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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Derivation of covariant derivative

I'm currently doing Introductory QFT and was confused about the origin of the additional terms in the covariant derivate. My understanding is as follows: If we begin with the Dirac Lagrangian ...
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Instantanous and uniform velocity and acceleration

If the mathemical expression of instantanous velocity is $d/t$, what is the mathematical expression of uniform velocity. If the mathematical expression of instantanous acceleration is $v/t$, what is ...
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Holding things constant in Statistical Physics for differentiation

I just want to know if the following is correct: If one wants to verify e.g. the Maxwell relation for the ideal gas $$\left(\frac{\partial T}{\partial V}\right)_{S,N}=- \left(\frac{\partial p}{\...
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Decay of the First Derivative of the Quantum Wave Function

I understand that the Hilbert space of all physical solutions of the Schrodinger equation have the property where $$ \lim_{x\to\infty}\Psi=0 $$ For one of my assignments, I wanted to use $$ \lim_{x\to\...
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At what point does the deviation from the minimum of a function become of first order rather than second order? [migrated]

I was reading Feynman's page on the Principle of Least Action, and he stated the following: "That’s a possible way. But we can do it better than that. When we have a quantity which has a minimum—...
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How does instantaneous velocity cause displacement in just one point? [closed]

I have a question. Falling object graph is curve shape right? And instantaneous velocity is tangent line but how does this velocity make displacement in distance? Because suppose instantaneous ...
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Does car move when instantaneous velocity is zero? [duplicate]

In 3blue1brown: derivative paradox. supposed car moving with: $S(t) = t^3$ And velocity is: $V(t) = 3t^2$ He asked when t = 0 velocity is 0 m/s , does that car move at that time ? And here his ...
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What is the correct formulation of momentum balance for a body of continuum?

What is the correct form of the momentum balance equation for a continuum body $\mathscr{B}$ whose particles are fixed, and occupies volume $V(t)$ at time $t$? \begin{align} &\frac{\mathrm{d}}{\...
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Tortoise coordinate derivative [closed]

According to https://en.m.wikipedia.org/wiki/Eddington–Finkelstein_coordinates the tortoise coordinate $r^*$ is defined as $r + 2GM\ln(\frac{r}{2GM} -1)$ It then says that $\frac{dr^*}{dr}$ = $\frac{1}...
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Is the time derivative of the adjoint equal to the adjoint of the time derivative?

This is hopefully straightforward. Starting from the Schrödinger equation as an axiom, one obtains the operator differential equation for the $U$ such that $| \psi(t) \rangle = U(t,t_0) | \psi(t_0) \...
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Why differentiation of Fourier operator is difficult? [closed]

I have a question when I read some papers about physics-informed neural networks. In the paper of physics-informed neural operator, they said "it is non-trivial to compute the derivatives for ...
4 votes
4 answers
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How to prove $\mathrm{Tr}[(\partial_\mu U)U^\dagger]=0$?

I am studying ChPT by referring to "A Primer for Chiral Perturbation Theory" by Stefan Scherer. I'm having a problem with the consideration of terms that appear in the Lagrangian. The ...
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Why did my rearrangement with chain rule end up equating velocity to position?

We all know acceleration is the time-derivative of velocity which in turn is the time-derivative of position. Vice versa: position is the integration of velocity and velocity itself is the integration ...
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What does $\partial_ν/\partial^2$ mean?

I found such notation in this article link, equations 24-25. I know that $\partial_μ$ is four-gradient, but it does not contain second-order derivatives. Only d'Alembert operator does, $\partial^μ\...
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If a body moves along a path (any path, not just circular) with constant speed, is it's tangential acceleration necessarily zero?

If a body moves along a path (any path, not just circular) with constant speed, is it's tangential acceleration necessarily zero? I could only find general proofs for the case of circular motion and ...
4 votes
2 answers
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Understanding this Lagrangian calculation

I was trying to understand this section of a Wikipedia article: $$0 = \delta \int \sqrt{2T} d\tau = \int \frac{\delta T}{\sqrt{2T}} d\tau = \frac{1}{c} \delta \int T d\tau$$ For the life of me, ...
1 vote
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Lienard-Wiechert Potential derivation in Wald's "Advanced Classical Electromagnetism" [closed]

I want to follow the Lienard-Wiechert potential derivation in Robert Wald's E-M book, page 179. I do not understand $dX(t_\text{ret})/dt$ on the right side. I assume the chain rule is applied and $x'^...
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How do I keep track of what to differentiate in a Dirac Hamiltonian/Lagrangian?

