All Questions
Tagged with differentiation notation
224 questions
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Deriving the Covariant Derivative of the Metric Tensor
First off, I did look through some other questions:
Covariant Derivative of Metric Tensor
Why is the covariant derivative of the metric tensor zero?
https://math.stackexchange.com/q/2174588/
But they ...
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2
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Index notation and total differential
I understand that the gradient $\partial_i$ is covariant.
Let f be a function of 3 variables
So I can write the total differential as
$$
df=\partial_1fdx^1+\partial_2fdx^2+\partial_3fdx^3 = \...
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3
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Showing the equivalence between the chain rule's Leibniz and Lagrange Notations
This may seem more math related but this question crossed my mind as I was reading the derivation of the Euler-Lagrange Equation.
In math, we were introduced to the Lagrange notation of the derivative ...
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Replacing infinitesimals with full vectors in a differential relation. Is it legit?
I'm reading Leonard Susskind's "Special Relativity and Classical Field Theory". On pg. 138 he generalizes a differential relation by replacing infinitesimals with full vectors like so:
Is this ...
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Can value of the variable be substituted in partial derivatives before taking the derivative?
I was going through the D'Alembert's solution for the wave equation using this pdf from University of British Columbia (UBC, Canada). Here's the link: https://www.math.ubc.ca/~ward/teaching/m316/...
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What does $\frac{Dv}{Dt}$ and $\frac{D\rho}{Dt}$ notion mean here?
Momentum conservation:
$$\rho\frac{Dv}{Dt}=\nabla\cdot\sigma+\rho g$$
Mass conservation:
$$\frac{D\rho}{Dt}+\rho\nabla\cdot v=0$$
What does $\frac{Dv}{Dt}$ and $\frac{D\rho}{Dt}$ notion mean here ...
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Expectation value of time derivative of operator vs. time derivative after operator
Problem 3.18 in Griffiths's Introduction to Quantum Mechanics (3rd ed.) asks to apply the generalised Ehrenfest theorem to operators like the Hamiltonian and momentum operator. The purpose of the ...
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What are Connection Forms in General Relativity?
I'm trying to follow an article by H. Ellis (1973), where he developed the first ever metric of a traversable Wormhole (more info here).
In pages 105-106 (the end of the 3rd page in the linked file ...
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Changing derivative to difference quotient
Can differential be changed to Delta or difference? In high school education, in the acceleration section of Newton's formula 2, acceleration is a change velocity (velocity difference) divided by a ...
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Textbook proof error? Runge Lenz
I was reading this proof in my textbook. They say that $$\vec{r} \cdot \dot{\vec{r}} = |\vec{r}||\dot{\vec{r}}|.$$ Doesn't that mean $\vec{r}$ is parallel to $\dot{\vec{r}}$, and if so, then the line ...
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4-velocity lowering index question
The 4-velocity in contravariant form is given by
$$V^\mu=\frac{dx^\mu}{d\tau}$$
for some general co-ordinates $x^\mu$ and proper time $\tau$.
Is the 4-velocity in covariant form given by
$$V_\nu=V^\...
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Why do they specify the differential of Work using an lowercase delta $\delta W$, instead of $dW$ [duplicate]
I was curious, why do they specify the differential of Work using an lowercase delta symbol $\delta$ as in "$\delta W$", instead of using a $d$, as in $dW$. For example:
$$\delta W=\vec{F} ...
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Meaning of the $\delta$ notation [duplicate]
Two examples I've seen of this so far are in statistical mechanics, when looking at the work done on a system:
$$dW=-\frac{1}{\beta}\frac{\partial \ln{Z}}{\partial V}\delta V$$
and in the Noether ...
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What does it mean "differentiation with respect to the coordinates of particle 1 or 2"?
I was reading Introduction to Quantum Mechanics by Griffiths. In Chapter 5, Identical Particles, I came across the notation $\nabla_1$ and $\nabla_2$. Griffiths writes that it means "differentiation ...
