All Questions
Tagged with partial-derivatives or differentiation
1,974 questions
0
votes
1
answer
130
views
Intuitive understanding of what coordinate basis elements of $T_p$ mean in relation to the manifold
I am trying to have a "visual" or intuitive understanding of the description that was made in my GR lecture about the object $\partial_\mu$. The following was said:
We have a manifold $M$, a ...
- 2,058
3
votes
1
answer
118
views
Covariant Derivative of a Vector Field in Flat Space but Curvillinear Coordinates
Let $\vec{V}$ be a vector in flat Euclidean space. In curvilliner coordinates, using Einstein summantion convention, $$\vec{V}=V^j\vec{e}_j$$ where $\vec{e}_j$ are the basis vectors and $V^j$ are the ...
- 12.9k
4
votes
2
answers
241
views
Derivative of density of state
I came across a reference here where in Eq. 2.12 one seems to be concerned with the derivative of the density of state which is given by
$$\begin{aligned}
-\operatorname{Tr} \delta^{\prime}(H-\mu) &...
- 1,726
0
votes
1
answer
142
views
Ricci and Einstein tensor
How from these two equations:
$$\nabla^\mu R_{\rho\mu}=\frac{1}{2}\nabla_\rho R$$
and
$$G_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}$$
follows
$$\nabla^\mu G_{\mu\nu}=0~?$$
Also, what is $\nabla^\mu$ ...
- 79
0
votes
1
answer
71
views
Do fundamental interactions alter spacetime?
I am currently studying field theories, particularly General Relativity and Classical Electrodynamics. In the former, when one has a particle subject to a gravitational field, instead of introducing a ...
- 2,646
1
vote
1
answer
115
views
Parallel transport in Yang-Mills
I'm currently reading through David Tong's "Gauge Theory" lecture notes, and came across the following parallel transport equation:
\begin{equation}
i \frac{dw}{d\tau} = \frac{dx^\mu(\tau)}{...
- 182
0
votes
1
answer
120
views
Problem with vector calculus - electric field from a dipole
While deriving the electric field from a dipole source, from the notes I am following I am required to process the following vector operation:
$$
\nabla \left(\frac{e^{jkr}}{r}\mathbf n\cdot \mathbf p\...
- 27
0
votes
3
answers
256
views
What does $\frac{\partial \mathscr{L}}{\partial A^\mu}$ mean?
Suppose we define a variation as
$$
\delta F \equiv F(x,a)-F(x,a=0)=\frac{\partial F}{\partial a}\bigg|_{a=0} a +\mathcal{O}(a^2),
$$
where $a$ is some continuous paramter and $x$ is a spacetime ...
- 345
0
votes
0
answers
153
views
Equation of motion, 4th order
I have the following term in my Lagrangian:
$$
L=V(r)((\Delta \phi )^2-5(\partial_i\partial_j \phi)^2).
$$
I am kind of confused about computing the equation of motion, I would say (is there a ...
- 515
4
votes
1
answer
496
views
Notation question about taking derivative wrt a vector
I just have a notational question about dirac notation used in a paper I tried to read a while back.
Say that $|\phi\rangle \in \mathcal{H}^A \otimes \mathcal{H}^B$ and $|i \rangle \in \mathcal{H}^A$ ...
- 576
4
votes
1
answer
590
views
Susskind's covariant derivative
Recently, I've been trying to get a better understanding on general relativity. As a non-physicist with a good mathematics background, I picked up on Susskind's course on general relativity (which is ...
- 141
2
votes
3
answers
296
views
Why does squaring the operator $-i\hbar \frac{\partial}{\partial x}$ result in a second derivative?
I understand that to calculate the expectation of any value of a quantity $Q(x,p)$ we integrate $$\int\Psi^{ \ \ast}[Q(x,-i\hbar \frac{\partial}{\partial x})]\Psi dx.$$
However for expectation value ...
- 391
1
vote
1
answer
150
views
Why is $\delta(\partial_\nu A^\mu)=\partial_\nu(\delta A^\mu)$? [closed]
Comment
Note that I am aware of the posts in:
Reasoning behind $\delta \dot q = \frac{d}{dt} \delta q$ in deriving E-L equations,
Least action principle : is $ \delta \frac{dx}{dt} = \frac{d \delta x}{...
- 345
1
vote
0
answers
100
views
The Smoothness Postulate in Lagrangian Mechanics [duplicate]
The Euler-Lagrange equation is central to Lagrangian mechanics:
$$
\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}_i}\right) - \frac{\partial L}{\partial q_i} = 0
$$
This equation's structure ...
- 11
0
votes
1
answer
195
views
Material Derivative of Infinitesimal Line Element
I am trying to self-learn fluid dynamics from online resources because I am interested in its application to astrophysical environments. I found lecture notes on astrophysical fluid dynamics on arXiv (...
- 11