deriving Definition and 1000 Threads

How did the d(mv)/dt become the other two? Can someone explain how do we derive for new formulas in physics?

So far, I have got the equations, ##u \cdot (\vec u \times \vec v) = 0## ##u_1a + u_2b + u_3c = 0## ##v_1a + v_2b + v_3c = 0## Could some please give me some guidance? Many thanks!

Margules suggested a power series formula for expressing the activity composition variation of a binary system. lnγ1=α1x2+(1/2)α2x2^2+(1/3)α3x2^3+... lnγ2=β1x1+(1/2)β2x1^2+(1/3)β3x1^3+... Applying the Gibbs-Duhem equation with ignoring coefficients αi's and βi's higher than i=3, we can obtain...

For this, Dose anybody please know of a better way to derive the formula without having ##c = \frac{\Delta Q}{m \Delta T}## then taking the limit of both sides at ##\Delta T## approaches zero? I thought ##\Delta Q## like ##\Delta W## was not physically meaningful since by definition ##Q## is...

This is the defining generator of the Lorentz group which is then divided into subgroups for rotations and boosts And I then want to find the commutation relation [J_m, J_n] (and [J_m, K_n] ). I'm following this derivation, but am having a hard time to understand all the steps: especially...

I am recently reading "Introduction to Electrodynamics, Forth Edition, David J. Griffiths " and have a problem with the derive of the curl of a magnetic field from Biot-Savart law. The images of pages (p.232~p233) are in the following: The second term in 5.55(page 233) is 0. I had known...

The standard derivation in obtaining a single wave equation involves making use of the heat equation with a Taylor expansion of the equation of state, then differentiating this equation and the continuity equation with respect to time, and combining with the divergence of the NS equation...

I could not find any derivations in the litterature, except for the expected value of the energy flux expression itself: $$\overline{\Phi_{effusion,\epsilon}} = \overline{\dot{N_{ef}}}\overline{\epsilon_{ef}}=\frac{3Nl}{2A}\sqrt{\frac{(k_BT)^3}{2\pi m}}$$ I've started off by calculating the...

Here is my epic fail at trying to derive the equation using Lagrange (this was my first time trying to use lagrangian mechanics except for when I memorized the derivation for a pendulum) $$L = \frac{m \dot r^2}{2} - \frac{k q_1 q_2}{r}$$ $$\frac{\partial L}{\partial r} = \frac{k q_1 q_2}{r^2}$$...

Hi guys, I can't seem to be able to get to $$ (\rho + p) \frac {d\Phi} {dr} = - \frac {dp} {dr} $$ from $$T^{\alpha\beta}_{\,\,\,\,;\beta} = 0$$ the only one of these 4 equations (in the case of a spherically symmetric static star) that does not identically vanish is that for ##\alpha=r##...

I consider a disc of thickness ## R d\theta ## as shown in the figure. Then, $$ dV = \pi R^2 sin^2 \theta R d\theta $$ ( Area of the disc * its thickness) Hence, $$ V = \int^{\pi}_{0} \pi R^2 sin^2 \theta R d\theta $$ $$ V = \frac 1 {2} {\pi}^2 R^3 $$ ....(1) While $$ V =...

Where did they get the equation in circled in red from? It does not seem that it can be derived from the graph below. Many thanks

Initial displacement is h above the ground ie ##s\left ( t =0\right )=h##. I've chosen the ground as the vertical origin with upwards as the positive direction. Gravity will therefore always act in negative direction throughout. Here are the graphs I which to reproduce from first principles...

Hi, I’m interested to understand some of the mechanics involved in meteorites that originate from the asteroid belt. I have researched several including the Barringer and the one in Northern Canada in 2008 that was caught on multiple CCTV cameras. They all have very similar velocities before...

Is there a way to obtain equation 9.42 (I is current, j is current density, and sigma is conductivity) in the following image (from Modern Electrodynamics by Andrew Zangwill, the part on electromotive force) besides using V=IR and substituting the line integral of j/conductivity for V? The...

For this problem, How did they get that formula shown? My working is, All the solutions wrote was, Many thanks!

How do one derive the relation E=hf?

We usually have an initial time and then find an equation for the variable final time. Can we derive a formula to calculate position with final time and variable initial time. ##v = v_i + a(t_f - t_i)## ##dx = v_idt + at_fdt - atdt## integrating ##x_f - x_i = v_i(t_f - t_i) + at_f(t_f - t) -...

