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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Field strength tensor written as commutator of covariant derivatives in QED
I am currently trying to understand the derivation of the relation
$$
\begin{equation}
F_{\mu\nu} = \frac{1}{iq}[D_{\mu}, D_{\nu}]\tag{1}\label{eq1}
\end{equation}
$$
in QED and I have trouble with ...
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Why is $(\partial_\mu F_{\alpha\beta})F^{\alpha\beta}=F_{\alpha\beta}\partial_\mu(F^{\alpha\beta})$?
I'm trying to prove that the divergence of the energy-momentum-tensor is zero by expressing it in terms of the field strength tensor: $\partial_\mu T^{\mu\nu}=0$.
In doing this, letting the derivative ...
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What is the difference between material time derivative and total time derivative of a tensor field? [closed]
I consider material coordintes as $(X_1(t),X_2(t),X_3(t),t_0),$ ($t_0$ arbitrary) and space coordinates as
$(x_1(t),x_2(t),x_3(t),t).$
$\textbf{Remark.}$ I am interested just in dim=3.
We consider a ...
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How to expand $(D_\mu\Phi)^\dagger(D^\mu\Phi)$ in $SU(2)$?
I would like to calculate the following expression:
$$(D_\mu\Phi)^\dagger(D^\mu\Phi)$$ where $$D_\mu\Phi = (\partial_\mu-\frac{ig}{2}\tau^aA_\mu^a)\Phi$$ and $A_\mu^a$ are the components of a real $SU(...
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I need an explanation for the time derivative omissions when solving for the Lagrangian of a system [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 ...
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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 ...
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Changing coordinate system [migrated]
Someone please explain how did we get second term in equation 2.15.
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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',...
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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 ...
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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 ...
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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 ...
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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 = -(...
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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)
$$
\rho c \Bigg[ \frac{\partial T(z,t)}{\...
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The conservative force [closed]
I read about the definition of the curl.
It's the measure of the rotation of the vector field around a specific point
I understand this, but I would like to know what does the "curl of the ...
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Differential form of Lorentz equations
A Lorentz transformation for a boost in the $x$ direction ($S'$ moves in $+x$, $v>0$) is given by:
$$ 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 ...
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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 $dr$...
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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 ...
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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 ...
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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 ...
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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 $$\vec r_t=x_t\cdot \hat x\ +\ y_t\cdot \hat y\ +\ z_t\cdot \hat z \\ \vec v_t=\frac{d\vec ...
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Covariant Directional Derivative
How is the covariant directional derivative $\frac{D}{d\lambda}=\frac{dx^{\mu}}{d\lambda}\nabla_{\mu}$ in GR related to acceleration? I am motivated to ask this question because I��ve seen it stated ...
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Laplace-Beltrami operator for a vector field
For a scalar field $\varphi$, the "wave" operator is defined as follows:
$$\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-...
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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)} =...
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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{...
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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 $\partial \Gamma$ ...
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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 ...
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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{\...
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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 ...
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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 ...
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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} \\
&...
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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 ...
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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....
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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?
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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 ...
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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 ...
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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:
$$\frac{dp^{\mu}}{d\tau}=\frac{d}{d\tau}(g^{\mu\...
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How is this deduced? (Differentiation of tensors)
In Schutz's An Introduction to General Relativity, he talked about how to differentiate tensors. Here is a step that I cannot understand.
$$\frac{d\mathbf{T}}{d\tau} = \left( T^{\alpha}_{\beta, \gamma}...
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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
$$D_{\mu}V^{\nu}=\partial_{...
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Solving divergence and curl equations numerically
I've recently come to learn about Jefimenko's general solution for Maxwell's equations as well as the FDTD method in electromagnetic optics, and that has got me thinking whether I myself can solve ...
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How does the chain rule work in sound wave analysis using fluid mechanics? $\tfrac{d x}{dt}\neq v$?
Context:
I am reading Landau & Lifshitz's book on Fluid mechanics. Specifically its section on Sound waves.
In section 101, the book's authors discuss about nonlinear traveling waves in one ...
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What's the physical meaning of Curl of Curl of a Vector Field?
The curl of curl of a vector field is, $$\nabla \times (\nabla \times \mathbf{A}) = \nabla (\nabla \cdot \mathbf{A}) - \nabla^2 \mathbf{A}$$
Now, curl means how much a vector field rotates ...
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Deriving Klein-Gordon equation from Euler-Lagrangian equation: Taking partial derivative inside [duplicate]
Lagrangian for Klein-Gordon equation is given by
$$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 ...
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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 ...
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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 ...
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