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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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I need an explanation for the time derivative omissions in Landau’s Mechanics: Chapter 1 [closed]

So I have been self-studying Landau and Lifshitz’s Mechanics for a little bit now, and I have been working through the problems, but Problem 3 is giving me some trouble. I solved the Lagrangian ...
Justyn's user avatar
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What happens if we differentiate spacetime with respect to time? [closed]

Essentially, what would differentiating space-time with respect to time provide us with? What are the constraints associated with such operations? Is it possible to obtain a useful physical quantity ...
Kimaya Deshpande's user avatar
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Changing coordinate system [migrated]

Someone please explain how did we get second term in equation 2.15.
Mr. Wayne's user avatar
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What are the operators here and how are these formulas derived? [closed]

In (23), are grad and div some kind of scalar operators comparing to $\nabla$ and $\nabla\times$? because tbh I dont know how $\text{curl}(\mu^{-1}\text{curl}\textbf{A})$ turns into $\text{div}\mu^{-1}...
user900476's user avatar
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What's the difference? $\nabla_\mu e_\nu=\Gamma_{\mu \nu}^\rho e_\rho~\text{ and }~\partial_\mu e_\nu=\Gamma_{\mu \nu}^\rho e_\rho~?$

What's the difference? $$\nabla_\mu e_\nu=\Gamma_{\mu \nu}^\rho e_\rho~\text{ and }~\partial_\mu e_\nu=\Gamma_{\mu \nu}^\rho e_\rho~?$$ In John Dirk Walecka's book 'Introduction to General Relativity',...
Jianbingshao's user avatar
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3 answers
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In equation (3) from lecture 7 in Leonard Susskind’s ‘Classical Mechanics’, should the derivatives be partial?

Here are the equations. ($V$ represents a potential function and $p$ represents momentum.) $$V(q_1,q_2) = V(aq_1 - bq_2)$$ $$\dot{p}_1 = -aV'(aq_1 - bq_2)$$ $$\dot{p}_2 = +bV'(aq_1 - bq_2)$$ Should ...
Bradley Peacock's user avatar
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3 answers
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Does (covariant) divergence-freeness of the stress-energy tensor ${T^{\mu\nu}}_{;\nu}=0$ follow from the Bianchi identity?

I'm working through Chap. $30$ of Dirac's "GTR" where he develops the "comprehensive action principle". He makes a very slick and mathematically elegant argument to show that the ...
Khun Chang's user avatar
2 votes
1 answer
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Total differential of internal energy $U$ in terms of $p$ and $T$ using first law of thermodynamics in Fermi's Thermodynamics

While reading pages 19-20 of Enrico Fermi's classic introductory text on Thermodynamics, I ran into two sources of confusion with his application of the First Law. Fermi introduces a peculiar notation ...
user104761's user avatar
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How do you differentiate $F^{αβ}$ with respect to $g_{μν}$?

I want to experiment with this relation (from Dirac's "General Theory of Relativity"): $$T^{μν} = -\left(2 \frac{∂L}{∂g_{μν}} + g^{μν} L \right)$$ using the electromagnetic Lagrangian $L = -(...
Khun Chang's user avatar
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Transformation to replace a Material derivative with a spatial derivative

In the technical paper referenced below, Gringarten et al. claim that the transient energy transport equation in a planar conduit (Eq. 1 in their paper) $$ \rho c \Bigg[ \frac{\partial T(z,t)}{\...
Sharat V Chandrasekhar's user avatar
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3 answers
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The conservative force [closed]

I read about the definition of the curl. It's the measure of the rotation of the vector field around a specific point I understand this, but I would like to know what does the "curl of the ...
Dirac-04's user avatar
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Covariant (absolute) derivative of a vector along a curve -- compare cartesian vs. polar coordinates [migrated]

BACKGROUND: Suppose $A^μ$ is a vector field and $x^μ(λ)$ is a curve in spacetime. A first guess at measuring the change in $A^μ$ along the curve might be $$\frac{dA^μ(x(λ))}{dλ} = \frac{∂A^μ}{∂x^ν} \...
Khun Chang's user avatar
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Differential form of Lorentz equations