In the boxed equation, how would you get the right hand side from the left hand side? We know that ##H(1,2) = H(2,1)##, but we first have to apply ##H(1,2)## to ##\psi(1,2)##, and then we would apply ##\hat{P}_{12}##; the result would not be ##H(2,1) \psi(2,1)##. ##\hat{P}_{12}## is the exchange...

Can you derive the formula for frequency observed from doppler effect with stationary person and moving sound source away from the person like this: ##v_t = v + v_s## where ##v_t## is the total velocity observed by stationary person from moving sound, v is velocity of sound and ##v_s## is...

My textbook is deriving wave speed on a string under tension with confusing thetas. It assumes ##\tan \theta_1 = \frac{-F_1}{F_T}## and ##\tan \theta_2 = \frac{F_2}{F_T}## which confuses me. I know for sure theta is the angle due to the position of y and x, ##\tan \theta = \frac{y}{x}##, but I...

Frustratingly, everything I read about deriving Avogadro's number uses the word "mole" somewhere in the explanation. Per Scientific American, for example, Robert Millikan divided the charge on a mole of electrons by the charge on a single electron to obtain a value of Avogadro’s number of...

While deriving Lorentz transformation equations, my professor assumes the following: As ##\beta \rightarrow 1,## $$-c^2t^2 + x^2 = k$$ approaches 0. That is, ##-c^2t^2 + x^2 = 0.## But the equation of the hyperbola is preserved in all inertial frames of reference. Why would ##-c^2t^2 + x^2##...

So I thought that the graph tries to tell us that the function is periodic after 2π interval. So I tried to derive its function from the graph as follows using the point slope equation form for the points (0,0) & (a,π): ##y= ({a}/{π})*x## I hope this function is alright and I just need to find...

In Zettili book, it is given that ## \nabla^2 \psi \left( \vec{r} \right) + \dfrac{1}{\hbar ^2} p^2 \left( \vec{r} \right) \psi ( \vec{r} ) =0 ## where ## \hbar## is very small and ##p## is classical momentum. Now they assumed the ansatz that ## \psi ( \vec{r} ) = A ( \vec{r} ) e^{i S( \vec{r} )...

The solution can be viewed here on page 41 https://usermanual.wiki/Document/Steven20H20Simon2020The20Oxford20Solid20State20Basics2C20Solution20ManualOxford20University20Press202015.1463186034/view What I have is $$\frac{\partial}{\partial \phi^{*}} (\frac{\sum_{n,m} \phi_{n}^{*}...

It is asking to derive the time-independent wave function and has managed to get the answer of and i am very confused as where (ix/a) and (-x^2/2a) came from ? Thanks.

I was following David tongs notes on GR, right after deriving the Euler Lagrange equation, he jumps into writing the Lagrangian of a free particle and then applying the EL equation to it, he mentions curved spaces by specifying the infinitesimal distance between any two points, ##x^i##and ##x^i...

Im not able to understand the derivation equations and all please. $$ \begin{aligned} \mathrm{HA}+& \mathrm{H}_{2} \mathrm{O} \rightleftharpoons \mathrm{A}^{-}+\mathrm{H}_{3} \mathrm{O}^{+} \\ K_{\mathrm{a}} &=\frac{\left[\mathrm{A}^{-}\right]\left[\mathrm{H}_{3}...

Is that possible to derive the Navier-Stokes equations with Lagrangian and Hamiltonian methods? If yes, how? and if it is not possible, why?

The statement "The net charge on every component in the system is always zero. Thus no component can collect a net excess of charge, although some components, as you will learn later, can hold equal but opposite separated charges" leads to believe that we are always dealing with charges of...

Summary:: Need opinions on how to go about deriving EDE equation Hello, I am an undergrad starting cosmo research under one of my professors and he assigned me to derive eqn 6 below. My plan now is to use eqn 4 to find the right side of eqn 5 and solve for omega d. I haven't tried this yet but...

As a part of my self study, I am trying to derive the Laplacian in spherical coordinates to gain a deeper understanding of the mathematics of quantum mechanics. For reference, this the sphere I am using, where ##r## is constant and ##\theta = \theta (x,y, z), \phi = \phi(x,y)##. Given the...

