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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What is the analogy between the gauge covariant derivative and the covariant derivative in General Relativity?

A particle in the Dirac field can be described with the following equation $$i\gamma^\mu\partial_\mu\psi-m\psi=0$$ This is if the particle is non-interacting. However, if we impose a local $U(1)$ ...
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121 views

Defining the covariant derivative on bitensors

Bitensors (tensors defined on two different points) are an extension of tensors found in some applications of general relativity, where objects such as the world function, parallel transport operator, ...
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1answer
833 views

Partial derivatives vs total derivatives in thermodynamics

The specific heat of a system is defined as $$C_z = T \left( \frac{\partial S}{\partial T} \right)_{z=\text{const}}.\tag{1}$$ Sometimes however, I find the same definition, but with total derivatives ...
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277 views

Schwarzian derivative from conformal factor

Suppose I have a 2D Lorentzian conformally flat metric $$ ds^2 = -\Omega(u, v) du dv.$$ I consider a conformal field theory whose stress-energy tensor $T_{ab}$ is known on the flat metric $$ds^2 = -...
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298 views

Laplacian and Dirac Delta function

Although I find it mathematically dubious, we said that $$\Delta \frac{1}{r} ~=~ -4\pi \delta^3({\bf r}).$$ Now, I was wondering is there a similar relation to the delta function if we look at $$\...
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83 views

Is there a sense in which one can interpret $\delta'(0) = 0$?

I wasn't sure whether to post this under physics or math (and landed on physics due to fear of being crucified for lack of rigor on math.stackexchange). In field theory, when we encounter divergences ...
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47 views

Lattice differentiation and Locality

Assume we define the locality of a theory in the following way: Assume we have a theory of real scalars, so this theory is non local if the action has terms like $$\int d^dx\,\phi(x)V(x-y)\phi(y).$$ ...
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144 views

Time derivative in rotating frame

In Goldstein (2ed) sec 4.9 - Rate of change of a vector, why does he say that the instantaneous angular velocity $\omega$ is not a derivative of any vector? $$ (d\textbf{G})_{space} = (d\textbf{G}...
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284 views

Connection between parallel transport and $SO(n)$ of vectors

I learned a few months ago that parallel transport or covariant derivative of vector along a close loop on Riemannian manifold just cause "rotation" of vector about some angle,but doesn't change it's ...
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270 views

Curl operator in Schwarzschild metric

I'm trying to write down the curl operator explicitly for a Schwarzschild metric in cylindrical coordinates. I am trying to use the general expression of the curl operator in orthogonal curvilinear ...
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745 views

How is Infinitesimal coordinate transformation related to Lie derivatives?

I am reading the book "Gravitaion and Cosmology" by S. Weinberg. In section 10.9, while discussing Lie derivatives of tensors of different ranks, he makes a general comment: The effect of an ...
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65 views

Is my geometric interpretation of $T \left(\frac{\partial S}{\partial T}\right)_P = \left(\frac{\delta Q}{dT}\right)_P$ correct?

I originally started writing this as just a question, but in the process of writing it I may have solved it myself. Still, I would very much appreciate if someone more knowledgeable than myself took a ...
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Proving that a map is differentiable

Let $H$ be a self-adjoint operator on $\mathcal{H}$, $\psi\in D(H)$ and $\beta\geq 1/2$. How can I see that $$ L \colon \mathbb{R}\to \mathcal{L}(D(\mathcal{N^\beta}),D(\mathcal{N^{\beta-1/2}})), \...
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How can I derive the covariant derivative of the Ricci tensor using the Ricci scalar?

\begin{align*} R&=g^{\mu\nu}R_{\mu\nu}\\ \Rightarrow \nabla^{\gamma} R&=\nabla^{\gamma}(g^{\mu\nu}R_{\mu\nu})\\ &=\nabla^{\gamma}(g^{\mu\nu})R_{\mu\nu}+g^{\mu\nu}\nabla ^{\gamma}(R_{\mu\nu}...
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1answer
212 views

Derivatives in the Lorentz Transformation

I am trying to better understand the Lorentz Transformation on a fundamental level and gain some intuition of it. In the Lorentz Transformation, the derivative of x' with respect to x must be a ...
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181 views

Explicit expression of gradient, laplacian, divergence and curl using covariant derivatives

For my course in General Relativity I am given the problem to find the expressions for the gradient, laplacian, divergence and curl in spherical coordinates using covariant derivatives. I have tried ...
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80 views

Why does invariance commute with partial derivative?

