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 the derivative of an eigenvalue equation in quantum mechanics?

Equation 2.1 is $$H_0(t)\vert n \rangle = E_n\vert n \rangle$$ Equation 2.8 is $$\langle m \vert \partial_tn \rangle = \frac{\langle m \vert \partial_tH_0 \vert n \rangle}{E_n-E_m}$$ In the following ...
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Scalar curvature from Riemannian metric

I want to compute the scalar curvature for points on an empirical manifold (sampled data). I have already an algorithm that learns the Riemannian metric and computes geodesics, so from the metric I ...
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Spherical and Cartesian forms of divergence [closed]

Suppose the electric field found in some region is $$\overrightarrow{E} = ar^3\vec{e}_r$$ in coordinates spherical (a is a constant). What is the charge density? So, using the spherical form of ...
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Question regarding Energy Interaction of two particles

https://imgur.com/s6RGUKb To give a context as to what I'm asking here ,I am talking about the energy of a two particle system (section 4.9 Taylor's Classical Mechanics) . My question is what does $\...
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What does divergence of scalar times vector vector field physically mean?

We know that: $\nabla \cdot (f \vec{A}) = f \nabla \cdot \vec{A} + \vec{A}\cdot(\nabla f)$ Now divergence of any vector field can be understood in terms of whether the concerning flux is outgoing ($\...
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What does $\dot x$ mean as an operator in quantum mechanics?

I've been looking at a paper titled "Feynman's proof of the Maxwell Equations" by Freeman Dyson (American Journal of Physics 58, 209 (1990); https://doi.org/10.1119/1.16188) and I'm confused ...
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Propagation Of Error Calculation Of Incompressible Continuity Equation

Hello I am having trouble deriving the correct equation for error in vertical velocity from the incompressible continuity equation. We are trying to estimate vertical velocity ($w$) from the ...
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Acceleration in terms of displacement

I am having problems understanding the derivation of acceleration in terms of displacement. The first step is fine: $$a(x) = \frac{\mathrm dv(x)}{\mathrm dt} = \frac{\mathrm dv(x)}{\mathrm dx} \frac{\...
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Action of Lie derivative on 1-forms

In Sean Carroll's spacetime and geometry appendix B he derives the action of the Lie derivative on 1-forms. He finds that $\mathcal{L}_X Y^\mu = [X, Y]^\mu$, which I believe is meant as $\mathcal{L}_X(...
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Two questions concerning dirac delta function and Hamiltonian

I'm trying to compute to quantities with Hamiltonian and Dirac delta function but I don't how to do it properly. I'm stuck calculating the following quantity $$ \frac{d}{dE} \left[ \theta(E-H(x,p;V)) ...
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Is gauge covariant derivative an ordinary covariant derivative?

The formal treatment of the gauge covariant derivative in most reference texts for students is too informal and too ad hoc, so that some general issues remain unclear. For example, the gauge covariant ...
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What is the Lie derivative of the field describing the change of mass?

I'm trying to understand Ch. 3.2 of the paper On Bubble Rings and Ink Chandeliers by Padilla et al.. I'm trying to understand the derivation of equation (15). Right now I'm stuck at the point where ...
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Question about derivative of transformation [duplicate]

I have a question that result equation is how it has derived. When we have a following equation, $$\frac{\mathrm dr}{\mathrm dt}=\frac{\delta r}{\delta t}+\omega \times r.$$ $\omega$ is angular ...
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Is $dx$ always positive?

When we refer to change in a quantity, we define it to be (final-initial). If it is positive it indicates an increase from the initial value and negative indicates a decrease. But when we take this to ...
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How to determine these derivatives wrt. matrix-valued fields?

We know strength tensor for $W$-boson field $$W_{\mu\nu}=\partial_\mu W_\nu-\partial_\nu W_\mu+\frac{ig}{2}[W_\mu, W_\nu]$$ Where $[W_\mu, W_\nu]=W_\mu W_\nu-W_\nu W_\mu$ is Lie bracket/commutator and ...
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Differentiation algebra with index notation

Consider $p_\mu p^\mu$ and let us differentiate it with respect to $p^\nu$. Then, $$\frac{\partial}{\partial p^\nu}(p_\mu p^\mu) = p^\mu \eta_{\mu \sigma}\delta^\sigma_\nu + p_\mu \delta_\nu^\mu = 2 ...
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Questions regarding the derivation of Euler-Lagrange Equation from Taylor's Classical Mechanics

I'm self-studying classical mechanics from Taylor's book. I saw his derivation of the Euler-Lagrange Equation and I'm confused about something, he created a 'wrong' function $$Y(x) = y(x)+\eta(x)\tag{...
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Trying to understand gravitational force equation

I don't understand the red underlined equations but I understand gravitational force equation in simpler form. Can anyone please explain the equations? Source An Introduction to Celestial Mechanics by ...
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Doubt on $SU(2)_{L} \times U(1)_{Y}$ covariant derivative and its action on a fermion

I) Introduction I.1) The mathematical structure is quite clear: given a spacetime $M$, and a Lie group $G$ (the gauge group), we can construct the Principal bundle $P^{G}_{M}$. The connection $1$-form ...
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What is $A'$ in the Reissner-Nordstrom metric?

