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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How to take derivative of density operator?

I was just trying to confirm to myself that the following density operator $$\rho(t) = e^{-iHt/\hbar} \rho(0) e^{iHt/\hbar}$$ fulfills the Liouville-von Neumann equation: $$\frac{d}{dt}\rho(t) = - \...
Physchem16's user avatar
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What's the difference between $\nabla\cdot(\rho v)$ and $\rho(\nabla\cdot v)$ as a physical intuition?

I'm currently learning on substantial derivatives in fluid mechanics and kind of understand how partial derivatives $\frac{(\partial\rho)}{(\partial t)}$ and substantial derivatives $\frac{(D\rho)}{(...
Lime nut's user avatar
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From where does the expression of the tangential accerelation come from?

I've seen so many times that the expression of the tangential acceleration is known to be: $$a_t=\ddot{s}$$ but from the expression of the acceleration in spherical coordinates, in the tangential ...
Ulshy's user avatar
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Sum of two state functions is not path independent

I am trying to explain the different physical meanings of the various thermodynamic potentials (before resorting to Legendre transforms, and without making appeals to statistical mechanics) and I ...
Emerson's user avatar
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Square Root of an Operator

NOTE: CROSS-POSTED ALSO ON MATH EXCHANGE I'm working with quantum mechanics and i have to solve an operator equation. I have the two canonical variables $S$ and $P$ which are then promoted to ...
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Is the derivative of the adjoint the adjoint of the derivative? [closed]

Does $\frac{d}{dx}\langle u|= (\frac{d}{dx}|u\rangle)^\dagger$? Here $|u\rangle$ is any vector in a Hilbert space and $\dagger$ is the adjoint/conjugate-transpose. It seems to me that this should not ...
Spenser Talkington's user avatar
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Derivation of covariant derivative by means of parallel transport

I've studied covariant derivative in many courses so far, but I got stuck on the definition given by the teacher notes of the exam of Topological QFT. I think that he improperly used the name "...
tpr's user avatar
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Why is the regular Taylor expansion so similar to the construction of the Hamiltonian from a Lagrangian/Legendre transform?

An example of a first order Taylor expansion of a function with two variables is given by: $$f\left(x,y\right)=f\left(x_0,y_0\right)+\left(x-x_0\right)\frac{\partial f}{\partial x}+\left(y-y_0\right)\...
bananenheld's user avatar
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Why does $\delta \vec{r} = \delta \vec{ \theta} \times \vec{r}$?

Hello fellow physicists, I was trying to understand some behavior on rotating objects, specifically about the formula $\vec{v} = \vec{\omega} \times \vec{r}$. The Book (Marion, J. B. (1965). Classical ...
Carrot Carron't's user avatar
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"Why is $n$ held constant when taking the time derivative in the course of the Van Kampen's system size expansion?"

I follow the notation used in "stochastic processes in physics and chemistry"(p.245) by Van Kampen. Left hand side of master equation is $$\frac{\partial P(n,t)}{\partial t}=\cdots.$$ We ...
kakaikiichi's user avatar
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Show that $dE/dt = -bv^2$ (Help with differentiation) [closed]

The question is: Show that $$dE/dt = -b (dx/dt)^2.$$ And the solution is: ...
Theo's user avatar
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Reduce multiplicative noise to additive noise with singular matrices

I have a stochastic differential equation as \begin{equation} \dot{\textbf{X}}=\textbf{A}\textbf{X}+\alpha(t)\textbf{B}\textbf{X}-\alpha^*(t)\textbf{B}^T\textbf{X} \end{equation} where $T$ ...
J.Agusti's user avatar
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How do we get Momentum operator from linear momentum? [duplicate]

How do we get the momentum operator $p=-i(h/2π) ∇)$?
Fredrick's user avatar
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3 answers
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Finding the vector potential

$$\nabla\times\mathbf{B}=\nabla\times\left(\nabla\times\mathbf{A}\right)=\nabla\left(\nabla\cdot\mathbf{A}\right)-\nabla^2\mathbf{A}=\mu_0\mathbf{J}\tag{5.62}$$ Whenever I try to work this out and ...
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Derivative of displacement in deriving expression for intensity of sound waves

I am currently working on deriving the expression for intensity of a sound wave: https://openstax.org/books/university-physics-volume-1/pages/17-3-sound-intensity The previously mentioned book states: ...
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Cylindrical incompressible Navier-Stokes, are the derivatives commuative?

I am currently trying to understand a paper by Eckhardt et al. (https://doi.org/10.1017/S0022112007005629). In it, a transformation is performed on the cylindrical incompressible Navier-Stokes ...
Yann's user avatar
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Physical Interpretation of Flowrate Time Derivative of Incompressible Fluid in Variable Volume Vessel with Single Inlet

I am an academic researcher who studies fluid mechanics of the left ventricle (the primary chamber of the heart that actually pumps blood to the rest of your body). The majority of my work focuses on ...
John's user avatar
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Confirming an action is invariant under a supersymmetric transformation

I am studying chapter 9 of the book Mirror Symmetry, available here. My question is relating to page 156/157 where Supersymmetry is being introduced for the first time in QFT in 0-dimensions. We are ...
Gleeson's user avatar
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Del operator confusion [closed]

The very first thing my textbook says is that the Del operator is defined as: $$\vec{\nabla}=\vec{a}^i\nabla_i$$ Where $\nabla_i$ is the covariant derivative and " $\vec{a}^i$ is the curvilinear ...
Krum Kutsarov's user avatar
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What is the intuition or the derivation of covariant derivative?