Goldstein 2nd ed. In its Appendix is given the derivation of Bertrands Theorem.Here ##x=u-u_0## is the deviation from circularity and ##J(u)=-\frac{m}{l^{2}} \frac{d}{d u} V\left(\frac{1}{u}\right)=-\frac{m}{l^{2} u^{2}} f\left(\frac{1}{u}\right)## If the R.H.S of A-10 was zero, the solution...

Hi all, I was thinking punching a round ball on a flat surface and seeing how I could determine a formula for force from it. I thought the following: 1. The ball will go further the harder I punch and thus force must be proportional to displacement d. 2. Ball will go further if it is lighter...

Hello! The paper I study is related to string theory and modified gravity theories topics. As they say in page 5 “The four-dimensional effective theory now follows by substituting Eq. (13) into the original action, Eq. (4)” I wonder how did they drive a 4- dimensional effective metric...

I found one article in 1993 talking about it.[Unacceptable reference deleted by the Mentors]

My mentor wants the derivation of this formula. Me a computer undergrad, unable to figure it out, and my final project are on a halt due to this, any help from the community is greatly appreciated!

I have no idea how to do this. I've tried conservation of mechanical energy and it didn't work. Ek = Kinetic Energy R = horizontal range of the ball h = height from which the ball is released

Hello all. I have a question about building the coherent transfer function and specifically how I would go about deriving the pupil function for this figure. I have not come across this in my class yet and am a bit stumped. Any help would be appreciated.

Deriving time dilation was easy: Imagine two events in frame O' at the same location. ##ds^2 = -c^2 dt'^2## The same viewed in O frame is: ##ds^2 = dx^2+dy^2 + dz^2 - c^2 dt^2## ##\Rightarrow dx^2+dy^2 + dz^2 - c^2 dt^2 = -c^2 dt'^2## ##\Rightarrow (\frac{dx}{dt})^2+(\frac{dy}{dt})^2+...

Hello, I'm in the process of deriving the Wassiljewa mixture model equation for a binary solution. I have to find an expression gE which represents the excess g term which is added to gIS, the ideal solution, to predict the g for a real solution. I have gotten up to a point but now I'm stuck...

I don't know how to do (a), so I decided to ignore it for now and just assume the result. Because ##j^a = 0## the Maxwell equations are ##\mathrm{d} \star F_{ab} = 0## and ##\mathrm{d} F_{ab} = 0##. For any two one forms, ##\frac{1}{2} \omega_a \wedge \eta_b = \omega_{[a} \eta_{b]}##, and so we...

To approach the problem I first studied section 1.3 and, more importantly, 1.4 of Osborn's notes. We first need to compute ##\partial_j \omega_i (x)## and ##\omega_i (x)\omega_i (x)## \begin{equation*} \partial_j \omega_i (x) = \delta_{ij} + \underbrace{\partial_j (g_{ilm})}_{=0}x_l x_m +...

Okay so I am learning Statistical mechanics from an Indian book "Thermal Physics,kinetic theory and statistical mechanics by Garg, Bansal and Ghosh". I have derived the MB distribution function, and have evaluated the parameters α and β. With its help I derived the expression for entropy...

1- Write down the complete MAXWELL equations in differential form and the material equations. 2- An infinitely extensive area is homogeneously filled with a material with a location-dependent permittivity. There are charges in the area. Give the Maxwell equations and material equations of...

Hello, I was watching a video lecture from MIT 8.04 (Allan Adams)– lecture #24 (around the 38 minute mark give or take) The topic is quantum computing, Dr. Adams is deriving / explaining how to get various computing operations. For the “NOT” operation he explains that the operator $$ U_{Not}...

Hello everyone, I am new here, so please let me know if I am doing something wrong regarding the formatting or the way I am asking for help. I did not really know how to start off, so first I tried to just write out all the ##\mu \nu \rho \sigma## combinations for which ##\epsilon \neq 0## and...

$$\nabla \times B(r)=\frac{\mu _0}{4\pi} \int \nabla \times J(r') \times \frac{ (r-r')}{|r-r|^3}dV'$$ using the vector identity: $$\nabla \times (A \times B) = (B \cdot \nabla)A - B(\nabla \cdot A) - (A \cdot \nabla )B + A(\nabla \cdot B)$$ ##A=J## and ##B=\frac{r-r'}{|r-r'|^3}## since...

In gravitational lensing, the image magnification is defined as the image area over the source area. But many texts also give it as the inverse of the determinant of the jacobian, A, of the of the lens equation. My question is how these are equivalent. The lens equation is...