All Questions
Tagged with differentiation differentiation or
261 questions
6
votes
3
answers
1k
views
Is $\dfrac{dx}{dt}$ a fraction or not?
I am new to calculus and during my mathematics class my sir defined $\dfrac{dx}{dt}$ as $$dx/dt=\lim_{t\to t_1}\dfrac{f(t)-f(t_1)}{t-t_1}$$ and my sir made a clear statement that
$\dfrac{dx}{dt}$ ...
- 480
5
votes
2
answers
950
views
Gauge covariant derivative on form
Let $e$ be a one-form gauge field that belongs to the adjoint representation of the gauge group, that is SO(1,2). It is defined as
\begin{equation}
e = e_{\alpha}^{A}T_Adx^{\alpha}.
\end{equation}
...
- 295
5
votes
1
answer
536
views
What is the definition of $\overleftrightarrow{\partial}$ in Dirac Lagrangian?
In my course, the teacher wrote the Dirac Lagrangian as :
$$ \mathcal{L}=\frac{i}{2} \bar{\psi}\gamma^{\mu}\overleftrightarrow{\partial_\mu} \psi -m \bar{\psi} \psi $$
I just would like to ...
- 1,560
5
votes
1
answer
1k
views
Is the inverse of the deformation gradient simply the deformation gradient of the inverse transformation?
If we have a continuum where the initial positions are denoted $X$ and the positions after some deformation are denoted $x$, the deformation gradient is defined:
$$ F = \frac{\partial x}{\partial X} $...
- 103
5
votes
1
answer
2k
views
Understanding notation: Derivative with respect to operator
I am currently trying to understand a set of lecture notes, where the notation is very poorly defined, unfortunately. In a "proof" that canonical quantisation works, the following Hamiltonian (...
- 864
5
votes
4
answers
307
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Newton's Law of Cooling: $\delta Q$ or $\mathrm{d}Q$?
In this popular answer, I invoked Newton's Law of Cooling/Heating:
$$\dot{q}=hA\Delta T\tag{1}$$
$$\dot{q}=\frac{\mathrm{d} Q}{\mathrm{d}t}\tag{2}$$
$$\dot{q}=\frac{\delta Q}{\mathrm{d}t}\tag{3}$$
$$\...
- 35.5k
5
votes
4
answers
5k
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Why are divergence and curl related to dot and cross product?
I've been reading Griffith's intro to electrodynamics and I've been a bit confused about his explanation of divergence and curl. I don't understand how divergence is the dot product of a gradient ...
4
votes
1
answer
702
views
Confused about the differential of a quantity
We know that by definition, the differential of a single variable function $f(x)$ is $$df(x)=\frac{df}{dx}dx$$
analogously, for a multi-variable function $f(x,y,z)$
$$df(x,y,z)=\frac{\partial f}{\...
- 1,047
4
votes
1
answer
3k
views
Differentiation of a vector with respect to a vector
Does differentiation of a vector with respect to a vector make any sense? Even if it makes sense, how does it make any physical meaning? I mean what is the physical interpretation?
- 4,984
4
votes
2
answers
594
views
Hamilton's Formulation and Independent Coordinates
In Lagrange's formulation we know that $q,\dot {q}$ are independent of each other i.e,
$$\frac { \partial q }{ \partial \dot { q } } =0.$$
My question is, is this true for $p$, $q$ in Hamilton's ...
- 588
4
votes
3
answers
792
views
If $\mathrm df$ is an inexact differential, how would the function $f$ look like?
I am studying thermodynamics and in the first chapter the concept of exact and inexact differentials were used to talk about the differences between internal energy, work and heat.
From Blundell and ...
- 1,545
4
votes
3
answers
1k
views
Lie derivative of a vector along itself
The Lie derivative for a covariant and contravariant vector is:
$$\mathcal{L}_U V^\mu=U^\nu\nabla_\nu V^\mu- V^\nu\nabla_\nu U^\mu$$
$$\mathcal{L}_U V_\mu=U^\nu\nabla_\nu V_\mu+ V_\nu\nabla_\mu U^\nu$$...
- 469
4
votes
1
answer
3k
views
Is the differential form of Faraday-Henry equation ( Curl(E)= - dB/dt) always valid?
My textbook suggests that the integral form of the law is evident from experiments, while the differential form can be obtained by considering a closed curve, constant in time, so that it is ...
4
votes
4
answers
5k
views
Why can't impulse be instantaneous?
We know from 2nd law of motion that $$\vec{F} = \frac{d\vec{p}}{dt}.$$
Now, a rate of change can be instantaneous. So, rate of change of momentum is instantaneous. But I doubt how can there be ...
user36790
4
votes
1
answer
18k
views
Index Notation with Del Operators
I'm having trouble with some concepts of Index Notation. (Einstein notation)
If I take the divergence of curl of a vector, $\nabla \cdot (\nabla \times \vec V)$ first I do the parenthesis:
$\...
- 105
4
votes
1
answer
3k
views
What is a covariant derivative in gauge theory?
