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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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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68 views

Differentiation of the determinant $g$

Let $g$ be the determinant of the metric tensor. I want to derive the following equation $g_{,\nu}=gg^{\lambda \mu}g_{\lambda \mu,\nu}$. It is said that $gg^{\lambda \mu}$ is a cofactor, but I can't ...
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240 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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174 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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603 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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100 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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1answer
2k views

Physical significance of divergence

In my textbook They considered a parallelopiped $ABCDEFGH$ with sides $dx,dy,dz$ parallel to $x,y,z$ axis respectively $\vec V$ represents the vector velocity of the fluid at the centre $P$ of f ...
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116 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 = ...
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1answer
194 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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163 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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157 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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125 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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163 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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325 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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1answer
571 views

Partial derivatives vs total derivatives in thermodynamics

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

H.Risken's book "The Fokker-Planck Equation" contains the following formula for the general 1D Fokker-Planck equation: $\frac{\partial W}{\partial t}=\left[-\frac{\partial}{\partial x}D^{(1)}(x)+\...
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1answer
41 views

Commutator of covariant derivatives to get the curvature/field strength

For notation and convention, please see Gauge theory formalism and Generalizing the covariant derivate for gauge theory. The covariant derivative can be used to construct curvatures (called field ...
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25 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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56 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
37 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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0answers
31 views

Differentiation of metric tensor in new coordinate

I want to understand the explicit meaning of $g_{\mu'\nu',\lambda}=0$ where unprimed coordinates are coordinates of the the original coordinate systems and primed ones are for new coordinate system. ...
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28 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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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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59 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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“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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1answer
94 views

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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1answer
77 views

Derivative of tensor product of quantum states

Recently I asked a question over at the math stack exchange: https://math.stackexchange.com/q/3210375/. However I figured I'd ask here too, seeing as the question originated in a physics course I'm ...
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50 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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3answers
90 views

Divergence, gradient and differentiation - radial irrotational fluid flow

Given a fluid with the steady spherically symmetric flow with only radial velocity $\vec v(r)$. We need to evaluate $ \vec v \cdot \nabla \vec v $. From vector calculus $$ \vec v \cdot \nabla( \vec ...
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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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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}(\...
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174 views

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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1answer
62 views

Double divergence of stress tensor for migration flux

I am looking to calculate migration as a function of time using equation in Image 1. SigmaP is the total particle stress tensor in the cylindrical coordinates (r, theta, z). I am only interested in ...
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1answer
45 views

Where is the error in this calculation of net curl for simple magnetic field?

I wasn't sure whether to post this on MSE, but PSE seems more appropriate. Let B be a static magnetic field in spherical coordinates, defined as $B=r\hat{\theta}$. Then, it's curl is $$\nabla \times ...
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51 views

Have fractional order differential models been explored as an alternative to standard gravitational field theory?

Since Einstein introduced his field equations and general theory of relativity, experimental evidence, at least on the cosmic scale has repeatedly supported the theory. Nevertheless, many seeking to ...
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1answer
71 views

Planck Blackbody Radiation: Is this an error in the textbook?

the textbook I am reading describes two forms of equations of Blackbody Radiation. $$d\rho(\nu, T) = d\rho_\nu(T)d\nu = \frac{8\pi h}{c^3}\ \frac{\nu^3d\nu}{e^{h\nu/k_BT}-1}\ . $$ Substituting $ c = \...
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1answer
97 views

What does lowercase-delta mean in Noether's first theorem?

Most expressions of Noether's Theorem I have come across do not use lowercase delta, but a couple sites do. I am confused....... Check out page 21 of the June 23 issue of 'Science News' ...
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86 views

Question about Covariant Derivatives in General Relativity

I'm following the differential approach of Schutz' book where vectores are geometrical objects written as \begin{equation} \vec{V}=V^a\ \vec{E}_a \end{equation} Where $V^a$ are the components of the ...
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41 views

Functional derivative of a symmetrized field

I'm confused whether a symmetrisation/antisymmetrization of a function with respect to its arguments, i.e., $$F(x_1,x_2,...,x_n)=\frac{1}{\sqrt{N!}}\sum_{\pi}\textrm{sgn}(\pi)~f(x_{\pi(1)},...,x_{\pi(...
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1answer
329 views

Partial Derivatives Relation on Thermodynamics and minus sign

My problem is to show that $$ \left( \frac{\partial C_{V}}{\partial V} \right)_{T} = -T \left[ \frac{\partial (\alpha/\kappa_{T})}{\partial T} \right]_{V}$$ where $C_{V}$ is molar specific ...
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1answer
61 views

Determining the change in radius of water flowing from a faucet - more general question on differentiation

I've been outside of the academic world for several years now, and I'm forcing myself to go back through old textbooks and resources and work through the information in there. I can tell I'm losing ...
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1answer
630 views

When can one omit a total time derivative in the Lagrangian formulation?

I am studying Lagrangian and Hamiltonian mechanics and i am using Landau & Lifshitz and Goldstein books. Both of them state that a modified lagrangian $$L'=L+\frac{df}{dt}$$ gives the same ...
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Does Wick's theorem still work for derivative fields

I am wondering if Wick's theorem still is useful for something like $$\langle0|T\ \partial\psi(x)\partial\psi(y)...\partial\psi(w)|0\rangle$$ can I say this things equals to all possible ...
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1answer
87 views

Mixing total and partial derivatives

I'm looking at a problem in kinetic theory, where the distribution function depends on two variables: time $t$ and energy $U$. $dU/dt \ne 0$, but there is a function $J(U, t)$ with $dJ/dt = 0$. While ...
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1answer
68 views

The dimensional analysis of the GR geodesic equation

The geodesic equation parametrized by the proper time contains two terms: $$ {d^{2}x^{\mu } \over ds^{2}}=-\Gamma ^{\mu }{}_{{\alpha \beta }}{dx^{\alpha } \over ds}{dx^{\beta } \over ds}\ $$ The ...
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59 views

Change/derivative of curvature in discrete time

I would like to calculate the change or derivative of curvature $\kappa$ of a moving object in discrete time. The curvature can be calculated according to: $$ \kappa = \frac{1}{R} = \frac{\dot{\theta}...
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245 views

Why does the material derivative and transport theorem look different?

Reynolds transport theorem says that $ \frac{d\int\phi}{dt}=\int\left(\frac{\partial\phi}{\partial t} + \nabla\cdot(\phi\otimes v) \right) $ Why is the material derivative not defined as what's ...
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1answer
94 views

Vector Derivative: General Case

From "An Introduction to Mechanics" by Kleppner & Kolenkow, SIE-2007, Chapter 1 (Vectors and Kinematics), Section 1.8 - "More about the derivative of a vector". In this section, towards the end, ...
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334 views

Partial derivative vs Total derivative

This is essentially a follow up to my question here since I seem to have some difficulties regarding the differences between partial and total derivatives. Consider a Lagrangian density $$\mathcal{...
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74 views

Why is this differential added instead of subtracted?

I was looking at a derivation of the Barometric formula which reads like this: Consider a flat disc of air of mass $\mathrm{d}m$ at distance $h$ above the ground of mass $\mathrm{d}m$ and thickness ...