All Questions
Tagged with differentiation notation
224 questions
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What do the subscripts represent in a Euler-Lagrange equation?
What do the subscripts $i$ and $j$ represent in the following Euler-Lagrange equation?
$$
(d/dx_i)(\partial L_d/\partial \psi_j,_i)-\partial L_d/\partial \psi_j =0
$$
- 103
2
votes
3
answers
195
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Notation of Maxwell relations
The Maxwell relations are often given as for example
$$\left(\frac{\partial T}{\partial V}\right)_S = -\left(\frac{\partial P}{\partial S}\right)_V.$$
What does the $S$ and the $V$ in the index of ...
- 123
2
votes
1
answer
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How is semicolon derivative notation defined for multiple derivatives?
I have a covector $\eta_\mu$. Then I have some notation which says $$\eta_{\alpha;\beta\gamma}$$ What does this mean? I understand that given a vector $A^\alpha$, that $$A^\alpha_{;\beta}=\nabla_\beta ...
- 491
2
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2
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241
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Misconception about index notation
I'm going to give an example in General Relativity but this is a question about index notation and coordinate transformations in general. In "Spacetime and Geometry" by Sean Caroll, there is this ...
- 4,737
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2
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123
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Physics of small values and differentials
In some formulas in physics having a ratio, for example $ pressure={F \over\ A}$, the denominator is chosen to be a small quantity ($\Delta A$) and is written like,
$$P= {\Delta F\over \Delta A}.$$
...
- 151
-1
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1
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111
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What is $\delta t$? [duplicate]
I'm confused whether it's difference between two times (i.e final and initial) or it represents very small time.
- 17
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2
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5k
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What does $(\mathbf{u}\cdot\nabla)\mathbf{u}$ mean in the Navier-Stokes equation?
I am studying the Navier-Stokes equations and I have the equation in the form:
$$\rho \dfrac{\partial{\mathbf{u}}}{\partial{t}} + \rho (\mathbf{u}\cdot\nabla)\mathbf{u} - \mu\nabla^2\mathbf{u} + \...
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4
answers
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Hamilton equations from Poisson bracket's formulation
Referring to Wikipedia we have that the equation of motion for a $f(q, p, t)$ comes from the formula
\begin{equation}
\frac{\mathrm{d}}{\mathrm{d}t} f(p, q, t) = \frac{\partial f}{\partial q} \frac{\...
- 315
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votes
2
answers
244
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What is the difference in the two notation? [duplicate]
I have read in Zeemansky's physics
$dQ=dU+pdV$ for first law of thermodynamics
But when I came across another book of thermal physics,it says
$δQ= dU +pdV$.
So what us the difference ?
user176937
5
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1
answer
536
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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
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3
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577
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What is meant by $dy/y$?
Consider the language in the following example:
What is meant by $dg$ and $dR$, and also by $dg/g$? Why does $dR/R=-2/100$ (negative for shrinks)? Is $4\%$ unity change? I mean $dg/g=4\%$ or $dg=...
- 93
4
votes
2
answers
758
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Ordinary vs. partial derivatives of kets and observables in Dirac formalism
I'm a bit confused as to when ordinary and partial derivatives are used in the Dirac formalism.
In the Schrödinger equation, for instance, Griffiths [3.85] uses ordinary derivatives:
$$ i \hbar \...
- 380
1
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1
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What does $ \vert \partial^{\alpha} v_o(x) \vert $ mean in the Navier-Stokes initial velocity condition?
The initial condition $\displaystyle \mathbf{v}_0(x)$ is assumed to be a smooth and divergence-free function such that, for every multi-index $\displaystyle \alpha$ and any $\displaystyle K>0$, ...
- 11
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2
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89
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Derivative of an Expression with respect to One Component of Strain
I recently come across a paper in which the notation of some equation confuses me a lot. Let's say, if I have an expression represented by delta $\delta_{jk},\delta_{jl}$, infinitesimal strain tensor $...
- 137
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1
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905
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On covariant derivative
Let us denote a 1 form on manifold M with $\eta$ which in a chart looks like $\eta=\eta_{\mu}dx^{\mu}$ where $\eta_{\mu}$ are smooth functions on M. Now given the coordinate vector fields $\frac{\...
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2
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2k
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Gradient of a wave function, notational confusion
I'm reading from "Quantum Physics for Dummies", by Steven Holzner. In chapter two, entitled "Entering the Matrix: Welcome to State Vectors", the author introduces the notation for a gradient of a wave ...
- 768
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2
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134
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Generalization of $F=mv\frac{dv}{dx}=\frac{m}{2}\frac{d}{dx}(v^2)$ to 3-dimensions in a compact notation
Starting from $F=ma=m\frac{dv}{dt}$, in 1-dimension, it is easy to show that $$F=mv\frac{dv}{dx}=\frac{m}{2}\frac{d}{dx}(v^2).\tag{1}$$ Can we generalize this formula in 3-dimensions? In 3D, $$\textbf{...