I've been studying electroweak theory and you need to keep the Lagrangian covariant by introducing covariant derivatives. What is a covariant derivative? And what does it mean to keep the Lagrangian ...
- 609
4
votes
1
answer
236
views
Understanding the use of $d$ and $\partial$ in thermodynamics
It seems a hundred variations of this question have been asked, and it's difficult to find which of those questions relates to exactly what I'm asking. My apologies if exactly this question has ...
- 292
4
votes
1
answer
381
views
Higgs mechanism in QED
I'm trying to understand the Higgs mechanics. For that matter, I'm exploring the possibility of giving mass to the photon in a gauge-invariant way. So, if we introduce a complex scalar field:
$$ \phi=...
- 2,937
4
votes
0
answers
1k
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 ...
- 437
4
votes
1
answer
401
views
Calculating equation of motion in gauge theories: using ordinary derivatives or covariant derivatives?
For general gauge theories, the total Lagrangian density is given as $$L=-\frac{1}{4}F^2+L_M(\psi, D\psi)$$ where $L_M(\psi, D\psi)$ is the matter field with the ordinary derivative replaced by the ...
- 1,706
4
votes
2
answers
3k
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Derivatives in the Lorentz Transformation
I am trying to better understand the Lorentz Transformation on a fundamental level and gain some intuition of it. In the Lorentz Transformation, the derivative of x' with respect to x must be a ...
- 375
4
votes
2
answers
987
views
Inner product between a state and its derivative (quantum mechanics)
I seem to have $\langle\varphi|\frac{d}{dt} \varphi\rangle=0$ for any ket $|\varphi\rangle$, which I doubt very much...
For any quantum state $|\varphi\rangle$, we know it's normalized and therefore $\...
- 78
4
votes
6
answers
856
views
How to understand instantaneous velocity concept [duplicate]
When I started learning instantaneous velocity it didn't make sense to me. I don't understand in real life why we can't measure instantaneous velocity and therefore why we use this concept.
Or is this ...
- 311
4
votes
2
answers
2k
views
Derivatives of Dirac delta function and equation of continuity for a single charge
For a single charge $e$ with position vector $\textbf R$, the charge density $\rho$ and and current density $\textbf{j}$ are given by:
\begin{equation} \rho(\textbf{r},t)= e\,\delta^3(r-\textbf{R}(t))...
- 1,412
3
votes
0
answers
184
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?
$$ (d\textbf{G})_{space} = (d\textbf{G}...
- 158
3
votes
1
answer
1k
views
Can Yang-Mills field strength be defined as covariant derivative squared?
In Yang-Mills theory the field strength tensor $F_{\mu \nu}$ can be calculated as
$$
\begin{equation}
F_{\mu\nu} \equiv \frac{i}{g} [D_\mu,D_\nu] = \partial_\mu A_\nu - \partial_\nu A_\mu -ig[A_\mu,A_\...
- 4,833
3
votes
2
answers
2k
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How would one show that a nonabelian field strength tensor transforms in a certain way under a local gauge transformation?
How would one show that the nonabelian ${F_{\mu\nu}}$ field strength tensor transforms as $${F_{\mu\nu}\to F_{\mu\nu}^{\prime}=UF_{\mu\nu}U^{-1}}$$ under a local gauge transformation? Rather than ...
- 217
3
votes
1
answer
2k
views
The role of the affine connection the geodesic equation
I apologise in advance that my knowledge of differential geometry and GR is very limited. In general relativity the equation of motion for a particle moving only under the influence of gravity is ...
- 155
3
votes
0
answers
378
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 = ...
- 4,574
3
votes
1
answer
557
views
Why $(\frac{\partial S}{\partial T})_P=(\frac{\partial S}{\partial T})_V+(\frac{\partial S}{\partial V})_T(\frac{\partial V}{\partial T})_P$?
In the thermodynamics book (Adkins) I'm using, the following relation is cited without reference but I am not sure where it comes from.
$$\left(\frac{\partial S}{\partial T}\right)_P=\left(\frac{\...
3
votes
1
answer
759
views
Neglecting second order differentials
I am currently doing some Lorentz invariance exercises considering infinitesimal Lorentz transformations, and have been told to neglect second order differentials.
It's not the first time I have come ...
- 6,137
3
votes
1
answer
340
views
Definition of exterior derivative
In Sean Carroll's GR book, pg 84, the exterior derivative $d$ is defined as
$$(dA)_{\mu_1\mu_2...\mu_{p+1}} = (p+1) \partial_{[\mu_1}A_{\mu_2...\mu_{p+1}]}\tag{2.76}$$
where $A$ is a $p$-form and the ...
- 4,276
3
votes
2
answers
639
views
The strange character of operator $\nabla$
I was first introduced to the mathematical operation gradient, divergence and curl not in Mathematics but during my studies of Electromagnetism. As you all know learning Maths from a Physics teacher ...
user240696
3
votes
2
answers
1k
views
Why is 4-velocity not defined as the covariant derivative of position instead of the regular time derivative? [duplicate]
The geodesic equation is usually written as
\begin{equation}
D_\tau u^\mu = 0
\end{equation}
where $D_\tau= u^\mu \nabla_\mu$ is the covariant proper time derivative and $u^\mu=\frac{dx^\mu}{d\tau}$ ...
