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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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
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Can I contract index in this expression?

I'm reading Carrol text on general relativity, on page 96 they arrive to the term \begin{equation} \frac{\partial x^{\mu}}{\partial x^{\mu '}}\frac{\partial x^{\lambda}}{\partial x^{\lambda '}}\frac{\...
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Covariant derivative in a basis

Reading through this paper, I saw that the energy momentum conservation: $$\nabla_\mu T^{\mu\nu}=0$$ can be evaluated as: $$\partial_t(\sqrt{-g}T^{t}_\nu)=-\partial_i(\sqrt{-g}T^{i}_\nu)+\sqrt{-g}T^...
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Connection between algebraic and analytic method of quantum harmonic oscillator

I am studying Quantum harmonic oscillator, There are 2 methods to solve Harmonic oscillator one is algebraic method and another is analytic method , Wave functions derived from 2 methods are ...
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How to take the derivative of potential?

The electric potential $\psi$ at a point inside the volume charge distribution is: $$\displaystyle \psi = \lim\limits_{\delta \to 0} \int_{V'-\delta} \rho'\ \dfrac{1}{|\mathbf{r}-\mathbf{r'}|} dV'.$$ ...
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Why derivatives appear in the form of gradients or divergence?

There are many physics problems whose mathematical equations have the same form. At these problems we always get an equation with a gradient. And the derivatives appear in the form of a gradient or a ...
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Derivative with respect to a coordiante differential (geodesic equation)

If the arc length is chosen to be the action integral, that is $$ S=\int \sqrt {g_{kn}\frac{dx^k}{ds} \frac{dx^n}{ds}} dx \tag{11.13} $$ Then Lagrangian is given by $$L=\sqrt {g_{kn}\frac{dx^k}{ds}...
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Verifying the valdity of interchanging $d/dt$ and $\partial/\partial q^j$ in deriving the Lagrange Equation [duplicate]

Goldstein mentions that we can interchange the derivative operator with respect to time with the derivative operator with respect to qj. I am having trouble figuring out why this is possible. ...
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What is the covariant derivative of a metric tensor $\nabla_{\mu} g^{\mu\nu}$ =?

What is the covariant derivative of a metric tensor, this particular one to be specific $\nabla_{\mu} g^{\mu\nu}$? Notice we've got repetitive indices here. Is it zero and has it got to do anything ...
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Why is the partial derivative a contravariant 4-vector?

The contravariant partial derivative is defined as following: $$\partial ^\mu = \frac{\partial}{\partial x_\mu}$$ where the index $\mu$ runs from 0 to 3. A contravariant vector under Lorentz ...
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Change of variables in gradient

Take two coordinates with $\mathbf r$ and $\mathbf r'$ and take a function $f(|\mathbf r - \mathbf{r'}|)$. In many electromagnetism derivations I see a conversion like this $$ \nabla_r f(|\mathbf r - \...
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Work-Kinetic energy theorem derivation

So I came across this derivation in the book Classical Mechanics by Herbert Goldstein. I don't follow from the second step onwards. I understand that there's a dot product, but how do you compute it? ...
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Landau's approach to collisions in plasma

I can't understand some steps in obtaining the collision term in the Boltzmann equation for plasma. For the first time it was made by L.D. Landau in his article "The kinetic equation in the case of ...
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Tensor Differentiation

In the book "Tensors, Relativity and Cosmology" the author derived Maxwell's Equation in covariant form using the EM field tensor Lagrangian $L=-\frac{1}{4}F^{jl}F_{jl}$ (source=0). One of the steps ...
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Is the derivative with respect to a fermion field Grassmann-odd?

Fermion fields anticommute because they are Grassmann numbers, that is, \begin{equation} \psi \chi = - \chi \psi. \end{equation} I was wondering whether derivatives with respect to Grassmann numbers ...
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General Relativity and cosmology [duplicate]

What is the physical meaning of Ricci scalar is a covariantly constant?
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Differentiating the Free Energy for the Chemical Potential

I wanted to ask a question about the partial differential of the Free Energy equation. I learnt to prove the Free Energy equation: \begin{aligned} \frac{F\left(N_{A}, N_{B}\right)}{k T}=& N_{...