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Covector basis derivation
On page 65 of Schutz's A first course in General Relativity, he introduces the notation $\phi_{,\alpha}=\partial\phi/\partial x^\alpha$. He then says that $x^\alpha_{\ \ ,\beta}=\delta^\alpha _{\ \ \ \...
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If $\mathrm df$ is an inexact differential, how would the function $f$ look like?
I am studying thermodynamics and in the first chapter the concept of exact and inexact differentials were used to talk about the differences between internal energy, work and heat.
From Blundell and ...
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Did R. Feynman know about the different notations for exact and inexact differentials? [closed]
I remember reading a long time ago, the story of a student taking R. Feynman for responsible of her (I think it was a woman, not sure though) fail at an exam of physics because what was written in her ...
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Expectation of partial time derivatives of $x$ in QM
In Ehrenfest theorem we know that
$$m\frac{d\left< x\right>}{dt}=\left< p\right>+m\left<\frac{\partial x}{\partial t}\right>.$$
So how can I exactly calculate a specific $\left<\...
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Commutation of position four-vector with spacetime derivatives
I am trying to understand a simple demonstration in Ashok Das' Lectures in QFT. He does the following on p. 134
$$[P_\mu,M_{\nu\lambda}]=[\partial_\mu ,x_\nu\partial_\lambda-x_\lambda\partial_\nu]=\...
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Is it reasonable and common to interpret $dt$ as a time point (a point in time)? [duplicate]
I heard some one talked about the instantaneous and average velocities.
He was using $\Delta t$ to denote a time frame, $dt$ denote a time point.
average velocities $\bar{v} = \dfrac{\Delta s}{\...
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What does $\Delta$ stand for? [duplicate]
Newton’s first law states that $\Delta v=0$ unless acted on by an external force, $F_{\mathrm{net}}\neq0$.
Can someone explain to me what the $\Delta v$ symbol means?
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$\delta Q = dU + \delta W$. Why is it $dU$ while others are partial differentials? [duplicate]
It is the first law of thermodynamics for a very small change in the state of the system.
It is in Heat thermodynamics and statistical physics by Brij Lal, Dr. N. Subrahmanyam, and P.S. Hemne.
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Confusing Total Derivative and Partial Derivative in Classical Field Theory - Noether Theorem
I'm really confused about total derivatives and partial derivatives.
My multivariable calculus book (Guidorizzi vol 2 Um Curso de Calculo) says that if I have a function like $f(a(u,v),b(u,v))$ then ...
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Index (Einstein summation) notation question
Question #1:
Lost as to how the second equality in the following equation holds —
$$\frac{\partial}{\partial \tau} (A^2) = \frac{\partial}{\partial \tau} (\eta_{\mu\nu}A^\mu A^\nu) = 2\eta_{\mu\nu}...
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Question about commutators acting on wavefunctions
Consider a commutator acting on a 1D wavefunction:
$$[\frac{\hbar}{i} \frac{d}{dx},x]\psi(x)=(\frac{\hbar}{i} \frac{d}{dx}x-x\frac{\hbar}{i} \frac{d}{dx})\psi(x).$$
Now does this mean
$\frac{\hbar}{...
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Vector calculus notation, maybe?
I just got a new book on turbomachinery that uses some notation I'm not familiar with.
$$ \nabla \lor \vec{W} = -2\vec{\Omega} $$
The del-(something)-vector format makes me think its vector calculus....
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Convective derivative vs total derivative
I was wondering what is the difference between the convective/material derivative and the total derivative. We were introduced to the notion of material derivative
$$ \frac{D\vec{u}}{Dt}=\frac{\...
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Notation for the divergence of a rank 2 tensor
I am studying advanced fluid mechanics and sometimes you see equations written in index notation like
$$ Dv_i= \partial_t v_i +v_j\partial_jv_i$$
but sometimes you find this arrow/vector notation (...
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What is the meaning of $d$? [duplicate]
What is the meaning of $d$? Is is Delta? If it is Delta, why is it then not $\Delta$? I am still confused with that. Can someone help explain it to me?
user224451
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Tensor Derivatives in Index Notation in Special Relativity
The energy-momentum tensor $T^{\mu\nu}$ is not uniquely defined because we can add a term $\partial_{\lambda}X^{\lambda\mu\nu}$ to it, where $X^{\lambda\mu\nu} = - X^{\mu\lambda\nu}$, and show that it ...
