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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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65 views

What is infinitesimal displacement?

This section is from the Openstax University Physics: Volume 1 online textbook. In physics, work is done on an object when energy is transferred to the object. In other words, work is done when a ...
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1answer
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Acceleration of speed of light [on hold]

If we consider the equation "$E=mc^2$" and if we differentiate the equation with respect to time, e.g. $$\frac{\mathrm dE}{\mathrm dt}=m\frac{\mathrm dc^2}{\mathrm dt},$$ we will get after ...
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Differentiation of metric tensor in new coordinate

I want to understand the explicit meaning of $g_{\mu'\nu',\lambda}=0$ where unprimed coordinates are coordinates of the the original coordinate systems and primed ones are for new coordinate system. ...
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2answers
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Differentiation of the determinant $g$

Let $g$ be the determinant of the metric tensor. I want to derive the following equation $g_{,\nu}=gg^{\lambda \mu}g_{\lambda \mu,\nu}$. It is said that $gg^{\lambda \mu}$ is a cofactor, but I can't ...
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The two definitions of the divergence of a vector field? [migrated]

Now, I am aware that the divergence of a vector field, $\vec{F}$, can be defined in two ways. What I don't understand is why do these equal each other formally? Definition 1: $$\text{div}\vec{F} = \...
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How to differentiate an operator in QM?

I recently started learning QFT and the lecturer wrote down some equations for a quantum harmonic oscillator: $$\hat H= \frac{\hat p^2}{2m}+\frac{1}{2}mw^2\hat x^2$$ $$\partial^2_t \hat x=-w^2\hat x$$ ...
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28 views

Delta over variable relations

In the text I'm reading, which discuss the heat transfer by convection in stars, it shows that the relation $\rho \propto \frac{P}{T}$ (equation of state for an ideal gas) implies $$\frac{\Delta\rho}{\...
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1answer
58 views

How to take derivative with respect to Lagrangian of complex field?

Basics: The Lagrangian in field theory was written as $$\frac{\partial \mathfrak{L}}{\partial \varphi}=\partial_\mu(\frac{\partial\mathfrak{L}}{\partial(\partial_\mu\varphi)})$$. Question 1: Is $\...
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How does vectorization affect $\nabla$?

The homework and exercise was to prove $\nabla \times {A}$ transform as a vector, and I've solved it thorough hard algebra. However, something occurred to my mind and I have a hard time to resolve it....
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3answers
107 views

Proof for Vector Identity

I am currently studying electrodynamic and came across the following vectoridentity, but I am unable to prove it: $$ \vec{f} \times ( \nabla \times \vec{f} ) -\vec{f}(\nabla\cdot\vec{f}) = \nabla \...
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1answer
64 views

Find $v(t)$ and $x(t)$, How do I treat $δt$? [closed]

We apply a force to a particle with a mass $m$ and inicial velocity $v_0$: $$ F(t) = \left \{ \begin{matrix} 0 & \mbox{ $t<t_0$} \\ \frac{p_0}{\delta t} & \mbox{ $t_0<t<t_0 +\...
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Fractional differential equations and Physics [duplicate]

Are the "fractional differential equations" have any real significance in respect to physics? or are they just stilted math?
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1answer
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What was the real need of the operators of divergence and curl?

As I'm advancing my study in Electromagnetism I'm getting introduced to more mathematical operators which are exclusively used in Electromagnetism and Fluid Dynamics only. Let me try to explain ...
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1answer
49 views

(Physics version of) Taylor expansion. In the the context of deriving a Lie groups generators (a Lie algebra from a Lie group)

Statement which I'm confused about: "Consider some n-dimensional Lie group whose elements depend on a set of parameters $\alpha = (\alpha_1 ... \alpha_n)$, such that $g(0) = e$ with e as the identity,...
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1answer
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Divergence of a vector multiplied by dot product [closed]

If I am correct, then $\operatorname{div} [(\vec A\cdot \vec B)\vec C] = (\vec A \cdot \vec B) \operatorname{div} \vec C + \vec C \cdot \nabla (\vec A\cdot\vec B)= (\vec A \cdot \vec B) \operatorname{...
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Converting velocity vector formula from Cartesian coordinate system to polar coordinate system

