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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