Questions tagged [differentiation]

Differentiation is the set of techniques and results from Differential Calculus, concerning the calculation of derivatives of functions or distributions.

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192 views

Landau's approach to collisions in plasma

I can't understand some steps in obtaining the collision term in the Boltzmann equation for plasma. For the first time it was made by L.D. Landau in his article "The kinetic equation in the case of ...
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Tensor Differentiation

In the book "Tensors, Relativity and Cosmology" the author derived Maxwell's Equation in covariant form using the EM field tensor Lagrangian $L=-\frac{1}{4}F^{jl}F_{jl}$ (source=0). One of the steps ...
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Is the derivative with respect to a fermion field Grassmann-odd?

Fermion fields anticommute because they are Grassmann numbers, that is, \begin{equation} \psi \chi = - \chi \psi. \end{equation} I was wondering whether derivatives with respect to Grassmann numbers ...
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General Relativity and cosmology [duplicate]

What is the physical meaning of Ricci scalar is a covariantly constant?
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Differentiating the Free Energy for the Chemical Potential

I wanted to ask a question about the partial differential of the Free Energy equation. I learnt to prove the Free Energy equation: \begin{aligned} \frac{F\left(N_{A}, N_{B}\right)}{k T}=& N_{...
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1answer
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$R$-Symmetry of gauge field

Suppose $V$ is a superfield scalar under R-transformations. This means that under an R-transformation $V\mapsto V'$ where $V'(x,\theta,\bar{\theta})=V(x,e^{-iK}\theta,e^{iK}\bar{\theta})$. What is ...
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Taylor expansion in derivation of Noether-theorem

In my classical mechanics lecture we derived the Noether-theorem for a coordinate transformation given by: $$ q_i(t) \rightarrow q^{'}_i(t)=q_i(t) + \delta q_i(t) = q_i(t) + \lambda I_i(q,\dot q,t).$$...
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Question about commutators acting on wavefunctions

Consider a commutator acting on a 1D wavefunction: $$[\frac{\hbar}{i} \frac{d}{dx},x]\psi(x)=(\frac{\hbar}{i} \frac{d}{dx}x-x\frac{\hbar}{i} \frac{d}{dx})\psi(x).$$ Now does this mean $\frac{\hbar}{...
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Expectation value of derivative of operator

I was given the following question: Let $A(\lambda)$ be a Hermitian operator, which is dependent on some real parameter $\lambda$. Let us denote the eigenvalues and corresponding eigenstates of $A$ ...
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Mathematical identity related to d'Alembert's Principle

In Hand & Finch's book on Analytical Mechanics, I came across this mathematical identity Eq. 1.19 in Chapter 1, page 5, which is related to the description of d'Alembert's principle: $$\dot{\vec{...
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The differentiation of a functional by a ket

I saw something very strange when I was studyng about the variational method. In the text, to minimize the functional $$E[\psi] = \frac{\langle \psi |\hat{H}|\psi \rangle}{\langle \psi |\psi \rangle},$...
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Why is force 0 either side of an inflexion point in neutral equilibrium?

In Tipler & Mosca 5th edition p173 it defines neutral equilibrium as a point in a U-x curve where $\frac{dU}{dx}=0$ and also $\frac{dU}{dx}=0$ for a small displacement either side of the point. ...
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How to derive kinematics equations using calculus? [closed]

I read derivation of kinematics equations using calculus: $$a=\frac{\text dv}{\text dt}$$ $$\implies \text dv=a\text dt$$ $$\implies \int_{v_0}^v\text dv=\int_0^t a\text dt$$ $$\implies v-v_0=at$$ $$\...
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Derivative of tensor product of quantum states

Recently I asked a question over at the math stack exchange: https://math.stackexchange.com/q/3210375/. However I figured I'd ask here too, seeing as the question originated in a physics course I'm ...
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In the Schrodinger Equation for the Hydrogen Atom does $\frac{{\partial}f(x)}{{\partial}x}$ equal $f(x)\frac{\partial}{{\partial}x}$? [closed]

I was looking at the Schrodinger Equation for the Hydrogen Atom, and saw it in the form $$\left(-\frac{\hbar^2}{2{\mu}r^2}\left(\frac{\partial}{{\partial}r}\left(r^2\frac{\partial}{{\partial}r}\right)+...
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Differentiability of electric field due to bounded volume charge distribution

In books on electromagnetism, one often sees expressions of Maxwell's equations like $\nabla \cdot \mathbf{E}$ and $\nabla \times \mathbf{E}$. These expressions make sense if $\mathbf{E}$ (which is ...
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1answer
55 views

Derivating operator acting on ket

I'm deducing a formula, and I used the "product rule" $\frac{\partial}{\partial t}(A|\phi>)=(\frac{\partial A}{\partial t})|\phi>+A\frac{\partial}{\partial t}|\phi>$. I'm actually getting the ...
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146 views

Partial Legendre transform: understanding a simple example

Consider the following function: $$f(x_1, x_2) = x_1^2x_2+x_1x_2^3$$ $f$ is a function of $(x_1, x_2)$. The conjugate variables $(u_1, u_2)$ to $(x_1, x_2)$ are $$u_1 = \partial f/ \partial x_1 = 2 ...
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Why does nature favour the Laplacian?