This question applies more generally to actions, but I am going to ask it for a specific example. I am getting confused when considering the invariance of the superstring action under Weyl ...
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1answer
133 views

Prove $D_\mu\phi^\dagger=(D_\mu\phi)^\dagger$

In gauge transformation, $D_\mu$ was defined to be $\partial_\mu-igA_\mu$. However, I have hard time to see that $D_\mu\phi^\dagger=(D_\mu\phi)^\dagger$ without ambiguity. (A comparable example in QED ...
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191 views

Uncertainty calculation - when to use absolute value bars?

I'm asking this because I saw (at least) two questions here on this Stack that seemed very similar and caused the same confusion to me in reading the answers to both. Suppose we have a formula: $A = ...
2
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1answer
446 views

Commutation relations for inverse d'Alembertian operator

Is there a commutation relation for the inverse d'Alembertian operator in general relativity? i.e. if we define $\Box = g^{\mu\nu}\nabla_\mu\nabla_\nu$ and $\Box \Box^{-1}X_{\alpha_1,\alpha_2...}=X_{\...
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227 views

Supercovariant Derivative action

My query is with Weinberg Vol3 equation just above 26.7.22 Weinberg follows Majorana Superfield formalism. Where, covariant derivative is defined as, $$D_{R\alpha}=-\epsilon_{\alpha \beta}\frac{\...
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182 views

Commutativity Variation and derivative

When in general is true that $\nabla_{\mu}\delta=\delta\nabla_{\mu}$ where $\nabla_{\mu}$ is covariant derivative? I was thinking this when one has to find the equations of motion for example in ...
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144 views

Dependence of scattering amplitudes on Mandelstam variables

It is well-known that scattering amplitudes in QFT are tensors, hence e.g. scalar amplitudes /written in momentum space/ depend only on the Mandelstam variables of the external momenta, involved in ...
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169 views

Problems while doing $\dfrac{\partial}{\partial(\partial_\mu \phi)}$ and $\dfrac{\partial}{\partial(\partial_\mu A_\mu)}$

In David Tong's lectures, he gives two Lagrangians as examples to derive the equations of motion: $$\mathcal{L} = \dfrac{1}{2}\eta^{\mu\nu}\,\partial_\mu\phi\,\partial_\nu \phi-\dfrac{1}{2}m^2\phi^2, ...
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363 views

Scale-invariant differential operator

For example, the differential operator Laplacian is $$\nabla^2 = \frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}.$$ My questions are: Is it scale-invariant? what is scale-...
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40 views

Calculating the variation of an operator in two different ways

Let $$ H_{T}=\dot{x}^{I}\frac{\partial}{\partial \psi^{I}}T(x,\psi) $$ and consider the transformation: $$ x^{I}\mapsto x^{I}+i\epsilon\psi^{I} \\ \psi^{I}\mapsto\psi^{I}-2\epsilon\dot{x}^{I} $$ where ...
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42 views

Lie derivatives and the tetrad formalism

I have been trying to learn about the tetrad formalism in general relativity and I understand the basic idea, but there is one issue that I can't seem to figure out: Is there a definition of a Lie ...
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24 views

Looking for differential operators satisfying a specific commutation relation with the Laplace operator

Consider the Laplace operator on some manifold, $\Delta=(\det g)^{-1/2} \frac{\partial}{\partial x^j} \left( (\det g)^{1/2}g^{jk} \frac{\partial}{\partial x^k}\right) $. I am looking for differential ...
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1answer
33 views

Expanding the product of two gauge covariant derivatives acting on Lie-algebra valued scalar fields

I would like to expand the term $$ \operatorname{Tr}(D_\mu \phi D^\mu \phi), $$ where $D_\mu=\partial_\mu+igA^aT^a$, and $\phi$ transforms in the adjoint of the gauge group. I expanded the above as $$ ...
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Box size $R$ as a thermodynamic variable vs position radius $r$ inside the box

Titles are difficult but I hope I can do a better job in the text. I am working with a spherical box where I want to have the box's radius $R$ and temperature at the wall $T$ to be the main ...
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1answer
78 views

Why do we ignore higher-order derivatives of entropy with energy in deriving the Boltzmann distribution?

