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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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29 views

Derivation of Perturbation Terms in Thermodynamic Perturbation Theory

In the "A critical evaluation of perturbation theories by Monte Carlo simulation of the first four perturbation terms in a Helmholtz energy expansion for the Lennard-Jones fluid" paper by T. van ...
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1answer
85 views

Did R. Feynman know about the different notations for exact and inexact differentials? [closed]

I remember reading a long time ago, the story of a student taking R. Feynman for responsible of her (I think it was a woman, not sure though) fail at an exam of physics because what was written in her ...
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41 views

Expectation of partial time derivatives of $x$ in QM

In Ehrenfest theorem we know that $$m\frac{d\left< x\right>}{dt}=\left< p\right>+m\left<\frac{\partial x}{\partial t}\right>.$$ So how can I exactly calculate a specific $\left<\...
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102 views

Differentiate the Lagrangian wrt. momentum?

Given $$ L=L(t, x_i, \dot x_i) $$ as a function of generalized coordinates/velocity, and $$ p_i:=\frac{\partial L}{\partial \dot x_i}, $$ how can we calculate $$\frac{\partial L}{\partial p_i}?$$
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Prove the Total Mechanical Energy of the System is Conserved via Differential Equations [closed]

Consider the dynamics of a particle P shown: Particle in 3D space with Radius r Newton's second law states that: $$\frac{d}{dt}(m\dot r) = \mathbf F$$ where, $\boldsymbol{r}$ is the position vector ...
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Non-uniform circular motion with constant radius of curvature

$\let\oldhat\hat \renewcommand{\vec}[1]{\mathbf{#1}} \renewcommand{\hat}[1]{\oldhat{\mathbf{#1}}}$ Suppose we have a car moving on a circular track of radius $b$ and speed $v=ct$, where $t$ is time ...
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2answers
2k views

How does instantaneous velocity or acceleration have any other numerical value than 0? [duplicate]

Instantaneous velocity is defined as the limit of average velocity as the time interval ∆t becomes infinitesimally small. Average velocity is defined as the change in position divided by the time ...
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2answers
64 views

Commutation of position four-vector with spacetime derivatives

I am trying to understand a simple demonstration in Ashok Das' Lectures in QFT. He does the following on p. 134 $$[P_\mu,M_{\nu\lambda}]=[\partial_\mu ,x_\nu\partial_\lambda-x_\lambda\partial_\nu]=\...
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Name for the set Displacement, Velocity, Acceleration, etc

Is there a name for the set Displacement, Velocity, Acceleration, Jerk, etc? The only name I can think of is 'Derivatives of displacement (wrt time)' which is rather long.
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Question about derivation of kinematics equations

Apologies if this has been asked before, but I browsed the sub and couldn't find something specific. I understand the derivation for one of the equations as follows: \begin{gather} \frac{dv}{dt} = a ...
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1answer
73 views

Show that $\partial_i A_j - \partial_j A_i = \epsilon_{ijk}B_k$

Let us start from $\textbf{B}=\nabla \times \textbf{A}$ and write its components $B_k=\epsilon_{ijk}\partial_i A_j$. I want to show that $\partial_i A_j - \partial_j A_i = \epsilon_{ijk}B_k$. I can ...
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131 views

Statistical physics is unable to prove that $TdS=d\overline{E}$

I will pose $k_B=1$. Suppose a system of statistical physics with the constraints: $$ \begin{align} 1&=\sum_{q\in\mathbb{Q}}\rho(q)\\ \overline{E}(\beta)&=\sum_{q\in\mathbb{Q}} E(q)\exp(-\...
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2answers
67 views

Is it reasonable and common to interpret $dt$ as a time point (a point in time)? [duplicate]

I heard some one talked about the instantaneous and average velocities. He was using $\Delta t$ to denote a time frame, $dt$ denote a time point. average velocities $\bar{v} = \dfrac{\Delta s}{\...
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If $dQ_p = dU_p + pdV = dH_p$, then how can $dQ_p / dT = \partial H_p / \partial T$

In the book of Kondepudi & Prigogine, Modern Theormodynamics, at page 65, (under constant pressure) $$dQ_p = dU_p + pdV = dH_p,$$ where $H_p$ is the entalpy at the constant pressure $p$. ...
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28 views

Electric field inside a homogenous distribution for slightly different Coulomb's law

I am trying to show that the electric field inside a homogeneous distribution of superficial charge is of the order of magnitude of $\delta$, with: $$V(\textbf{r})=\int d^3\textbf{r'}\frac{\rho(\...
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Is my thinking correct for partial derivatives and tensors?