The three-dimensional Laplacian can be defined as $$\nabla^2=\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2}.$$ Expressed in spherical coordinates, it ...
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Vector calculus notation, maybe?

I just got a new book on turbomachinery that uses some notation I'm not familiar with. $$ \nabla \lor \vec{W} = -2\vec{\Omega} $$ The del-(something)-vector format makes me think its vector calculus....
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What happens to velocity when Time equals zero?

I am not formally educated in Science but natural questions have always intrigued me.The way I put it is that I am married to Commerce but Science has been a childhood love. Now I have this very basic ...
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Is curvature the exterior covariant derivative of the connection?

Let $P\to M$ be a $G$-principal bundle, $G$ a topological group, $\omega$ the connection and $V$ a vector space. We define $d_\omega: \Omega^k_G(P, V)\to\Omega^{k+1}_G(P, V)$ the exterior covariant ...
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How to efficiently compute the commutator $[\hat{r},\nabla^2]$?

Given a system with Hamiltonian $ \hat{H} = \frac {\hat{p} ^2}{2m} + \hat{V}(r)$ in a certain state $|\psi \rangle$, I want to find if $\langle r \rangle$ varies with time. From $$ i \hbar\frac {d ...
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Variation under infinitesimal reparametrization

Say that under the infinitesimal reparametrization $\sigma \rightarrow \sigma' = \sigma-\xi(\sigma)$, $x^\mu$ transforms as a scalar, i.e. $x'^\mu(\sigma')=x^\mu(\sigma)$. I would like to show the ...
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Does it make sense to speak in a total derivative of a functional? Part II

I am trying to derive the Noether theorem from the following integral action: \begin{equation} S=\int_{\mathbb{\Omega}}d^{D}x~\mathcal{L}\left( \phi_{r},\partial_{\nu}% \phi_{r},x\right) , \tag{II.1}\...
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318 views

Does it make sense to speak in a total derivative of a functional? Part I

I would like to consider the problem of the total derivative of a given functional \begin{equation} \mathcal{L}\bigg[\phi\big(x,y,z,t\big),\frac{\partial{\phi}}{\partial{x}}\big(x,y,z,t\big),\frac{\...
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303 views

Convective derivative vs total derivative

I was wondering what is the difference between the convective/material derivative and the total derivative. We were introduced to the notion of material derivative $$ \frac{D\vec{u}}{Dt}=\frac{\...
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1answer
75 views

Derivation of gradient of the expectation of local energy

Background: In Variational Monte Carlo, given a Hamiltonian $H$ and a wave function $\psi_\alpha$ dependent on some parameter(s) $\alpha$, we have defined a quantity known as the local energy, $$E_L =...
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A problem in newtonian physics [closed]

Hello! I have solved problem 89 using analytic way, i.e, the length of the string will always be constant, so repeated substitutions and differentiating will make way(i got answer as "c"). But is ...
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Vector calculus simplification in calculation of generalized force

Consider a system of $N$ particles subject to forces $\vec F_i\ (i=1\dots N)$ that derive from a potential $V$. My lecture notes propose a simple proof that $$Q_j = -\frac{\partial V}{\partial q_j}$$ ...
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Divergence, gradient and differentiation - radial irrotational fluid flow

Given a fluid with the steady spherically symmetric flow with only radial velocity $\vec v(r)$. We need to evaluate $ \vec v \cdot \nabla \vec v $. From vector calculus $$ \vec v \cdot \nabla( \vec ...
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Generally Covariant Dirac equation: The spin connection

Wikipedia, an answer on stackexchange and a few papers in the Arxiv I've found all have different definitions of the spin connection found in the Dirac equation. Can anyone please tell me what the ...
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Partial derivative of $PV=nRT$ with respect to $T$

We all know for an ideal gas $PV=nRT$ then, if we differentiate it partially w.r.t $T$ shouldn't we get $$\frac{\partial}{\partial T}(PV)=P\left(\frac{\partial V}{\partial T}\right)_P+V\left(\frac{\...
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Covariant derivative of a composite field and the chain rule

I have a gauge theory with some rather strange covariant derivatives and I am wondering how they act on a composite field like $\psi= \phi\psi'$. In my setup, the covariant derivative acting on a ...
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Showing the form of the covariant derivative of $\phi$, if $\phi$ transforms as the adjoint representation of $SU(n)$