I am taking my first course in statistical mechanics, one point that I don't really get is the justification for ignoring higher-order derivatives of entropy w.r.t energy. We began the course by ...
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54 views

Affine Connection

On page 74 of Weinberg's General Relativity textbook he writes the following: Equation 3.2.4: $$\Gamma_{\mu \nu}^{\lambda} \equiv \frac{\partial x^\lambda}{\partial \xi^\alpha}\frac{\partial^2 \xi^\...
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1answer
42 views

Is double derivative of a tensor component with respect to time itself a tensor?

Can you please clear my doubt that if we take double derivative of a tensor component with respect to time, will the resulting quantity be a tensor or not?
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2answers
113 views

Invariance of differential operators

How do we prove that del operator is invariant under any kind of change of coordinates, specifically under galilean transformations? I am getting an extra term containing the relative velocity of two ...
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1answer
67 views

What is the difference between zero and an infinitesimal number?

In a standard Atwood machine physics problem, the string going over the pulley is considered massless. So does that imply mass = 0 or mass = dm? General question: what is the difference between 0 and ...
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43 views

Graph of $dQ/dt$ discontinuous

Charge is quantised then why how do we define $dQ/dt$ as current when graph of $Q$ v/s $x$ will be discontinuous and hence non-differentiable. Is it an approximation we use?
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Question about proportionally rules

I don't think context is needed but to make sure: I'm doing a homework exercise on binary system. P is the orbital period and E the energy of the system. The following is in the solution when trying ...
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101 views

Lie derivative of the non-coordinate metric being 0

I'm trying to answer a question about a the Lie derivative of a metric in a non-coordinate basis. Here, the $C^{a}_{bc}$ are from the Lie derivative of one basis vector with respect to another, or ...
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26 views

Divergence of a vector which has explicit and implicit position dependence

I am doing EMT and I am trying to calculate the divergence of this current density given as, $$\vec{J}(\vec{r}', t_r) = \vec{J}(\vec{r}', t - \frac{|\vec{r}-\vec{r}'|}{c})$$ for $\vec{r} = (x,y,z)$ ...
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What the correct way to write the discrete kinetic energy operator?

For a bit of context, I am making simulations of a quantum algorithm that is meant to variationally find the ground state of a quantum harmonic oscillator potential. In one dimension, we know that $\...
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58 views

Differentiating the four-velocity contracted with itself

Let us denote $u^\mu$ as the contravarient component of a four velocity at a point in some coordinate system for a pseudo-Riemannian manifold. I want to examine the following equation. $$\partial_\nu(...
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34 views

Guoy method derivation, or, how not to break math?

I start from $d\vec{F}=\nabla(\vec{m}\vec{B})$, $\vec{M}=\frac{d\vec{m_i}}{dV}$, and $\vec{M}=\frac{\chi_v}{\mu_o}\vec{B}$. I write out the definition of the force, determine we're only interested ...
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62 views

From Maxwell equations to wave equation: what is the meaning of the differentiation?

Starting from the Maxwell's equation in empty space and without charges and currents, we can write the two following equations: $\bar \nabla \times \bar E= -{{\partial \bar B}\over{\partial t}}$ $\...
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2answers
62 views

Product rule of variations

I am deriving the Einstein equation using the Einstein-Hilbert action: It is obvious that the variation in the Riemann Tensor is calculated from a variational product rule. What is not obvious to ...
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30 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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62 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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320 views

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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68 views

Variation under infinitesimal reparametrization

Say that under the infinitesimal reparametrization $\sigma \rightarrow \sigma' = \sigma-\xi(\sigma)$, $x^\mu$ transforms as a scalar, i.e. $x'^\mu(\sigma')=x^\mu(\sigma)$. I would like to show the ...
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2answers
70 views

Calculate heat production rates in well-mixed batch fermenter

I have a formula to calculate the heat production rates in well-mixed batch fermenter: $$V * c_p * \rho * \frac{dT_j}{dt} = F * c_p * \rho * (T_0 - T_j) + U * A * (T_R - T_j) + r_Q * V$$ U [kJ/($m^2$ ...
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64 views

Does the proper four-acceleration $A^{\mu} = (0,0)?$

Let the proper four-position vector $x^{\mu}(\tau) = (0, \tau)$. Differentiating this successively wrt $\tau$ I get the four-velocity $u^{\mu}(\tau) = (0, 1)$ and then the four-acceleration $A^{\mu}(\...