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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Moving coordinate systems

Derive a formula for $\frac{d^3 \vec A}{dt^3}$ in terms of starred derivatives relative to a rotating coordinate system.
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Is derivative of a primary operator primary?

For primary operators we have $$T(z)O(w)=\dfrac{hO(w)}{(z-w)^2} + \dfrac{\partial_wO(w)}{z-w}+\cdots$$ (the ordering problem can be ignored by setting $|z|>|w|$.) If we apply $\partial_w$, then a $(...
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Deduction of Kinetic energy operator in quantum mechanics

In Chapters 1 and 2 of Introduction to Quantum Mechanics Third edition, Griffiths and Schroeter state that to get kinetic energy operator one replaces momentum with $p\rightarrow -i\hbar\,\partial/\...
GedankenExperimentalist's user avatar
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Are eigenvalues of slashed covariant derivative real?

I am trying to demonstrate that the slashed covariant derivative $$ \gamma^\mu D_\mu = \gamma^\mu(\partial_\mu -iA_\mu) $$ has real eigenvalues: $$ \gamma^\mu D_\mu \varphi_m(x)=\lambda_m \varphi_m(x)...
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Derivation of lagrange equation in classical mechanics

I'm currently working on classical mechanics and I am stuck in a part of the derivation of the lagrange equation with generalized coordinates. I just cant figure it out and don't know if it's just ...
Jan Oreel's user avatar
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How to take the second-order gauge covariant derivative in quantum field theory?

I am studying quantum field theory and gauge theory, and I am confused about how to take the second-order gauge covariant derivative of a field. (1) The first way is to write the second order gauge ...
Ruan's user avatar
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Is there no sense of 'absolute' in the universe?

Imagine we are talking about electric potential (e.g. gravitational potential or electric potential or whatever, it doesn't matter), then we have: \begin{equation} dV = \textbf{E} \cdot d\textbf{l}, \...
Bruce M's user avatar
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Finding condition for Adiabaticity

I have a differential equation describing a resonator that looks like this: $$ \frac{d\alpha(t)}{dt} = [j a - b]\alpha(t) + \sqrt b e^{jct}$$ where I can solve it putting: $$\alpha(t) = \alpha e^{jct}$...
SiPh's user avatar
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What does the notation $(k \cdot \nabla ) v$ mean? [duplicate]

I am reading a paper and it uses a notation I am not too familiar about. Although I saw it used elsewhere, I don't remember the meaning of it and I don't want to misinterpret it and realize after ...
tommy1996q's user avatar
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Are we allowed to cancel the units of a derivative?

Since the volume of a sphere $v(r)=\frac{4}{3} \pi r^{3} \left[m^{3}\right]$, its derivative relative to the radius is: $$ \frac{dv}{dr} =4\pi r^{2} \left[\frac{m^{3}}{m}\right] $$ Which is also a ...
Stanislav Bashkyrtsev's user avatar
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Isomorphism of the tangent space and the space of directional derivatives [closed]

I have already constructed the tangent space to a manifold, denoted $T_pM$, and I have a good basis for it $\{\hat e_{(\mu)}\}$. (I followed the method of equivalence classes of curves tangent at $p$....
hodop smith's user avatar
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Can the different differentiation notations be equated and do they have an integral definition? [closed]

Are these all equivalent and is there an extension of this to other notation? Does anyone have a clear and concise chart equating the different notation dialects? I am also curious if there are more ...
Kenneth Mikolaichik's user avatar
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Application of Fermi-Walker derivative to specific problem

I am now reading about the tetrad formalism in GR and I am starting (how not) with the Wikipedia Article: Frame fields in general relativity. In this article, as an example, they show how tetrads can ...
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Spectrum of Harmonic Oscillator with Ladder Operators [closed]

Is the ladder operator trick specific to the harmonic oscillator or can it be generalized to arbitrary second order operators? If yes, what is the general mathematical theory behind it? Can all ...
Bondo's user avatar
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Time taken for a rocket to travel upwards [closed]

My doubt is a rather silly, simple one but i cant seem to understand what's wrong. Let's assume a rocket is moving up with a constant acceleration of a, is moving strictly vertically(no gravity turns, ...
Star Gazer's user avatar
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Why does $e^{-H}\partial_j e^{H} = \partial_j + \partial_jH$?

I apologize if this is a dumb question but I have really thought about this a while and I can’t understand it. I have tried to prove this using the power series of the exponential function but I did ...
JosephSanders's user avatar
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Dirac Delta applied to the gradient of a function

The supplementary section of a paper I am reading uses the "substitution" property of the Dirac delta function : $$\mathbf{v}\left(\mathbf{x}_j\right)\delta\left(\mathbf{x}-\mathbf{x}_j\...
haricash's user avatar
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How to use differentials in thermodynamics? [closed]

I wonder how to use and manipulate differential forms in thermodynamics. I see for $ U= αPV$, it is written $dU = αPdV + αVdP$ But how this works in terms of differentiation? (Proof)
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When to use multivariable chain rule in thermodynamics?

If I take $U(P,V)$, I can do: $$ \frac{dU}{dT} = \frac{\partial U }{\partial P } \frac{dP}{dT}+\frac{\partial U }{\partial V} \frac{dV}{dT} \tag1$$ But, I see the following used in books, $ dU = ...
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Problem in calculation of spherically symmetric Laplacian in electrodynamics

I have come across the following operation in two electrodynamics textbooks, which I find problematic: When evaluating an integral over a Laplacian in a spherically symmetric function, the radial term ...
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Covariant derivative to the metric determinant?