So I was transforming the affine connection and I ended up with a term like this: $$ \frac{\partial^2 x'^a}{\partial x'^b \partial x^p} $$ where $x$ and $x'$ are two different coordinate systems ...
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3answers
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What does $\Delta$ stand for? [duplicate]

Newton’s first law states that $\Delta v=0$ unless acted on by an external force, $F_{\mathrm{net}}\neq0$. Can someone explain to me what the $\Delta v$ symbol means?
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Question about using Liouville's theorem to calculate time evolution of ensemble average

With the Liouville's theorem $$\frac{{d\rho }}{{dt}} = \frac{{\partial \rho }}{{\partial t}} + \sum\limits_{a = 1}^{3N} {(\frac{{\partial \rho }}{{\partial {p_a}}}\frac{{d{p_a}}}{{dt}} + \frac{{\...
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1answer
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What is $\frac{d}{d\psi}\langle\psi| \hat{O} | \psi\rangle$?

I would like to know what is the derivative of an expectation value with respect to the molecular state $$\frac{d}{d\psi}\langle\psi| \hat{\mathbf{O}} | \psi\rangle$$ Note that here $|\psi\rangle$ ...
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The equation of motion for a scalar field in curved spacetime in terms of the covariant derivative

The equation of motion for a scalar field in curved spacetime $$\frac{\partial\mathcal{L}}{\partial\phi}=\frac{1}{\sqrt{-g}}\partial_{\mu}\left[\sqrt{-g}\frac{\partial\mathcal{L}}{\partial\left(\...
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1answer
91 views

Meaning of time derivative of the Lorentz factor $\gamma$?

This question about the Lorentz factor $\gamma$ in special relativity. I know what $\gamma$ means and how to drive. I'm wondering if I have time derivative of $\gamma$, what dose it mean conceptually?
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$\delta Q = dU + \delta W$. Why is it $dU$ while others are partial differentials? [duplicate]

It is the first law of thermodynamics for a very small change in the state of the system. It is in Heat thermodynamics and statistical physics by Brij Lal, Dr. N. Subrahmanyam, and P.S. Hemne.
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The use of the commutators in quantum mechanics: explanations [duplicate]

Considering that I've never studied quantum mechanics before I have need to understand the operator commutator. My start is: $[A,B]=AB-BA \tag{a}$ Now, why must be $$\left[\frac{\partial }{\partial ...
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2answers
111 views

What does it mean to velocity be a derivative of position? Isn't position a point, not a function? [closed]

So in mass-spring simulation I encountered that one simulates particles by using positions and velocities of particles etc. By mechanics: Velocity is the derivative of position. But, isn't "...
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2answers
103 views

Confusing Total Derivative and Partial Derivative in Classical Field Theory - Noether Theorem

I'm really confused about total derivatives and partial derivatives. My multivariable calculus book (Guidorizzi vol 2 Um Curso de Calculo) says that if I have a function like $f(a(u,v),b(u,v))$ then ...
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76 views

Terminology for time derivative of speed (not velocity)

Is there any standard terminology for the derivative of the magnitude of velocity with respect to time (suitable for use in first-year Calculus)? The word ‘acceleration’, in its technical sense, is ...
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70 views

Is there a generalization of ${{\partial }_{\alpha }}\left( \sqrt{-g}{{V}^{a}} \right)=\sqrt{-g}{{\nabla }_{a}}{{V}^{a}}$ for arbitrary connection?

I am studying general relativity and I am trying to understand how to perform variation of the Einstein–Hilbert action with respect to the metric ${{g}_{\mu \nu }}$ and an arbitrary connection ${{\...
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223 views

Differentiation of a ket vector with respect to a spatial dimension

Consider a state $|\psi\rangle$. While discussing the Schroedinger equation, we say $$\hat{H}|\Psi(t)\rangle=i\hbar\frac{\partial}{\partial t}|\Psi(t)\rangle$$ We also define the hamiltonian operator ...
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91 views

Why does the integral symbol disappear when applying a functional derivative?

it is known that variation is defined by following: but could anyone tell me why the integral symbol disappears after following functional derivative?
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Derivative with respect to vector

How in Lagrangian and Hamiltonian mechanics we take derivatives with respect to velocity and momentum respectively if they are vectors? Can we take derivative with respect to a vector?
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A doubt regarding Modelling physical phenomena and position uncertainty

For example, in velocity, when we say $v=\frac{dx}{dt}$, there is no proof for it. Its almost like an axiom. Something taken to be true, without a proof. How do I know that for every $x=f(t)$, $v=f'(t)...
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321 views

How to prove that the covariant derivative obeys the product rule [closed]

In General Relativity the covariant derivative of contravariant vectors $A^\mu$ is: \begin{equation} \nabla_\mu A^\nu=\partial_\mu A^\nu+\Gamma^\nu_{\mu\alpha}A^\alpha \end{equation} where $\Gamma^\...
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Covariant derivative with respect to commutator

I have some confusion with the notion of $\nabla_{[A, B]}\bf{v}$, that expression, with a commutator of vector fields as the subindex of the connection appears for instance in the definition of the ...
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1answer
78 views

States for derivatives of wave function?

