All Questions
Tagged with vector-fieldsdifferentiation
210 questions
4votes
1answer
84views
Understanding condition of Hypersurface orthogonality
Wald at page#436 under the heading B.3.2 wrote The dual formulation of Frobenius's theorem gives a useful criterion for when a vector field $\mathcal{E}^a$ is hypersurface orthogonal. Letting $T^*$ ...
- 595
1vote
0answers
60views
Fermi-Walker transport in arbitrary coordinates
I am having a hard time working with the Fermi-Walker transport equation in arbitrary coordinates. Background: The problem I was trying to analyze is that of an electron treated as a classical ...
- 598
1vote
0answers
35views
Fermi-Walker transport of proper acceleration vector field along timeline congruence's worldlines
If we take a generic irrotational/zero vorticity timelike congruence, do the 4-velocity and the direction of proper acceleration $($i.e. the vector in that direction at each point with norm $1$$)$ ...
- 1,999
1vote
0answers
66views
Potential typo on covariant derivative in Schutz's General Relativity?
In the latest (3rd) edition of Schutz's "A First Course in General Relativity", there is an equation below equation 5.63 on page 130 (which I think should be clear without context) about ...
0votes
0answers
38views
Four-divergence of a vector [duplicate]
The covariant derivatives of a four-vector is $$ \nabla_{\nu}U_{\mu} = \partial_{\nu}U_{\mu} - \Gamma^{\lambda}_{\mu\nu}U_{\lambda} $$ It has the following identity: $$ \nabla_{\mu}U^{\mu} = \frac{\...
5votes
4answers
660views
Vector triple product with $\nabla$ operator
I came across the following expression in several books (especially in plasma physics literature while deriving the magnetic pressure): $$(\mathbf{\nabla} \times \mathbf{B})\times \mathbf{B} = \left(\...
- 61
1vote
1answer
112views
Proving a Superfunction Identity
I am trying to figure out the proof of the identity given between equations (1.11.7) and (1.11.8) in ref. [1], i.e. \begin{align} \Phi'(e^{-K}\,z\,e^K)=e^{-K}\Phi'(z) \tag{1} \end{align} where $z=(...
- 49
5votes
1answer
345views
Divergence of vector field term-wise
In a spacetime $(M, g)$ the following identity for the divergence of a vector field $X$ holds $$ \nabla_{\mu} X^{\mu} = \frac{1}{\sqrt{-\det g}} \, \partial_{\mu} \big( \sqrt{- \det g} \ X^{\mu} \big)...
- 813
1vote
1answer
88views
Meaning of colon symbol $:$ in optics
When I was reading some early days nonlinear optics paper/textbooks (particularly between 1960-1985), I often see expressions such as: $\chi^{(2)}:\textbf{E}\textbf{E}$ or $\nabla\textbf{E}:\partial \...
2votes
1answer
434views
How can the divergence of a cylinder with uniform magnetic field be non-zero?
When I'm calculating the divergence of a cylinder with uniform magnetic field ($B=K=\text{constant}$) according to the formula of divergence in cylindrical coordinates I'm getting the same constant ...
- 31
1vote
3answers
135views
The conservative force [closed]
I read about the definition of the curl. It's the measure of the rotation of the vector field around a specific point I understand this, but I would like to know what does the "curl of the ...
- 11
1vote
0answers
106views
A preposterous abuse of notation involving Helmholtz decomposition theorem
Take what I am about to present with a light heart, since the mathmetically inclined may find it too out-of-the-world and devastating. The above diagram (this is drawn by me, but the original is very ...
3votes
1answer
151views
A theorem on page 72 in The Large Scale Structure of Space-Time [closed]
In chapter 3 of the book, page 72, a static observer is defined as $V^{a}\equiv f^{-1}K^{a}$, where $K^{a}$ is a timelike Killing vector field and $f^{2}=-K^{a}K_{a}$. Then, Hawking & Ellis claim ...
3votes
1answer
194views
Laplace-Beltrami operator for a vector field
For a scalar field $\varphi$, the "wave" operator is defined as follows: $$\Box \varphi \equiv g^{ab}\nabla_a\nabla_b~\varphi = \frac{1}{\sqrt{|g|}}\partial_a\left\{\sqrt{|g|}~g^{ab}~\...
- 653
-2votes
1answer
269views
Exact definition of divergence. Is it really the dot product of nabla operator with a vector? [closed]
I was trying to understand the derivation for divergence in cylindrical and spherical coordinate system, and I am a bit confused here. https://www.gradplus.pro/deriving-divergence-in-cylindrical-and-...
- 33