- 12.3k
3
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1
answer
504
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$\delta$ differential notation
Various textbooks that I am currently consulting (including Spacecraft Dynamics and Control An Introduction - Anton H.J. De Ruiter | Christopher J. Damaren | James R. Forbes Section 1.4, page 32) use $...
- 258
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3
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Are indices conventionally raised inside or outside of partial derivatives in general relativity?
If $A_\mu$ is a one-form, then is there a widely accepted convention among physicists about whether the notation $$\partial_\mu A^\mu \tag{1}$$ means "the partial-derivative four-divergence of the ...
- 49.4k
2
votes
1
answer
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Two different meanings of $\nabla$ with subscript?
I am trying to understand the meaning of $\nabla$ when it appears with subscript. I have found two separate Physics SE answers that imply different meanings.
The notation $\vec \nabla_B$ means ...
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1
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Partial Derivative and Dirac Notation [duplicate]
Does the partial derivative of $\langle x'|\alpha\rangle$ with respect to $x'$ equal $|\alpha\rangle$? Why?
Note: $|\alpha\rangle$ is an arbitrary ket, $x'$ is an eigenvalue, and $\langle x'|$ is an ...
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Why do we write $(v\cdot \nabla) v$ instead of $v \cdot (\nabla v)$ for $v_j \frac{\partial}{\partial x_j} v_i$ in the material derivative?
Suppose I have a steady flow and I want to find the rate of change of pressure of a bit of fluid. This depends on the velocity of the fluid and the pressure gradient,
$$\frac{\mathrm{d}P}{\mathrm{d} ...
- 53.7k
0
votes
1
answer
141
views
The integrals of the motion
If I have a given hamiltonian $H$ and some $f$ which claims to be an integral of the motion and I have this identity $$\frac {d}{dt} = \{\;,H\} + \frac {\partial}{\partial t}$$ where $\{\;,H\}$ is ...
user148787
1
vote
1
answer
1k
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Lagrange density for massless scalar field [duplicate]
I am reading a book on QFT which is stating the following.
For a massless scalar field $\phi$ the simplest possible Lagrangian is given by
$$
\mathcal{L}(x) = \frac{1}{2} \partial^\mu\phi \partial_\...
- 1,259
0
votes
1
answer
309
views
What does index $\mu$ in $\partial_{\mu}$ mean? [duplicate]
I am a beginner in QFT, and am reading it from Quantum Field Theory Demystified by David McGowan, a Tata McGraw-Hill publication.
Here, in this book, the author at one point, while explaining ...
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2
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What does $\partial_{\mu}$ mean?
I've stumbled across the following notation a couple times reading physics articles on wikipedia:
$$\partial_{\mu}$$
But what does it mean? They don't clarify.
Source: https://en.wikipedia.org/wiki/...
0
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1
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187
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Density operator as a function of time
Given the density operator $\rho = \sum_iw_i | \alpha^{i} \rangle \langle \alpha^{i}|$, how does the density operator change with time. Apparently I should get $$i \hbar \frac{\partial \rho}{\partial ...
- 1,053
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1
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388
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Scalar Field Theories
The Lagrangian density for a single real scalar field theory is \begin{equation}\mathcal{L}=\frac{1}{2}(\partial_{\mu}\phi)^{2}-V(\phi)\end{equation} I have often seen this written \begin{equation}\...
- 1,229
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3
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Is the shorthand $ \partial_{\mu} $ strictly a partial derivative in field theory?
The Euler-Lagrange equation for particles is given by
$$ \frac{d}{dt}\frac{\partial L}{\partial \dot{q}} = \frac{\partial L}{\partial q},\tag{1}$$
and for fields it is
$$ \partial_{\mu} \frac{\...
- 4,064
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1
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124
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$x'^i_j x^j_k = n\delta^i_k$ rather than $1\delta^i_j$?
These are my calculations
$$x'^i_j x^j_k = \sum_{j=1}^n \frac{\partial x'^i}{\partial x^j}\frac{\partial x^j}{\partial x'^k} = \sum_{j=1}^n \frac{\partial x'^i}{\partial x'^k} =n \delta^i_k\ne \delta^...
- 107
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1
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953
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Derivative with tensor indices
I have trouble figuring out derivatives in tensor notation in SR. I haven't been able to find a simple recipe for writing down a solution. For example what would be the solution to the following ...
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2
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209
views
Very simple index notation question
Trying to understand index notation in the context of spacetime. If I have $x^{\mu}$
and then set $\mu=\phi$
(for example), is it acceptable to then write $x^{\phi}$
or should I just write $\phi$...
- 3,089
-1
votes
1
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104
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Meaning of the notation $\partial_{\log x}$
I am reading this paper, and there is the notation $\partial_{\log x}$ in (6.21) on page 17. What does this notation mean?
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3
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Why is $\frac{d^2}{dx^2}=\left(\frac{d}{dx}\right)^2$ justified in the equation for the square of the momentum operator?