I have a little question about converting Velocity formula that is derived as, $$\vec{V}=\frac{d\vec{r}}{dt}=\frac{dx}{dt}\hat{x}+\frac{dy}{dt}\hat{y}+\frac{dz}{dt}\hat{z}$$ in Cartesian Coordinate ...
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0answers
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Derive ballistic equation using conservation of energy, stuck with (dx/dt)^2 [closed]

When observing a falling particle, it's easy to derive the trajectory using the following reasoning: the particle is subject to one external force mg and by taking ...
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1answer
36 views

Meaning of normal acceleration?

acceleration means the rate of change in velocity (vector quantity) and the differentiation means to divide a certain quantity into small elements (i.e $dx$) as we do to find the acceleration at any ...
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3answers
97 views

If $\mathrm df$ is an inexact differential, how would the function $f$ look like?

I am studying thermodynamics and in the first chapter the concept of exact and inexact differentials were used to talk about the differences between internal energy, work and heat. From Blundell and ...
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4answers
106 views

Why the need for defining the velocity as a derivative? [closed]

Something intuitive and fundamental as the concept of velocity (of a particle for example) in classical physics is defined as a derivative, a concept to me quite vague and strange, although i know its ...
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How is this equality true? Lagrangian mechanics

So this equality was given as a part of how to derive the Lagrangian equality from Newton's second law. How it is true? $$\frac{d\pmb{p}_k}{dt}\cdot\frac{\partial\pmb{r}_k}{\partial q_i} = \frac{d}{...
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2answers
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Leibniz Rule for Covariant derivatives

I recently came across a video by prof Fredrick Schuller on general relativity where he defines the leibniz rule to be, $\nabla_X (T(\omega,Y))=\nabla_XT(\omega,Y)+T(\nabla_{X} \omega,Y)+T(\omega,\...
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Is rate of change of internal energy with respect to volume at constant temperature always zero?

According to substitutions of some equations in "Heat and Thermodnamics by Zemansky", $\left(\frac{\partial U}{\partial V}\right)_T$ turns out to be zero. First $T\ dS$ relation is: $$TdS=C_v\ dT+T\...
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1answer
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Why do we neglect $\Delta t^2(\frac{d\vec{r}}{dt}\frac{d\vec{\hat{r}}}{dt})$ at Taylor Expansion?

I'm just started to Ankara University Physics Department two weeks ago. I have missed my 2 hours of PHY105 course that is the last week Wednesdey. The subject that i missed was Derivatives of Vectors. ...
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1answer
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Do Equal Time Commutation / Anticommutation relations automatically also apply for spatial derivatives?

The question is basically in the title. My naive thought was that when a commutation relation holds for all field operators $\Psi(\vec{x})$ (by "all" I mean "at all positions $\vec{x}$") on a fixed ...
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Cauchy-Rienmann equations [migrated]

This may be a stupid question but I was reading the proof of Cauchy-Riemann equations from Wolfram MathWorld. Here, they seem to have put $\frac{\partial x}{\partial z} = \frac{1}{2}$. Shouldn't this ...
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51 views

Math question about point transformations

I am trying to prove the classic problem to showcase Lagrangian's scalar invariant property. Namely, that if you have $x_i = \{ x_1, ...., x_n; t \}$ , you can then represent $L(x_1,....,\dot{x_1},.....
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1answer
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Computing total derivative of Kinetic Energy w.r.t time

I am confused as to how to take the total derivative $\frac{dKE}{dt}$, where $KE$ is the kinetic energy. I know that $KE = 1/2 *m * \dot{\vec r} \cdot \dot{\vec r}$. From here, if I take derivative ...
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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
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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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1answer
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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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2answers
101 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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2answers
103 views

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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1answer
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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
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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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0answers
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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
71 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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1answer
130 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
66 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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2answers
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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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0answers
27 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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0answers
56 views

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
93 views

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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1answer
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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
78 views

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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1answer
135 views

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(\...
2
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1answer
80 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.