I am reading the paper “alternatives to dark matter and dark energy”, but cannot obtain one specific equation no matter how I tried. So I wrote an email to the author, the following is what he replies ...
user392063's user avatar
3 votes
4 answers
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Variation of a function

I'm studying calculus of variations and Lagrangian mechanics and i don't understand something about the variational operator Let's say for example that i got a Lagrangian $L [x(t), \dot{x}(t), t] $ ...
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How do I get a derivative of the field inside of the path integral?

I am trying to find the 3-gluon vertex rule in QCD by finding the amplitude of a 1-2 gluon scattering process. I want to find the generating functional of the interaction by taking the functional ...
bradas128's user avatar
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Help deriving the integral representation of the derivative of the exponential map [duplicate]

I'm trying to derive the following equation using a Riemann sum formulation only $$ \frac{d}{d t}e^{A(t)} = \int_0^1 ds \,\,\,\,e^{sA(t)}(\frac{d A(t)}{dt})e^{(1-s)A(t)}. $$ The book is using einstein ...
Giovanni Brown's user avatar
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1 answer
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Sufficient condition for conservation of conjugate momentum

Is the following statement true? If $\frac{\partial \dot{q}}{\partial q}=0$, then the conjugate momentum $p_q$ is conserved. We know that conjugate momentum of $q$ is conserved if $\frac{\partial L}{\...
Rainbow's user avatar
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D'Alembert Solution to 1+1D wave equation - integration step

I am working through d'Alembert's solution to the 1+1D wave equation using the substitution of canonical coordinates. I have an initial condition of: $$u_{t}(x,0) = g(x) $$ with a general solution ...
Alexander Savadelis's user avatar
2 votes
1 answer
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Closed interval in variation of a field

Let's suppose that $\psi$ is a field so that $$\psi\in\Gamma^\infty(\pi):=\{\psi\in C^\infty(M,E)\ |\ \pi\circ\phi=I_M\}$$ where $E$ is a spacetime bundle over spacetime $M$ and $\pi : E\rightarrow M$ ...
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What is the correct term for $\nabla\phi$? Co-vector or 1-form or both?

In the olden days, $\nabla\phi$ was used to be called a covariant vector (Weinberg used this language in his book Gravitation & Cosmology). But this terminology is considered bad for several ...
Solidification's user avatar
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Time derivative of a "general" vector $\vec A$ in an accelerating frame: what about e.g. velocity $\vec v$?

According to Morin "Classical Mechanics" (Section 10.1, page 459), the derivative of a general vector $\vec A$ in an accelerating frame may be given as $$\frac{d\vec A}{dt}=\frac{\delta \vec ...
klonedrekt's user avatar
1 vote
1 answer
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Definition of the left-right derivative symbol in the Klein-Gordon scalar product [duplicate]

At the start of QFT, studying the Klein-Gordon scalar field, it is often mentioned that the following is the definition of the scalar product in the space of the solutions: $$\langle f _{\vec{k}}|f_{\...
Noumeno's user avatar
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Spherical coordinate of a vector when divergence of the vector is zero

$\nabla \cdot \mathbf{\delta u_{perp}} = 0$ where $\mathbf{\delta u_{perp}}$ is a function of both x and y coordinates and perpendicular to z axis. Moreover, $\delta u_{perp}$ along z axis is $0$. I ...
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Double covariant derivative of a mixed tensor

Let's say, we have a mixed tensor of type (2,1) denoted by $T^{mn}{}_p$ and the goal is to find the expression of $[\nabla_a, \nabla_b] T^{mn}{}_p$ in terms of fundamental tensors. Firstly, I am ...
raf's user avatar
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Applications of time derivative of unit vector

A math methods textbook I'm currently reading went into great detail deriving the following expression for the time derivative of a generic unit vector $\hat{r}$. $$ \frac{d\hat{r}}{dt} = \frac{1}{r^2}...
quantumNeko's user avatar
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Angle for maximum range for projectile from elevation [closed]

A gun of muzzle speed v is situated at a height h above a horizontal plane. At what angle,should the gun be fired to get the maximum horizontal range. I got R = ((vsinx+((v^2)sin^2x+2gh)^(1/2))vcosx)/...
LiikeWhy's user avatar
1 vote
2 answers
51 views

How to calculate wave equation from a stretched string?

I am reading "Introduction to Electrodynamics" [Griffiths] and in section 9.1.1, there is an explanation for why a stretched string supports wave motion. It begins as follows: It identifies ...
shafe's user avatar
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Why is it wrong to find centripetal acceleration using change of velocity over change of time?

This question asks to find the centripetal acceleration by giving the initial and final velocity over the change of time. As shown, my book combined two rules to find the acceleration. I utterly ...
Manar's user avatar
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4 answers
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How to find the double covariant derivative of a general vector?

I have been reading through Carroll's GR textbook and there is a line in the derivation of the Riemann tensor that I do not understand. $$\nabla_\mu \nabla_\nu V^\rho=\partial_\mu(\nabla_\nu V^\rho) - ...
Chris G's user avatar
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Laplace transform: How to evaluate partial derivative in the denominator of a fraction?

I am solving a differential equation using the Laplace transform. However, to evaluate it I need to evaluate some strange terms. Specifically, I have a partial derivative in the denominator of the ...
J.Agusti's user avatar
2 votes
2 answers
202 views

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
1 vote
2 answers
118 views

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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1 answer
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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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1 answer
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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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3 answers
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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 "...
polology's user avatar
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1 answer
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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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1 answer
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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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1 answer
63 views

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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3 votes
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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
2 votes
3 answers
107 views

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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