So I was reading this paper on the Reissner-Nordstrom metric and on it they define $A$ as: But they don't define $A'$. Yet $A'$ still ends up in other equations like defining the Ricci tensors: So ...
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Goldstein: derivation of work-energy theorem

I am reading "Classical Mechanics-Third Edition; Herbert Goldstein, Charles P. Poole, John L. Safko" and in the first chapter I came across the work-energy theorem (paraphrased) as follows: ...
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What problems would arise in physics by treating infinitesimals as ~1 (in units much smaller than the measurement precision) rather than ~0?

I have a hopefully simple/ignorant question. The difference quotient, where $h$ is an "infinitesimally" small value: $$f'(x) = \frac{f(x + h) - f(x)}{h}$$ What problems (if any) would arise ...
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Wavefunctions and Hamilton-Jacobi equation [duplicate]

I was reading this paper, Wavefunctions and Hamilton-Jacobi Equation. The author started with the Hamiltonian formalism and then came up with a connection to the Schrodinger equation. There was a ...
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Wavefunction from the Hamilton-Jacobi formalism [closed]

I was reading this paper, Wavefunctions and Hamilton-Jacobi Equation by Sabrina Pasterski. The author started with the Hamiltonian formalism and then came up with a connection to the Schrodinger ...
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Implications of Galilei-Invariance on a time-independent potential

I'm trying to compute a result shown in my classical mechanics lecture on my own. Namely, consider that a system composed of $n$ particles follows a law of force $m_k\ddot{\vec{x_k}} = \vec{F_k}(\vec{...
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Einstein field equations from covariant derivative of a general linear gauge transformation

A general linear transformation is given by \begin{align} \psi'(x) \to g \psi(x) g^{-1}, \end{align} The gauge-covariant derivative associated with this transformation is \begin{align} D_\mu \psi=\...
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Differentiation [closed]

Why is $$\frac{d}{dt}v^2=2v\frac{dv}{dt},$$ When: $$\frac{d}{dx}x^2=2x,$$ where $v$ is velocity? I don't understand why the variable $x^2$ has the derivative of $2x$, whereas the variable velocity has ...
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Explicit calculate covariant derivative for spinor field

I want to explicitly calculate the covariant derivative for spinor-fields for a given metric (to investigate the 1d dirac equation in curved spacetime): $\begin{align*} g_{\mu\nu}=\frac{L^2}{cos^2(\...
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Is every $dm$ piece unequal when using integration of a non-uniformly dense object?

When we want to find the total charge of an object or total mass, usually we start off with a setup such as: $$ m = \int dm \:\;\:\text{or} \:\;\:q = \int dq$$ in which we then use (and to keep it ...
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Why is instantaneous velocity tangent to trajectory?

Trajectory is the path of an object through space as a function of time. However, in many trajectory plots, when the movement is planar, a horizontal position axis and a vertical position axis are ...
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In Radius of Curvature calculation why do I have to assume $\text{d}^2x/\text{d}t^2=0$?

Recently, I was calculating the radius of curvature of projectile trajectories at certain points. There are two ways to do the same: Given the velocity and acceleration of the particle at some point, ...
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Commutator between covariant derivative and a field

I have field as an element of a Lie algebra as $\Phi = \phi^at^a$ and I want to calculate the commutator $$[D_{\mu}, \Phi],$$ with $$D_{\mu} = \partial_{\mu} + igA^a_{\mu}t^a,$$ the covariant ...
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Determine the meaning of a gradient of a graph [closed]

How do you determine the gradient of a graph in physics, such as how with a velocity-time graph, the gradient is acceleration. I want to know the general method for figuring out what the differential ...
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2 votes
1 answer
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Proof that the Euler-Lagrange equations hold in any set of coordinates if they hold in one

This is a question about a specific proof presented in the book Introduction to Classical Mechanics by David Morin. I have highlighted the relevant portion in the picture below. In the remark, he ...
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Finding the Euler-Lagrange equation for a scalar field

Consider a scalar field with the following lagrangian density: $$\mathscr{L}=-\frac{1}{2} \partial_{\mu} \phi \partial^{\mu} \phi-V(\phi).$$ I want to find the corresponding Euler-Lagrange equation, ...
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Higher dimension derivatives

In the case of higher dimensions (e.g. 4+1 dimensions) how would the 5 derivative ($\partial_5$) change? For example if $\\X=(x^{\mu},z)$, would the 5 derivative change as $$\partial_5\partial^5X=\...
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What is the meaning of $\nabla _{\mu}\nabla _{\nu}\phi(r)$ in general relativity?