- 4,737
3
votes
1
answer
458
views
Partial derivatives when the function's inputs are dependent on the same variable
at 53:13 of this lecture by mit ocw, the prof. Moungi Bowendi writes,
$$ (\frac{\partial U}{\partial T})_{p} = (\frac{\partial U}{\partial T})_{v} + (\frac{\partial U}{\partial T})_{T} (\frac{\...
- 8,040
3
votes
1
answer
454
views
Heaviside-Feynman formula derivation
I want to discuss derivation of Feynman-Heaviside formula.
The topic has already been discussed here but I can not put there any question that's why I'm making new post.
Deriving Heaviside-Feynman ...
3
votes
1
answer
742
views
Gauge covariant derivative and Leibniz rule
Let's say I've got 2 different fields $a, b$ and I want to compute its covariant derivative $D_\mu = \partial_\mu + iA_\mu^a T^a$ where $\{A_\mu^a\}$ is the set of gauge fields and $\{T^a\}$ the ...
- 1,607
3
votes
2
answers
2k
views
Physical meaning of harmonic function?
In complex numbers, we define a harmonic function as a twice continuously differentiable function such that the Laplace operator acting on it gives zero. Can anybody explain me the physical ...
- 1,558
2
votes
2
answers
536
views
Exponential of an operator shifted by the derivative operator
Let $p(x)$ and $f(x)$ be sufficiently smooth functions and $D=\frac{d}{dx}$. It is easy to show that $$e^{p(x)D}f(x)=f(e^{p(x)D}x).\tag{1}$$
If $p(x)=a \in \mathbb{R}$ , we have the shift operator as $...
- 213
2
votes
1
answer
105
views
Work-Kinetic energy theorem derivation
So I came across this derivation in the book Classical Mechanics by Herbert Goldstein. I don't follow from the second step onwards. I understand that there's a dot product, but how do you compute it? ...
- 87
2
votes
1
answer
152
views
Confirming an action is invariant under a supersymmetric transformation
I am studying chapter 9 of the book Mirror Symmetry, available here. My question is relating to page 156/157 where Supersymmetry is being introduced for the first time in QFT in 0-dimensions.
We are ...
- 213
2
votes
4
answers
979
views
Question about infinitesimals
In physics for example in electrostatics we consider infinitesimal quantities like $dq$ which means a very small charge which we integrate over the entire body. Now the meaning of $dy$ or $dx$ means a ...
- 317
2
votes
5
answers
348
views
Why does $\vec{r}\cdot\dot{\vec{r}}=r\dot{r}$?
Why is $$\vec{r}\cdot\dot{\vec{r}}=r\dot {r}$$ true? Before saying anything, I have seen the proofs using spherical coordinates for $$\dot{\vec {r}}= \dot{r}\vec{u_r}+r\dot{\theta}\vec{u_\theta}+r\sin\...
- 69
2
votes
1
answer
184
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Definition of time derivative of angular momentum (Wittenburg, "Dynamics of Multibody Systems")
In Wittenburg, "Dynamics of Multibody Systems", 2e, 2008, p.45, (3.33), the definition of time derivative of angular momentum with respect to a reference coordinate system, in its most ...
- 183
2
votes
2
answers
339
views
Different definitions of exterior derivative
In Sean Carroll's GR book, pg 84, the exterior derivative $d$ is defined as
$$(dA)_{\mu_1\mu_2...\mu_{p+1}} = (p+1) \partial_{[\mu_1}A_{\mu_2...\mu_{p+1}]}\tag{2.76}$$
where $A$ is a $p$-form and the ...
- 4,276
2
votes
1
answer
877
views
Hermitian conjugate of 4-derivative $\partial_\mu$
I want to find the hermitian conjugate of 4-derivative $\partial_\mu$ for the real scalar Lagrangian defined as
$$\mathcal{L} = \frac{1}{2} (\partial_\mu \phi)^\dagger(\partial^\mu\phi) - \frac{1}{2}...
- 634
2
votes
1
answer
913
views
Expectation value of time derivative of operator vs. time derivative after operator
Problem 3.18 in Griffiths's Introduction to Quantum Mechanics (3rd ed.) asks to apply the generalised Ehrenfest theorem to operators like the Hamiltonian and momentum operator. The purpose of the ...
- 357
2
votes
2
answers
1k
views
Action of Lie derivative on 1-forms
In Sean Carroll's spacetime and geometry appendix B he derives the action of the Lie derivative on 1-forms. He finds that $\mathcal{L}_X Y^\mu = [X, Y]^\mu$, which I believe is meant as $\mathcal{L}_X(...
- 2,068
2
votes
1
answer
535
views
Expectation value of derivative of operator
I was given the following question:
Let $A(\lambda)$ be a Hermitian operator, which is dependent on some real parameter $\lambda$. Let us denote the eigenvalues and corresponding eigenstates of $A$ ...
- 353
2
votes
0
answers
86
views
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 ...
- 3,159