I asked this question in mathematics but the answer I got was a bit too abstract for me so I hope that my fellow physicists can give me more of an intuition or an easier explaination of my question. ...
Krum Kutsarov's user avatar
4 votes
2 answers
142 views

Looking for the geometric meaning of the curl of Killing vector fields

From Killing equation $$\nabla_\nu \xi_\mu + \nabla_\mu \xi_\nu = 0$$ it can be shown that $\nabla_\nu \xi_\mu$ is antisymmetric. From it we can construct an antisymmetric tensor $\mathcal{A}_{\mu\nu}$...
Pau Bañón Pérez's user avatar
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What is the gauge covariant derivative?

What is the gauge covariant derivative in layman's? Does it describe the kinetic energy of the gauge fields?
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Integral of derivative of delta function gives strange answer [closed]

So I've been doing some QM and I keep coming across the following type of integrals: $$ \int f(x) \frac{\partial}{\partial x} \delta(x-x') dx. $$ I know that I should integrate by parts but then I ...
Gytis Vejelis's user avatar
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Covariant derivative of metric determinant with torsion

I have some troubles taking the covariant derivative of the metric determinant with torsion. Let's suppose that we take a metric such that $\nabla_\mu g_{\nu\rho}=0$. My reasoning is the following. ...
Physics Koan's user avatar
3 votes
3 answers
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Is the rate of change of duration a valid quantity?

I was wondering that, if the duration of a recurring event varies as time goes on, what would the magnitude of this quantity be measured in? For instance, if the time for an oscillation of a weighted ...
Nasser Kessas's user avatar
1 vote
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Lie derivative of a one-form

I am going through Nakahara's textbook on geometry and topology in physics. Intuitively, I understand the definition of a lie derivative of a vector field $$\mathcal{L}_xY = \lim_{\epsilon_\to 0}\frac{...
Souroy's user avatar
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The BRST variation of the gauge fixing condition

Following Polchinski volume I, p 126 onwards, The BRST variation of fields $\phi^{i}$ is given by $$\delta_{B} \phi_{i} = - i \epsilon c^{\alpha} {\delta}_{\alpha}\phi_{i} \; .\tag{4.2.6a}$$ My ...
unifymchn_MCR's user avatar
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2 answers
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About the traditional explanation of the continuity of the first derivative of a 1D wavefunction

I would like to receive some clarifications about the traditional explanation of the continuity of the first derivative of a 1D wavefunction (E.g. see the very clear answer by @ZeroTheHero ...
Valter Moretti's user avatar
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Proving entropy is an exact differential [duplicate]

I have been trying to understand something that my professor explained and I have tried everything (books, my classmates' notes, internet) but I can't understand, can you please help retrieve the ...
mlp's user avatar
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What are some ways to derive $\left( \boldsymbol{E}\cdot \boldsymbol{E} \right) \nabla =\frac{1}{2}\nabla \boldsymbol{E}^2$?

For each of the two reference books the constant equations are as follows: $$ \boldsymbol{E}\times \left( \nabla \times \boldsymbol{E} \right) =-\left( \boldsymbol{E}\cdot \nabla \right) \boldsymbol{E}...
Vancheers's user avatar
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1 answer
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Cross factor for dependent terms in a differential?

How do you derive a cross factor to decouple differentials into independent differentials? For example: $$ d(PV)= PdV+VdP $$ $$ PV=\int{PdV}+\int{VdP} $$ Obviously dP and dV are related. Do you simply ...
ChemEng's user avatar
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5 answers
192 views

Why does $\vec{r}\cdot\dot{\vec{r}}=r\dot{r}$?

Why is $$\vec{r}\cdot\dot{\vec{r}}=r\dot {r}$$ true? Before saying anything, I have seen the proofs using spherical coordinates for $$\dot{\vec {r}}= \dot{r}\vec{u_r}+r\dot{\theta}\vec{u_\theta}+r\sin\...
Ulshy's user avatar
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3 votes
2 answers
220 views

Meaning of the differential entropy

The definition of differential (or continuous) entropy is problematic. As a matter of fact, differential entropy can be negative, can diverge and is not invariant with respect to linear ...
Upax's user avatar
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2 answers
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Velocity to Acceleration negative line [closed]

Is the velocity line in below 0 is a different acceleration line? For example from 0 - 6s and from 10 - 17s. It has the same slope.
Howard Tran's user avatar
2 votes
2 answers
140 views

How to calculate the rotation at a singularity?