I know the covariant derivative of a tensor is $$\nabla_{\mu} V_{\nu}=\partial_\mu V_\nu-\Gamma_{\mu\nu}^{\lambda}V_{\lambda}$$ Now I want to obtain $\nabla_{\mu}\nabla_{\nu}\Phi(x)$ where $\Phi(x)$...
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Taking the second time derivative of a scalar field

Given some scalar field $\phi(x,y,x,t)$, taking its first total derivative we get: $$\frac{d\phi}{dt}=\frac{\partial\phi}{\partial t}+\frac{\partial\phi}{\partial x}\frac{dx}{dt}+\frac{\partial\phi}{\...
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Question about Wald's example of a "derivative operator"

I am trying to study Wald's book on General Relativity. I thought the first two chapters were ok, but I'm completely stuck in chapter 3, on page 32 where he gives an example of a "derivative ...
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Why can we change $dt$ with $(dt/dp)_s dp$?

In my homework assignment there's the following question: A general thermodynamic system is being compressed isentropically from pressure $P_i$ to $P_f$ while keeping the number of particles constant....
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Standing Wave Equation: Why does assuming a small slope $du/dx$ not make $d^2u/dx^2$ negligible as well?

Referencing the above image, just change the label for $y$-axis to $u$-axis.^ Following the derivation of the standing wave equation: https://www.youtube.com/watch?v=IAut5Y-Ns7g&t=1324s So if ...
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Simple difference between module of velocity and time derivative of module of position [duplicate]

What is the conceptually difference between the two: $$\frac{d|\vec{r}|}{dt}=\frac{\vec{r}\cdot\frac{d\vec{r}}{dt}}{|\vec{r}|}\neq|\dot{\vec{r}}|\equiv \bigg|\frac{d\vec{r}}{dt}\bigg|$$ ...
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Why are formulas in Physics represented in form of differentiation?

Mostly formulas are represented in differential form as we learn more about the concepts of physics , for ex. $I = \frac {q}{t}$ is also written as $\frac {dq}{dt}$ A explanation would help .
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Variation of the Lagrangian

In Tong's QFT notes at the bottom of page 14, it is claimed that if a change $x\mapsto x-\epsilon$ is made, the Lagrangian changes in the following way: $$\mathcal L(x)\rightarrow \mathcal L(x)+\...
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4 votes
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What does $\delta/\delta t$-derivative represent in tensor calculus?

Some texts, such as Pavel Grinfeld's, talk about a $\delta/\delta t$-derivative whose role (in trajectory analysis of particles using tensor calculus) is pretty obscure to me. For example, the ...
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7 answers
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What is the instant velocity? [duplicate]

The velocity is the variation rate of the position correct? So does it make sense to talk about velocity without time?
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Currently self-studying QFT and The Standard Model by Schwartz and I'm stuck at equation 1.5 in Part 1 regarding black-body radiation

So basically the equation is basically a derivation of Planck's radiation law and I can't somehow find any resources as to how he derived it by adding a derivative inside. Planck says that each mode ...
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Non-parallel light diffraction

Does light (and in general any kind of wave) diffract only when the wave fronts are parallel? Like if you did the double slit experiment when the waves were coming from a point source close to the ...
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What are the spaces in which quantum fields belong and how does that affect the hermitian conjugate of $\partial_{\mu}$?

In this post, I claimed in my answer that $\partial_{\mu}^{\dagger}=\partial_{\mu}$. The reason I claimed that is because $D^{\dagger}_{\mu}=\partial_{\mu}-iqA_{\mu}$ and I assumed that $$(\partial_{\...
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4 votes
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Generalizing Fermi-Walker Derivative/Transport to General Vector Bundles

The usual definition of the Fermi derivative (eg as given in Hawking and Ellis) is to consider a Lorentzian manifold $(M,g)$ and a unit timelike curve $\gamma$, a smooth vector field $X$ along $\gamma$...
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