A Lorentz transformation for a boost in the $x$ direction ($S'$ moves in $+x$, $v>0$) is given by: $$ t'=\gamma\left(t-v\frac{x}{c^2}\right),~x'=\gamma(x-vt)$$ In the derivation of the addition of ...
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Commutation in the Local Gauge Transformations

Let's say that I have a Unitary Local Gauge Transformation $U$, in which the Lie Generators are $T$: $$ \partial_{\mu} U = \partial_{\mu} e^{-i T^{a} \alpha_{a}(x)} = U \partial_{\mu} \left( -i T^{a} \...
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Why must a constraint force be normal?

If we impose that a particle follows a holonomic constraint, so that it always remains on a surface defined by some function $f(x_1,x_2,x_3)=0$ with $f:\mathbb{R^3}\rightarrow\mathbb{R}$, we get a ...
16π Cent's user avatar
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Why do I get two different expression for $dV$ by different methods?

So, I was taught that if we have to find the component for a very small change in volume say $dV$ then it is equal to the product of total surface of the object say $s$ and the small thickness say $dr$...
Madly_Maths's user avatar
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A preposterous abuse of notation involving Helmholtz decomposition theorem

Take what I am about to present with a light heart, since the mathmetically inclined may find it too out-of-the-world and devastating. The above diagram (this is drawn by me, but the original is very ...
Jonathan Huang's user avatar
3 votes
1 answer
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A theorem on page 72 in The Large Scale Structure of Space-Time [closed]

In chapter 3 of the book, page 72, a static observer is defined as $V^{a}\equiv f^{-1}K^{a}$, where $K^{a}$ is a timelike Killing vector field and $f^{2}=-K^{a}K_{a}$. Then, Hawking & Ellis claim ...
Rui-Xin Yang's user avatar
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7 answers
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How does the result of derivative become different from average ratio calculation?

Lets give an example. Velocity, $v=ds/dt$. If we know the value of $s$ (displacement) and $t$ (time), we can instantly find the value of $v$. But then this $v$ will be the average velocity. Now ...
Arafat's user avatar
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2 answers
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Problem with resources, Walter Lewin's third lecture

I've watched Walter's third lecture in 8.01 and I have a small problem with the last part, where he says that $$\vec r_t=x_t\cdot \hat x\ +\ y_t\cdot \hat y\ +\ z_t\cdot \hat z \\ \vec v_t=\frac{d\vec ...
0 votes
1 answer
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Covariant Directional Derivative

How is the covariant directional derivative $\frac{D}{d\lambda}=\frac{dx^{\mu}}{d\lambda}\nabla_{\mu}$ in GR related to acceleration? I am motivated to ask this question because I’ve seen it stated ...
ICOR's user avatar
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Laplace-Beltrami operator for a vector field

For a scalar field $\varphi$, the "wave" operator is defined as follows: $$\Box \varphi \equiv g^{ab}\nabla_a\nabla_b~\varphi = \frac{1}{\sqrt{|g|}}\partial_a\left\{\sqrt{|g|}~g^{ab}~\...
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Exact definition of divergence. Is it really the dot product of nabla operator with a vector? [closed]

I was trying to understand the derivation for divergence in cylindrical and spherical coordinate system, and I am a bit confused here. https://www.gradplus.pro/deriving-divergence-in-cylindrical-and-...
DocAi's user avatar
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Writing a single partial derivative as a Jacobian

I was looking here: https://en.wikipedia.org/wiki/Maxwell_relations#Derivation_based_on_Jacobians And am confused at: If I follow the definition of a Jacobian, $\frac{\partial (T,S)}{\partial (V,S)} =...
psychgiraffe's user avatar
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When can I commute the 4-gradient and the "space-time" integral?