Given a wave function $\psi_t(x)$. The quantum state of a system at time t can be written as the sum of basis states multiplied by the amplitude: $$|t\rangle = \int \psi_t(x)|x\rangle dx^3$$ What ...
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649 views

Confused about the differential of a quantity

We know that by definition, the differential of a single variable function $f(x)$ is $$df(x)=\frac{df}{dx}dx$$ analogously, for a multi-variable function $f(x,y,z)$ $$df(x,y,z)=\frac{\partial f}{\...
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5answers
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Equation of distance and time

How is this equation derived? $$r = r_0 + ut + at²/2$$ where $r_0$ is the initial position of particle and $r$ is the position of the particle after all the motion it has undergone, $a$ and $t$ ...
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Why the continuous arrangement of point masses (particles) at infinitesimal separations leads to a extended system?

I am basically talking in terms of Newtonian mechanics. The Newton's laws started with a good and easy assumption of particles as point masses. This assumption clearly reformed physics and a great ...
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Torsion form and exterior covariant derivative

The torsion form can be defined as the exterior covariant derivative of a solder form, $\Theta=d_\omega\theta$. This derivative is always in the fundamental representation of the algebra $\mathfrak g$ ...
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“Chain Rule” for functional derivatives in the context of a derivation of the geodesic equation by the stationary proper-time principle

I have been working on deriving the geodesic action via finding the stationary points of the proper-time integral for a massive point particle. Consider a space-time manifold $M$ ($\dim M=4)$ equipped ...
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1answer
86 views

Is the Lie derivative along the normal well-defined?

This question is cross-posted at https://math.stackexchange.com/q/3274757/247251 Let $(\Sigma, q)$ be a non-degenerate submanifold of a Lorentzian manifold $(M,g)$. Let $N$ be the section of $T\Sigma ...
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Regd. derivation of some equations in “Bertrand Spacetimes” by Pelick

We are going through "Bertrand Spacetimes" by Dr Perlick, in which he first gave the idea of a new class of spacetimes named as Bertrand spacetimes after the well-known Bertrand's Theorem in Classical ...
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Where is the frame information in a time derivative of a physical vector in a moving frame using limits?

In the equation: $$\left(\!\frac{d \vec r}{dt}\!\right)_{\!1}= \left(\!\frac{d\vec r}{dt}\!\right)_{\!0} + \vec\omega_{01}\wedge\vec r$$ How could this be translated to a mathematical definition of ...
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1answer
45 views

Gibbs-Helmoltz equations

Problem: Show that $$E = T^2 \frac{\delta (S-ET)}{\delta T}$$ Attempt: $$ E = T^2 \frac{\delta (S-ET)}{\delta T} = T^2 \frac{\delta (S)}{\delta T} - \frac{\delta (ET)}{\delta T} =T^2 \frac{\delta (S)...
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Generalized Coordinates Property for a System of Particles

I"m looking at "Principles of Dynamics: Second Edition" by Donald T Greenwood. I'm trying to figure out how he obtains Eq. (6-64) $$\frac{\partial\dot x_j}{\partial\dot q_i} = \frac{\partial x_j}{\...
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1answer
52 views

Divergence of a tensor

On pg.70 of Dalarsson's "Tensors, Relativity and Cosmology" For a mixed tensor of contravariant order 2 and covariant order 1 $(T^{mn}_{p,m})$, the divergence with respect to m is defined as:$$T^{...
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2answers
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On the derivation of Ward-Takahashi identity

I am reading Weinberg's QFT book and in 10.4 he introduced a derivation of Ward-Takahashi identity (where $T$ is the time ordering): $$\begin{align} \frac{\partial}{\partial x^\mu}T{\{J^\mu(x)\Psi_n(y)...
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63 views

Box operator in FLRW metric

Definition of box operator in curved space time is $g^{\mu \nu}\partial_{\mu}\partial_{\nu}$ and in FLRW metric $g_{\mu \nu}$ is $diag(1 ,-a^2(t)$ $,-a^2(t),-a^2(t) )$ so the box operator should be $\...
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63 views

How to differentiate exponentials of operators?

Suppose we have $$e^{At}e^{Bt}=F(t),$$ where $$A, B$$ - operators that do not commute. Now I need to take the derivative $$dF(t)/dt.$$ In which order do I write the operators? $$dF(t)/dt = Ae^{At}e^{...
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69 views

Index (Einstein summation) notation question

Question #1: Lost as to how the second equality in the following equation holds — $$\frac{\partial}{\partial \tau} (A^2) = \frac{\partial}{\partial \tau} (\eta_{\mu\nu}A^\mu A^\nu) = 2\eta_{\mu\nu}...
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53 views

Momentum operator dot a vector

Why is $P \cdot A = A \cdot P -i\hbar\nabla \cdot A$? I was just replacing $P=-i\hbar\nabla $ so I didn't get the first term on the right side