An electrodynamics lecture asks me to prove that $$ \nabla \times \left( \frac{\vec{M} \times \vec{x}}{ |\vec{x}|^3} \right) = \frac{8 \pi}{3} \vec{M} \delta^3(\vec{x})- \frac{\vec{M}}{|\vec{x}|^3}+ \...
F L's user avatar
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1 vote
3 answers
176 views

Entropy as an exact differential

I have been trying to prove that $dS$ is an exact differential given this definition: If you have a function of two variables $z = f(x, y)$ for which $dz$ can be expressed as: $$dz = \left(\frac{\...
mlp's user avatar
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1 answer
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Maxwell's relations and adiabats

I was trying to understand problem regarding finding the adiabatic modulus given the isothermal young's modulus. I'm still an amateur in thermodynamics. I just didn't understand the final step where ...
ilawid's user avatar
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1 vote
1 answer
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Weird derivative with respect to inverse temperature identity in Tong's statistical physics lecture notes

While reading David Tong's Statistical Physics lecture notes (https://www.damtp.cam.ac.uk/user/tong/statphys.html) I came across this weird identity in page 26 (at the end of the 1.3.4 free energy ...
duodenum's user avatar
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1 answer
54 views

Tensor Index Manipulation

I am trying to study General Relativity and I thought about starting with some index gymnastics. I found a worksheet online and I am stuck with a simple problem. I have to prove that $$\partial_{\mu} ...
 Paranoid's user avatar
1 vote
1 answer
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Equality of variables for small values of time, when the time derivative of the variables are equal to one another

It is given that $\frac{ds}{dt} = \frac{d\theta}{dt}$, i.e. the time derivative of s and $\theta$ are equal to each other. Does it follow that for small values of $t$, $\Delta s ≈ \Delta \theta$? My ...
Joel Sam Johnson's user avatar
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2 answers
62 views

How do extreme points work in Statistical Mechanics?

Suppose that I have an $S,V,N$ ensemble. Every variable is a function of the other variable: $U(S,V,N)$, $S(U,V,N)$, $V(S,U,N)$ and $N(S,U,V)$. The functions are everywhere differentiable. But there ...
Riemann's user avatar
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-2 votes
2 answers
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Why does $\vec{a}=\vec{\omega}\times \vec{r}$ as well as the velocity does?

Today I came in class and in one of the problems the teacher used $\vec{a}=\vec{\omega}\times \vec{r}$ which made me very confused because I don't know where it comes from, it seems pulled out of thin ...
Ulshy's user avatar
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8 votes
2 answers
690 views

How does the covariant derivative satisfy the Leibniz rule?

In Carroll's "Spacetime and Geometry", he states on page 95 (section 3.2) that the covariant derivative, $\nabla$, is a map from $\left(k, l\right)$ tensor fields to $\left(k, l+1\right)$ ...
MBar2269's user avatar
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1 answer
97 views

How to evaluate the action of a fractional differential momentum operator?

I need to understand how a fractional operator works if before being applied on a test function it acts on another (well known)function: $$(-i\hbar\frac{\partial}{\partial x}\cdot\delta(x))^\epsilon\...
Cuntista's user avatar
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From where does the uncertainty formula come from? [duplicate]

The uncertainty formula is the one used in the laboratories to find the uncertainty of a variable. Say $X$ is a function of $Y$ and $Z$ such that $X=X(Y,Z)$ then it's uncertainty can be found with: $$\...
Ulshy's user avatar
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1 vote
1 answer
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Is the Lie derivative in a coordinate direction covariant?

Considering a partial derivative of a vector field $w^a$ in x-direction (also called here 1-direction) I can write it as $$\frac{ \partial w^a}{\partial x^1 } = \partial_1 w^a - \Gamma^a_{1c} w^c + \...
Frederic Thomas's user avatar
0 votes
1 answer
57 views

How to understand the derivatives in wave equation?

I am looking at the derivation of the wave equation, but I am stuck on the math. Specifically, in the following: How do they get the equivalence between $\frac{\partial}{\partial z} (\frac{dg}{du}) = ...
reesespieces's user avatar
2 votes
6 answers
205 views

Lagrangian - How can we differentiate with respect to time if $v$ not a function of time?

In the Lagrangian itself, we know that $v$ and $q$ don't depend on $t$ (i.e - they are not functions of $t$ - i.e., $L(q,v,t)$ is a state function.) Imagine $L = \frac{1}{2}mv^2 - mgq$ Euler-Lagrange ...
Giorgi's user avatar
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0 answers
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How does the divergence change under a change of frame (with geometric algebra)?

I'm trying to prove equations (85) and (86) from Hestenes' paper Gauge Theory Gravity with Geometric Calculus (ResearchGate version). $$ \dot{\nabla}^\prime \cdot \dot{\underline{f}}(A) = J_f^{-1}[ (\...
foghorn's user avatar
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1 vote
1 answer
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Variation of Torsion-Free Spin Connection

In the book 'Supergravity' by Freedman and van Proeyen, in exercise (7.27) it is written To calculate [the variation $\delta\omega_{\mu ab}$ of the torsion-free spin connection], consider the ...
vyali's user avatar
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