) +...

Hi guys, suppose we have a function ##C(x, y)## into the real numbers. Suppose also that ##y=y(x)##, i.e. ##y## is a function of ##x##. Now in my script, I have a term ##\nabla_x C(x_0, y(x_0)) ##. From my point of view, this means that you take the partial derivative of ##C(x,y)## with...