The square of the momentum operator $\hat p$ from the time independent Schrödinger equation is $$\hat p^2=-\hbar^2\frac{d^2}{dx^2}\tag{1}$$ in the one dimensional case.
So if we solve this equation ...
- 2,520
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2
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243
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What is this notation with an un-sandwiched comma in the subscript?
I have a scalar deflection potential (in the study of weak lensing) and in the book (Schneider, Kochanek and Wambsganss's Gravitational Lensing: Strong, Weak and Micro) I have the following passage:
...
- 4,637
7
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4
answers
16k
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Is Del (or Nabla) an operator or a vector?
Is Del (or Nabla, $\nabla$) an operator or a vector ?
\begin{equation*}
\nabla\equiv\frac{\partial}{\partial x}\vec{i}+\frac{\partial}{\partial y}\vec{j}+\frac{\partial}{\partial z}\vec{k}
\end{...
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2
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185
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Tensor index question
I am looking at the solution in the book "Problem book in Relativity and Gravitation" for problem 10.6. I don't think I need to go into the details of the problem (I will do so if need be) because I ...
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2
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678
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Derivative with respect to the spacetime derivative of a field $\phi$
I've encountered the following notation several times (for example, when discussing Noether's Theorem):
$$\frac{\partial L}{\partial(\partial_\mu \phi)}$$
And it's not immediately clear to me what ...
- 2,135
2
votes
3
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814
views
Notation in thermodynamics derivatives
In Yung Kuo Lim's book of exercises in thermodynamics and Stat. Physics I have found more than once the following notation for partial derivatives (ex. 1081 page 79):
$$ \left(\frac{\partial T}{\...
- 175
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1
answer
2k
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Use of infinitesimals in physics [duplicate]
I want to ask about infinitesimals and non-standard analysis. In physics we always use $\mathrm dx,~\mathrm dv,~\mathrm dt$ etc. as infinitesimal quantities. When we deduce equations in physics, when ...
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2
answers
2k
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Variation of Lagrangian density $\mathcal{L}$ w.r.t $x^{\mu}$
If a function $f(x(t),y(t))$ has no explicit dependence on the variable $t$, then $\frac{\partial f}{\partial t}=0$.
In quantum field theory, the Lagrangian density $\mathcal{L}(\phi,\partial_\mu\phi)...
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0
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359
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Different subscripts for $\nabla$ operators while deriving force on system of many particles
Consider a system of 4 particles in an external conservative field. So force acting on each particle is derived from potential energy $U(x,y,z)$of the particle+field system:
Total (external) force on ...
- 1,034
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1
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Calculating motion of equation in tensor form
for the Lagrangian density $$\mathscr{L}=\frac{1}{2}(\partial_{\mu}A^{\mu})^2$$
how can I get this $$\frac{\partial{\mathscr{L}}}{\partial(\partial_{\mu}A_\nu)}=(\partial_\rho A^\rho)\eta^{\mu\nu}$$
...
- 337
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0
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583
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
$$\mathcal{...
- 1,674
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votes
2
answers
1k
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Partial derivatives vs total derivatives in thermodynamics
The specific heat of a system is defined as
$$C_z = T \left( \frac{\partial S}{\partial T} \right)_{z=\text{const}}.\tag{1}$$
Sometimes however, I find the same definition, but with total derivatives ...
- 428
1
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1
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562
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Difference between $dM/dt $ and $\partial M/\partial t$ [duplicate]
$\frac{dM}{dt} = 0$ represents a constant of motion $M.$ Why not $\frac{\partial M}{\partial t}$ represent a constant of motion $M$?
- 71
2
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1
answer
192
views
What does the zero in the differential operator $\partial_0$ mean?
I have noticed the differential operator $\partial_0$ in a lot of equations whilst studying quantum field theory. I am used to the notation $\partial_x$ meaning $ \frac{d}{dx} \\\\ $ etc. but just a ...
- 57
0
votes
0
answers
69
views
Usage of delta operator [duplicate]
So I've always thought that "$\Delta$" when applied to an n-tuple or scalar was the change of that n-tuple or scalar relative to a previous state in time and proportional to the amount of time or $\...
- 217
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1
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186
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What is the difference between $\frac{DA^\mu}{D\lambda}$ and $\frac{DA^\mu}{d\lambda}$?
I earlier asked this question How can you have $\frac{DA^\mu}{d\tau}$? I am now wondering:
What is the difference between $\frac{DA^\mu}{D\lambda}$ and $\frac{DA^\mu}{d\lambda}$?
In the linked ...
0
votes
1
answer
119
views
How can you have $\frac{DA^\mu}{d\tau}$?
If a covariant derivative is given by:
$$D_\nu A^\mu=\partial_\nu A^\mu +\Gamma^\mu_{\nu \lambda} A^{\lambda}$$
Then how does $\frac{DA^\mu}{d\tau}$ make any sense? Since there are no 'differentials' ...