Let's say I have the following situation $$I = \dfrac{\partial}{\partial x^{\alpha}}\int e^{k_{\mu}x^{\mu}} \;d^4k$$ Would I be able to commute the integral and the partial derivative? If so, why is ...
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Material to Study the Definition, Algebra, and Use of Infinitesimals in Physics [closed]

This is going to be a rather general question about suggestions on best supplementary material to properly explain the use of infinitesimals (or differentials?) for the purposes of integration or ...
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Differentiation of a product of functions

If I have three (vector)functions, all dependent on different (complex)variables: \begin{equation} a = X^{\mu_1}(z_1, \bar{z}_1), b = X^{\mu_2}(z_2, \bar{z}_2), c= X^{\mu_3}(z_3, \bar{z}_3) \end{...
j_stoney's user avatar
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0 answers
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Partial derivatives of Christoffel symbols to Covariant derivatives

I wanted to express this thing: $g^{ab}\partial_c\Gamma^c_{ab} - g^{ab}\partial_a\Gamma^c_{cb}$, in terms of a covariant derivative. I figured out that if you swap $a$ and $c$ in the $\partial \Gamma$ ...
Stargazer's user avatar
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0 answers
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Taylor expansion of scalar function for a coordinate infinitesimal transformation (Poincaré group)

For a coordinate infinitesimal transformation of the form $x^{\prime \mu} = x^{\mu} + a^{\mu} + \omega^{\mu}_{ \ \nu}x^{\nu}$, we want to derive its effect on a space of scalar functions $f(x)$. This ...
SweetTomato's user avatar
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1 answer
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What does this equation for density mean?

What does this equation for density mean? $$\rho = \lim_{\Delta V\to\varepsilon^3} \ \frac{\Delta m}{\Delta V}$$
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Component notation and matrix notation for gradient of vector

I'm trying to understand vector and tensor notation, but I'm coming across some difficulties. Say I have vector $\vec{u}$ and I compute its gradient $\nabla \vec{u}$. Then I get a tensor $\frac{\...
John Vector's user avatar
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1 answer
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Space-for-time Derivative Substitution in Solving for Elliptical Orbit

I am currently working on a simulation of the Newton's Cannonball thought experiment, in which a stone is launched horizontally from atop a tall mountain at a high speed (in the absence of air) and ...
Oscar Jaroker's user avatar
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5 answers
137 views

Why $VdP$ term omitted in isothermal Work?

Context: I'm asking about classical thermodynamics, that is "ideal gas", closed system, reversible processes etc. Why is the $VdP$ term omitted in calculation of work during isothermal ...
coobit's user avatar
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1 vote
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A covariant derivative computation in General Relativity [duplicate]

I am trying to compute $\nabla^\mu\nabla^\nu R_{\mu\nu}$. I proceed as follows: \begin{align} \nabla^\mu\nabla^\nu R_{\mu\nu}&=g^{\mu\rho}g^{\nu\lambda}\nabla_\rho\nabla_\lambda R_{\mu\nu} \\ &...
vyali's user avatar
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1 answer
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Time derivative of moment of inertia tensor

Suppose that I have some fluid in a fixed volume. Its moment of inertia is given by $I=\int\rho r^2dV$. The derivative of $I$ is given by $\dot{I}=\int\frac{\partial\rho}{\partial t}r^2dV$. Why do we ...
James's user avatar
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2 votes
1 answer
102 views

The Abelian versus the non-Abelian commutator of covariant derivatives in field theory

In the case of Abelian symmetry, the covariant derivative is defined as $D_\mu\equiv \partial_\mu + ieA_\mu$, where $e$ is an arbitrary constant and the vector field, $A_\mu$ is a called a gauge field....
Sirius Black's user avatar
1 vote
1 answer
68 views

Covariant derivative for spin-2 field

I have mostly seen the concept of covariant derivative with regard to spin-1 fields. Is it possible to define the covariant derivative for spin-2 fields as well?
physics_2015's user avatar
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1 answer
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In Lagrangian mechanics, do we need to filter out impossible solutions after solving?

