All Questions
Tagged with differentiation notation
224 questions
2
votes
4
answers
129
views
Differentials and small changes in thermodynamics
This may seem like an elementary question, but I'm a bit confused right now about this. From the first and second laws of thermodynamics, and from the definition of enthalpy (per unit mass), we have ...
-3
votes
1
answer
393
views
What is $δx$ used in physics? [duplicate]
I know that:
1) Change in $x$ ie., $Δx$, when $\lim Δx→0$, then $Δx$ is replaced by $dx$.
2) I also know that $∂x$ is used in partial derivative.
Then what is $δx$?
Is $dx$ and $δx$ is just the ...
0
votes
1
answer
859
views
Variations of Kinematic equations
So I recently decided to start learning physics, and have been using various online resources to learn. What I always find are different ways to write the same equation. Now I realize this might be a ...
-1
votes
1
answer
51
views
Assistance interpreting equation
Given a position function of a particle:
$$
\mathbf r=r\,\hat{\mathbf r}\left(\theta\right),
$$
where $\hat{\mathbf r}(θ)$ is the polar unit vector, to find the velocity, we take the derivative which ...
35
votes
2
answers
5k
views
Symbols of derivatives
What is the exact use of the symbols $\partial$, $\delta$ and $\mathrm{d}$ in derivatives in physics? How are they different and when are they used? It would be nice to get that settled once and for ...
0
votes
2
answers
232
views
$\partial$ used for both total and partial derivative
I am currently going through Introduction to Electrodynamics by Griffiths. 4th ed.
In the book p.16 problem 1.14, I noticed an expression like this:
For $f(y,z)$ and $\bar{y}(y,z)$,
$$\frac{\...
1
vote
0
answers
583
views
Partial derivative vs Total derivative
This is essentially a follow up to my question here since I seem to have some difficulties regarding the differences between partial and total derivatives.
Consider a Lagrangian density
$$\mathcal{...
2
votes
1
answer
3k
views
Trying to understand the difference between $\Delta t$ and $dt$ [duplicate]
I'm trying to gain a more conceptual understanding of derivatives and would appreciate your feedback on this.
Say I have a quantity, $x$, at time $t$. Now $x$ moves to a different location $x'$ in ...
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 (...
-1
votes
1
answer
651
views
What is difference between $d\vec{l}$ and $\vec{dl}$? [closed]
What is difference between $d\vec{l}$ and $\vec{dl}$? $d$ means differential as usual.
0
votes
1
answer
111
views
What do the subscripts represent in a Euler-Lagrange equation?
What do the subscripts $i$ and $j$ represent in the following Euler-Lagrange equation?
$$
(d/dx_i)(\partial L_d/\partial \psi_j,_i)-\partial L_d/\partial \psi_j =0
$$
2
votes
1
answer
2k
views
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 $$A^\alpha_{;\beta}=\nabla_\beta ...
2
votes
2
answers
241
views
Misconception about index notation
I'm going to give an example in General Relativity but this is a question about index notation and coordinate transformations in general. In "Spacetime and Geometry" by Sean Caroll, there is this ...
1
vote
2
answers
123
views
Physics of small values and differentials
In some formulas in physics having a ratio, for example $ pressure={F \over\ A}$, the denominator is chosen to be a small quantity ($\Delta A$) and is written like,
$$P= {\Delta F\over \Delta A}.$$
...
-1
votes
1
answer
111
views
What is $\delta t$? [duplicate]
I'm confused whether it's difference between two times (i.e final and initial) or it represents very small time.
1
vote
1
answer
562
views
Difference between $dM/dt $ and $\partial M/\partial t$ [duplicate]
$\frac{dM}{dt} = 0$ represents a constant of motion $M.$ Why not $\frac{\partial M}{\partial t}$ represent a constant of motion $M$?
6
votes
2
answers
5k
views
What does $(\mathbf{u}\cdot\nabla)\mathbf{u}$ mean in the Navier-Stokes equation?
I am studying the Navier-Stokes equations and I have the equation in the form:
$$\rho \dfrac{\partial{\mathbf{u}}}{\partial{t}} + \rho (\mathbf{u}\cdot\nabla)\mathbf{u} - \mu\nabla^2\mathbf{u} + \...
0
votes
2
answers
244
views
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 ?
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 ...
0
votes
3
answers
577
views
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=...
4
votes
2
answers
758
views
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 \...
1
vote
1
answer
155
views
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$, ...
0
votes
2
answers
89
views
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 $...
0
votes
1
answer
905
views
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{\...
0
votes
1
answer
1k
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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 ...
6
votes
2
answers
2k
views
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)...
0
votes
2
answers
2k
views
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 ...
3
votes
2
answers
134
views
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{...
27
votes
3
answers
24k
views
Derivative with respect to a vector is a gradient?
I've encountered in some books (and even completed an exercise from the Goldstein by using it), a strange notation that seems to work exactly like a gradient, I have tried to look for an explanation ...
3
votes
1
answer
504
views
$\delta$ 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 $...
8
votes
3
answers
3k
views
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 ...
2
votes
1
answer
5k
views
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 ...
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 ...
1
vote
1
answer
1k
views
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
$$
\mathcal{L}(x) = \frac{1}{2} \partial^\mu\phi \partial_\...
0
votes
1
answer
309
views
What does index $\mu$ in $\partial_{\mu}$ mean? [duplicate]
I am a beginner in QFT, and am reading it from Quantum Field Theory Demystified by David McGowan, a Tata McGraw-Hill publication.
Here, in this book, the author at one point, while explaining ...
3
votes
2
answers
2k
views
What does $\partial_{\mu}$ mean?
I've stumbled across the following notation a couple times reading physics articles on wikipedia:
$$\partial_{\mu}$$
But what does it mean? They don't clarify.
Source: https://en.wikipedia.org/wiki/...
0
votes
1
answer
187
views
Density operator as a function of time
Given the density operator $\rho = \sum_iw_i | \alpha^{i} \rangle \langle \alpha^{i}|$, how does the density operator change with time. Apparently I should get $$i \hbar \frac{\partial \rho}{\partial ...
-1
votes
1
answer
124
views
$x'^i_j x^j_k = n\delta^i_k$ 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^...
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:
$\...
0
votes
1
answer
953
views
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 ...
7
votes
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{...
0
votes
2
answers
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$...
-1
votes
1
answer
104
views
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?
18
votes
2
answers
16k
views
Do derivatives of operators act on the operator itself or are they "added to the tail" of operators?
How do derivatives of operators work? Do they act on the terms in the derivative or do they just get "added to the tail"? Is there a conceptual way to understand this?
For example: say you had the ...
1
vote
3
answers
301
views
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}$$ in the one dimensional case.
So if we solve this equation ...
2
votes
3
answers
814
views
Notation in thermodynamics derivatives
In Yung Kuo Lim's book of exercises in thermodynamics and Stat. Physics I have found more than once the following notation for partial derivatives (ex. 1081 page 79):
$$ \left(\frac{\partial T}{\...
0
votes
2
answers
678
views
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 ...
1
vote
2
answers
185
views
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 ...
0
votes
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 ...
0
votes
0
answers
359
views
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 ...