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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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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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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Covariant derivative of the vielbein determinant

The vielbein postulate says that $$\nabla_\mu e_v^{\,a}=\partial_{\mu}e_\nu^{\,a}+\omega_{\mu\,\, b}^{\,\,a}\,e^b_\nu-\Gamma^\sigma_{\mu\nu}\,e^{\,a}_\sigma=0.$$ $\nabla$ is the coordinate covariant ...
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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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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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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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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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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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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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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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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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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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Mayard's mechanical differentiator

I've recently been reading an article on a mechanical diferentiator (Nouvelles solutions de calcul grapho-mécanique. Dérivographe et planimètre). The author describes a mechanical device, which, given ...
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Confusion regarding 4-Velocity Derivative Identity (for conservation of energy momentum tensor) in Carroll's Spacetime and Geometry

During Carroll's discussion of the energy-momentum tensor for a perfect fluid (page 36), he writes out that its divergence should be zero. He then expands this as follows: $$\partial_\mu T^{\mu\nu} = \...
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Dirac delta function representations in physics

The most common representation of the Dirac delta function in physics is $$\delta(x)= \frac{1}{2\pi}\int_{-\infty}^{\infty}dk \,e^{ikx}.$$ My question is in which sense is it a faithful representation ...
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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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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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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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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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Thermodynamics and differential forms

In Potter's Thermodynamics: Demystified (page 68), the author wrote: Since only two independent variables are necessary to establish the state of a simple system, the specific internal energy can ...
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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 = ...
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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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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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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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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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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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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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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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1 answer
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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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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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1 answer
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Convective derivative N-S

This is probably an easy answer, but I've not been able to find it yet - Why in some formulations of the N-S equations (for example here https://www.grc.nasa.gov/www/k-12/airplane/nseqs.html), is the $...
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Are covariant derivative on associated bundles exterior covariant derivatives?

The gauge covariant derivative we encounter in gauge theory $D\psi = d\psi + A\wedge \psi$ is a covariant derivative on the associated vector bundle, right? Here $\psi$ is the matter field, $A$ the ...
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Covariant derivative with an upper index in terms of Christoffel symbols

I have encountered expression $$\frac{1}{2}\left(2 \dot{g}_{\mu}{}^{\lambda ; \mu}-\dot{g}_{\mu}{}^{\mu ; \lambda}\right)$$ in a GR paper. Here we assume to be working with the de Sitter metric $g$ ...
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Question regarding adiabatic process (Ex 12.2 in Blundell's Concepts in Thermal Physics)

It can be derived from first law of thermodynamics that $$dQ=\bigg(\frac{\partial U}{\partial T}\bigg)_V dT +\bigg[\bigg(\frac{\partial U}{\partial V}\bigg)_T+p\bigg]dV$$ On page 119 of the book, a ...
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About Poisson's equation and its meaning

So, by working out a physics equation, I ended up with Poisson's equation: $$ \nabla^2A=G. $$ Now the thing is, $A$ is not some sort of potential, but a physical quantity, energy in my case. If the ...
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Covariant derivative in spherical coordinates

Let's say I have a 4-vector $A^{\nu}$ and I take its covariant derivative (I'm using cartesian coordinates), so: $\nabla_{\mu} A^{\nu}=\partial_{\mu}A^{\nu}+\Gamma^{\nu}_{\mu \alpha} A^{\alpha}$ But ...
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Derivative of Function of Unitary matrices

I need some help in understanding derivative of function of matrices, Unitary matrices in my case. I am studying lattice-qcd, there i need to take derivative of Wilson gauge action, $S[U]$ w.r.t link $...
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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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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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Bianchi identity of gauge theory

How to prove Bianchi identity? \begin{align*} \varepsilon^{\mu\nu\rho\sigma}D_{\nu}F_{\rho\sigma}=0 \end{align*} using Jacobi identity; \begin{align*} \epsilon^{\mu\nu\rho\sigma}[D_{\mu},[D_{\rho},D_{\...
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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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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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1 answer
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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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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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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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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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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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