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Reason why dot notation isn't used for time derivatives in Maxwell's equations [closed]
Maxwell's equations seem to be usually written:
\begin{align}
\nabla \cdot \mathbf{E} &= \rho/\epsilon_0,\\
\nabla \cdot \mathbf{B} &= 0,\\
\nabla \times \mathbf{E} &= -\frac{\partial \...
user113773
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Lagrange equations in a conservative system, understanding $\nabla_i$
For a system of multiple particles with conservative forces: $\mathbf{F}_i = - \nabla_i V$, with $V \equiv V(\mathbf{r}_1,\dots,\mathbf{r}_N)$ the potential in function of the position of the $N$ ...
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Gauge-covariance of the Yang-Mills field strength $F_{\mu\nu}^a$
Accordingly to Yang-Mills theories, after the introduction of a covariant derivative such that
$$D_\mu = \partial_\mu - igA_\mu, \tag1$$
you can built the kinetic term for the gauge potential $A_\...
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Notation and concepts of Yang Mills Theory
I am studying loop quantum gravity using the book by Pullin and Gambini. I am having some trouble understanding and getting past the chapter on Yang Mills theory, mainly because I am confused about ...
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What does $\delta$ represents in FLUCTUATION-DISSIPATION THEOREM?
i am trying to follow the following tutorial. I keep seeing $\delta$ over functions such as $\delta F(x)=F(x)-\langle F(x)\rangle_t$ (Eq 14.4) in this and in other tutorials and questions here. What ...
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Why is $\partial_{\mu}x^{\nu} = \delta^{\nu}_{\mu}$?
In Blundell's book on QFT, one can find the following
Is this because of:
$$\partial_{\mu}x^{\nu} = \partial_{\mu}x^{\nu^{'}} \partial_{\nu^{'}}x^{\mu}$$
$$\partial_{\mu}x^{\nu} = \Lambda_{\mu}^{\...
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Operator $A$ only act on the neighboured state or operator but not the entire expression?
In state vector formalism $A|\psi(x)><u(x)|=(A|\psi(x)>)<u(x)|$, where $A$ only act on $|\psi(x)>$
However, in terms of wave formalism, suppose $A$ is the well known $\frac{d}{dx}$.
...
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Is $\nabla=\nabla'$? Nabla operator acting on $r^n$
I have been taught that
$$\nabla r^n =\text{gradient of }r^n =n r^{n-1}\ \hat{\boldsymbol r}$$
but in introduction to electrodynamics by Griffith (4th edition) on page 173, $\nabla' r^n$ is given by $-...
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Abuse of Calculus [duplicate]
I'm following Professor R. Shankar's Fundamentals of Physics course on YouTube.
There I saw him doing manipulations of Calculus I never saw before.
Here it goes,
$$\newcommand\deriv[2]{\frac{\mathrm ...
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Non-differential total exact element notation question [duplicate]
I have questions regarding the notation in thermodynamics books of "d bar" (instead of delta) for the non-differential total exact elements like for work $\delta W$.
When did it appear?
Where are ...
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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 ...
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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 ...
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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 ...
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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 ...
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What does lowercase-delta mean in Noether's first theorem?
Most expressions of Noether's Theorem I have come across do not use lowercase delta, but a couple sites do. I am confused....... Check out page 21 of the June 23 issue of 'Science News' ...
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Lorentz transformation of the Klein-Gordon equation
In the Lorentz transformation of the field $\partial_\mu\phi(x)$ (Peskin, p.36)
\begin{eqnarray}
\partial_\mu\phi(x)\to\partial_\mu(\phi(\Lambda^{-1}x))=(\Lambda^{-1})^\nu_{\phantom{\nu}\mu}(\...
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2
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$\partial$ 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{\...
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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 ...
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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 (...
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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.
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