Suppose we have the dirac Hamiltonian: $$ H = \int d^3y\bar\psi(y)_b(-i\gamma^k\partial_k+m)_{bc}\psi(y)_c. $$ My question is should I think the derivative operator $\partial_k$ is acting on the ...
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Proof of differentiate form of dynamical semigroups

I am studying some basics of the pure mathematical background for open quantum systems from Angel Rivas`s book which is "Open quantum systems, an introduction". Here is a theorem (Page 6, ...
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How can a definite acceleration integral be useful in mechanics and why is an indefinite integral not used?

We have an acceleration function and in order to find the displacement function, it would be logical to take an indefinite integral 2 times. Then we would get a function. Why is it proposed here to ...
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Sine and Cosine Functions [closed]

So long story short, We were given a windmill to experiment with and a sensor could sense the Voltage produced and graph it concerning time. We decided to make a sine wave out of the positive and ...
3 votes
1 answer
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Some basic questions about applying operator in quantum mechanics [closed]

Given a momentum operator $$ \def\bra #1{\langle#1|} \def\ket #1{|#1\rangle} \def\braket #1{\langle#1\rangle} $$ $$ P \equiv - i \hbar \frac{\partial}{\partial x}, $$ I want to calculate $\bra{a} P \...
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Why the first-order derivative is missing when composing a Hamiltonian of simple harmonic oscillator by the lowering and the raising operators? [closed]

Given the lowering operator ($a$) and the raising operator ($a^\dagger$) $$\begin{align*} a &= \frac{1}{\sqrt{2m \hbar \omega}}\left(-i \hbar \frac{\partial}{\partial x} - i m \omega x\right) \\ a^...
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Material derivative of circulation, Kelvin's theorem

i'm trying to work with the material derivative $$\frac{D}{Dt}=\frac{\partial}{\partial t}+(\vec{u}\cdot \vec{\nabla}) $$ of the circulation $\Gamma=\oint_{C=\partial S}\vec{u}\cdot d\vec{l}$, that ...
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Alternate derivation of the covariant derivative of a contravariant vector

In Dirac's “General Theory of Relativity”, he derives the covariant derivative of a contravariant vector (his Eq. (10.7)): $$ A^\mu_{: \sigma} = A^\mu_{, \sigma} + \Gamma^\mu_{\alpha \sigma}A^\alpha $$...
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Why area under a curve equals the sum of values of function/quantities taken elementally? [duplicate]

Background: I was taught basic formulas of differentiation and integration when I started learning physics. However, it wasn't taught through intuitions and concepts. It was like : " Hey these ...
2 votes
2 answers
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Why the derivative of the coordinate of a volume control is not zero?

When deducing the Navier-Stokes equation, for conservation of momentum, in an Eulerian frame (a control volume) the derivative of fluid velocity $U_{(t)}$ is calculated $$\frac{\mathrm{dU} }{\mathrm{...
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Does this particular notation for derivatives imply anything in particular? [duplicate]

In some physics textbooks (and in those of other sciences that use physics, like soil science), I've seen some derivatives written as: $$\frac{\delta f}{\delta t} $$ Which is a bit strange. Does this ...
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Derivatives of the lagrangian of generalized coordinates [closed]

I know that $$U= \frac{1}{2} \sum_{j,k} A_{jk} q_j q_k \quad \quad T= \frac{1}{2} \sum_{j,k} m_{jk} \dot{q}_j \dot{q}_k $$ and the lagrangian is $$ \frac{\partial U}{\partial q_k} - \frac{d}{dt} \...
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What is the physical meaning of non-commuting tetrads?

I'm reading about the tetrad formalism in GR and one main difference between the coordinate and the tetrad frame is that coordinate derivatives commute $\partial_\mu \partial_\nu = \partial_\nu \...
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2 answers
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The treatment of infinitesimal quantities [duplicate]

Please be advised that my question is different from some of the existing threads like this one. I have long been convinced that if we are to question the value of something which we ultimately are ...
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What does this vertical line notation mean?

Here is the definition of the Noether momentum in my script. $$I = \left.\frac{\partial L}{\partial \dot{x}} \frac{d x}{d \alpha} \right|_{\alpha=0} = \frac{\partial L}{\partial \dot{x}} = m \dot{x} = ...
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Fraction with components of Lorentz transformation

I want to show how partial derivative transforms under a Lorentz transformation. Since the partial derivative has a fixed definition with respect to the $x$-coordinate it stays unchanged: $\partial_\...
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Invariance over Galilean transformation

I want to prove that the Wave Equation is not invariant under Galilean Transformation. I'm having a little trouble with it but this is my attempt. 1. First of all, what does it mean by "not ...
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How is the Material derivative related to Hamiltonian and Lagrangian?