The principle behind Lagrangian mechanics is that the true path is one that makes the action stationary. Of course, there are many absurd paths that are not physically realizable as paths. For ...
Cort Ammon's user avatar
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1 vote
1 answer
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Does the divergence theorem require the covariant derivative to be metric compatible?

I know this is more of a mathematical question, but it arises in the context of general relativity and uses its language so I thought it would be best to ask it here. I understand that the divergence ...
P. C. Spaniel's user avatar
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1 answer
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Question about the derivative of contravariant momentum 4-vector wrt proper time

I'm confused about an expression I saw without further explanation. It is the total derivative of the contravariant momentum 4-vector wrt proper time: $$\frac{dp^{\mu}}{d\tau}=\frac{d}{d\tau}(g^{\mu\...
Il Guercio's user avatar
1 vote
1 answer
106 views

How is this deduced? (Differentiation of tensors)

In Schutz's An Introduction to General Relativity, he talked about how to differentiate tensors. Here is a step that I cannot understand. $$\frac{d\mathbf{T}}{d\tau} = \left( T^{\alpha}_{\beta, \gamma}...
Gene's user avatar
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1 answer
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What is the relation between gauge field and Levi-Civita connection?

In field theory, covariant derivative is something like $$D_{\mu}\phi=(\partial_{\mu}-igA_{\mu})\phi$$ while in differential geometry, covariant derivative is something like $$D_{\mu}V^{\nu}=\partial_{...
Baoquan Feng's user avatar
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1 answer
81 views

Solving divergence and curl equations numerically

I've recently come to learn about Jefimenko's general solution for Maxwell's equations as well as the FDTD method in electromagnetic optics, and that has got me thinking whether I myself can solve ...
Lagrangiano's user avatar
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2 votes
1 answer
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How does the chain rule work in sound wave analysis using fluid mechanics? $\tfrac{d x}{dt}\neq v$?

Context: I am reading Landau & Lifshitz's book on Fluid mechanics. Specifically its section on Sound waves. In section 101, the book's authors discuss about nonlinear traveling waves in one ...
asal's user avatar
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3 votes
1 answer
175 views

What's the physical meaning of Curl of Curl of a Vector Field?

The curl of curl of a vector field is, $$\nabla \times (\nabla \times \mathbf{A}) = \nabla (\nabla \cdot \mathbf{A}) - \nabla^2 \mathbf{A}$$ Now, curl means how much a vector field rotates ...
Plague's user avatar
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0 answers
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Deriving Klein-Gordon equation from Euler-Lagrangian equation: Taking partial derivative inside [duplicate]

Lagrangian for Klein-Gordon equation is given by $$L=\frac{1}{2}\partial_\mu \phi \partial^\mu\phi - m^2\phi^2/2.$$ To derive Klein-Gordon Equation I have to Compute derivative in Euler-Lagrange ...
Vivek's user avatar
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2 answers
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How to calculate the final position of a particle under variable accelaration and its instantenous velocity?

I'm a first-semester physics student who was recently on a train. On a screen, it said the instantaneous velocity of the train was 176 km / h. We had 4 min left until our destination. I wanted to ...
jazzblaster's user avatar
1 vote
2 answers
120 views

Question regarding error analysis of focal length of a lens [duplicate]

The question in whose context i am asking this question is as follows In an experiment for determination of the focal length of a thin convex lens, the distance of the object from the lens is $10 \pm ...
koiboi's user avatar
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2 answers
81 views

Solving a PDE using $x-vt$ as a variable

So I was reading this Landau and Lifshitz paper: https://doi.org/10.1016/B978-0-08-036364-6.50008-9 The article can also be found without a paywall by just searching its title, "On the Theory of ...
Andreas Christophilopoulos's user avatar
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0 answers
52 views

Partial derivative operator

It's mentioned in this paper that if $\partial^i \partial^j$ applied on an equation like: $$ x \delta_{ij} + (\nabla^2 \delta_{ij}- \partial_i \partial_j) y =0 $$ It yields a couple of equations: $$ ...
Dr. phy's user avatar
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