I want to show that if $\phi$ transforms as the adjoint representation of SU(n), its covariant derivative is given by $\textbf{D}_\mu \phi = \partial_\mu \phi + i [\textbf{A}_\mu, \phi]$. (Exercise in ...
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Change of variable in function

Suppose I have a function $h(\theta)$ measuring the height of a piston, with $\theta = \omega t$. I would like to know the vertical acceleration of this piston as $\omega$ changes at the point $\theta ...
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49 views

Speed is different when differentiating a function and when not differentiating

I have the function, $S(t) = t^2$. When Finding speed $= V = \frac{dS}{dt}$, we get $V = 2t$. Now If, I don't differentiate it and simply put $V = \frac{Distance(S)}{Time(t)} = \frac{t^2}{t}$ We ...
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Functional derivative commutes with total derivative

I have a question about a rule from the calculus of variations. Assume we consider the space of differentiable functions on $C^1(\mathbb{R})$ (or for the sake of simplicity the smooth functions $C^{\...
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Is the partial derivative in the Dirac equation in curved space contracted with a tetrad?

The Dirac Equation in Curved spacetime makes a difference between Lorentzian indicies and Covariant indicies. In the equation we find a $\partial_\mu$. Is this actually $e^a_\mu\partial_a$ where $e$ ...
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Why, when going from special to general relativity, do we just replace partial derivatives with covariant derivatives?

I've come across several references to the idea that to upgrade a law of physics to general relativity all you have to do is replace any partial derivatives with covariant derivatives. I understand ...
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Derivation of Covariant derivative for fermionic fields

I've been reading about the Dirac equation in curved spacetime and understand the nature of the verbien, but am wondering what the relationship is between the two definitions of the Fermionic ...
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Field momentum of Klein-Gordon Lagrangian

Given the Lagrangian $L$ of the field $\phi$ the field momentum $\Pi$ reads: $$L_{KG}=-\frac{1}{2}\partial^\mu\phi\partial_\mu\phi-\frac{1}{2}m^2\phi^2$$ $$\Pi=\frac{\partial L}{\partial(\partial_\...
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1answer
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Thermodynamics - please check my proof that $\partial C_p/\partial p$ = 0 for an ideal gas

Prove $$\left(\frac{\partial C_p}{\partial p}\right)_T = 0$$ for an ideal gas. All the $\partial$s are partial derivatives Please check to see if this makes sense. We know that $$C_p = \left(\...
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1answer
63 views

Deciding which given function represents wave function? [closed]

I have been given four functions: $u(x, t) = (x + 3t^3)^{1/3}$ $u(x,t) = 2\cos(5t)\sin(5x)$ $u(x,t)=(x−5t)^2 + \exp(\cos(5t+x))$ $u(x,t)=x^2\cos(7x−t)$ None of them seems to satisfy the ...
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How does the partial derivative of a tensor of rank $n$ creates a tensor of rank $n+1$? (cartesian coordinates)

The partial derivative of a tensor of rank $n$, $T_{...i}$, with respect to $x^j$ can be expressed using the transformation rule: \begin{equation} \frac{\partial}{\partial x^j}T'_{...i}=\frac{\...
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What does $\nabla_{[a}\Omega_{b]}=\nabla_{[a}\nabla_{b]} t=0$ represent?

In Sec. 2.3 of Baumgarte and Shapiro's "Numerical Relativity", we find this statement: From $t$ we can define the 1-form $$\Omega_a = \nabla_a t, \tag{2.19}$$ which is closed by construction, $$\...
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How to pick a boundary layer coordinate or stretching transformation

I am following Introduction to Perturbation Methods by Holmes and am unsure how I to pick the power in my boundary layer coordinate if my governing equation is the Laplace equation given by \begin{...
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Time derivative of the continuity equation - accelerated flow

Given a fluid with accelerated motion $\vec a= d \vec v / d t$ (in one direction). The question is to write the continuity equation using the acceleration value. The continuity equation reads $$\frac{...
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1answer
86 views

Explanation of the identities $\rho=\rho(\delta)$ ($\delta$ function) and $\rho=q\,\delta(\bar{r})$

Poisson's equation is $$\boldsymbol{\nabla}^{2}\varphi=-4\pi k_{e}\,\rho, \tag{*}$$ that in the case of a point charge $q$, already with spherical symmetry, has as solution \begin{equation} \varphi(...
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1answer
111 views

Divergence of $ \frac{1}{s}\hat{s}$ in cylindrical coordinates

In Griffiths' electrodynamics, the divergence of $\frac{1}{r^{2}}\hat{r}$ is evaluated in spherical coordinates to be $4\pi\delta(r)$. I encounter the same problem in case of $\frac{1}{s}\hat{s}$ ...

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