The material derivative is: $$ \frac{Df}{Dt} = \frac{\partial f}{\partial t} + \dot{\vec{x}} \cdot \nabla f $$ Where $f(\vec{x}, t)$ could be a scalar field or a vector field. If we look at the ...
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Proving the relation $\frac 1 2 \left[\nabla^2,r \right] = \frac 1 r + \frac \partial {\partial r}$ (quantum mechanics exercise) [closed]

I'm trying to prove this relation in my quantum mechanics exercise book $$\frac 1 2 \left[\nabla^2,r \right] = \frac 1 r + \frac \partial {\partial r}.$$ Here's my attempt: Expand the Laplacian ...
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1 answer
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How to define differentiation of a time-dependent vectors with respect to a specific reference frame in a coordinate-free manner?

It is usual in classical mechanics to introduce the derivative of a time-dependent vector with respect to a reference frame. This is accomplished through the use of a basis that is fixed with respect ...
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Divergence not defined

I’m currently working on the practice problems in Introduction to Electrodynamics by Griffiths. I got confused by the solution to this problem. What does “ill-defined divergence” even mean? I ...
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How to calculate the correlation function like $\langle \partial_i \phi(x) \partial_j\phi(0) \rangle$ by Gaussian path integral?

From the standard text book about quantum field theory, we know that if we consider $$\mathcal{L}=\frac{1}{2}(\partial_{\mu} \phi)^2-\frac{m^2}{2}\phi^2,$$ the partition function of this Gaussian ...
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Derivation of a partial derivative equation by Albert Einstein in Special Theory of Relativity

I was reading Albert Einstein's "On the Electrodynamics of Moving Bodies". In section "Kinematical Part", on $3 (Theory of the transformation of coordinates and times from a ...
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Connection between covariant derivative operators upon conformal compactification

I'm having trouble determining the connection between two covariant derivative operators. These are: the one associated with the original space-time (and thus with the metric $ \tilde{g}_{ab}$) and ...
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Partial derivatives in thermodynamics: general mathematical procedure [closed]

In the lecture notes (thermodynamics) the following mathematical identity is often used: $$ \left(\frac{\partial A}{\partial X}\right)_Z = \left(\frac{\partial A}{\partial X}\right)_Y + \left(\frac{\...
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Is the contracted Christoffel symbol a tensor?

The coordinate transformation law (from coordinates x to coordinates y) for the Christoffel symbol is: $$\Gamma^i_{kl}(y)=\frac{\partial y^i}{\partial x^m} \frac{\partial x^n}{\partial y^k} \frac{\...
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Use Index Notation properly when indices are already used in identifying which bases is the matrix metric calculated with respect to

I am wondering how to apply the usual linear algebra to the rather unfamiliar case of 'matrices' with indices in special relativity or even general relativity. In particular, consider$$f=\sqrt{-\det\...
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Christoffel symbol with conformal time not equal to with cosmic time one when making a change of coordinates for d'Alembertian

I think I am having a misunderstanding that would be nice to clear up. The covariant d'Alembertian $$ \Box \phi = g^{\mu\nu}\nabla_\mu\partial_\nu \phi= \left(\partial^2 + \Gamma^\mu_{\mu\lambda}\...
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2 answers
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Covariant Derivative and Cartesian Coordiantes

I have a somewhat weird question regarding some problems I have been encountering lately when dealing with transformations of the general covariant derivative to cartesian coordinates and back. In the ...
4 votes
1 answer
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Is there a quick way to calculate the derivative of a quantity that uses Einstein's summation convention?

Consider $F_{\mu\nu}=\partial_{\mu}A_\nu-\partial_\nu A_\mu$, I am trying to understand how to fast calculate $$\frac{\partial(F_{\mu\nu}F^{\mu\nu})}{\partial (\partial_\alpha A_\beta)}$$ without ...
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A question on $\frac{\partial {\bf v}_i}{\partial\dot{q_j}}$ in the derivation of Lagrangian Equations

As an undergraduate, I'm only getting introduced to Lagrangian Mechanics and the book that I'm currently referring to is Classical Mechanics by Herbert Goldstein. My question concerns a mathematical ...
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Partial Differential with independent quantities held constant meaning?

$$ \mu_{JT}=\left(\frac{\partial T}{\partial P}\right)_H= \frac{V}{C_p}(\alpha T -1) $$ and $$\left(\frac{\partial T}{\partial P}\right)_H \left(\frac{\partial H}{\partial T}\right)_P \left(\frac{\...
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Covariant derivative vs Ordinary derivative

How correct is the statement: Covariant derivative, denoted by $\nabla_u v$ = Ordinary derivarive (denoted by $u(v)$) - Normal components, also called second fundamental form. If the former statement ...
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