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Variation of Torsion-Free Spin Connection

In the book 'Supergravity' by Freedman and van Proeyen, in exercise (7.27) it is written To calculate [the variation $\delta\omega_{\mu ab}$ of the torsion-free spin connection], consider the ...
vyali's user avatar
  • 372
3 votes
2 answers
201 views

What is difference between an infinitesimal displacement $dx$ and a basis one-form given by the gradient of a coordinate function?

In general relativity, we introduce the line element as $$ds^2=g_{\mu \nu}dx^{\mu}dx^{\nu}\tag{1}$$ which is used to get the length of a path and $dx$ is an infinitesimal displacement But for a ...
Mahtab's user avatar
  • 634
2 votes
1 answer
258 views

Infinitesimal coordinate transformation and Lie derivative

I need to prove that under an infinitesimal coordinate transformation $x^{'\mu}=x^\mu-\xi^\mu(x)$, the variation of a vector $U^\mu(x)$ is $$\delta U^\mu(x)=U^{'\mu}(x)-U^\mu(x)=\mathcal{L}_\xi U^\mu$$...
vyali's user avatar
  • 372
1 vote
0 answers
64 views

How to transform a partial derivative to a directional derivative with respect to some affine parameter?

Suppose an affine parameter $\lambda$ is defined along a null geodesic with $dx^\mu/d\lambda=k^\mu$. How could I write the partial derivative $\partial f/\partial x^\mu$ by using $df/d\lambda$? If $k^\...
Haorong Wu's user avatar
9 votes
1 answer
598 views

Inverse of the covariant derivative

Given the covariant derivative of some tensor, for the sake of this example a covariant vector: $$\nabla_\mu A_\nu$$ Is there a well-defined inverse operation on the covariant derivative such that it ...
Tachyon's user avatar
  • 613
4 votes
2 answers
510 views

Converting differential to gradient

Landau & Lifschitz's fluid mechanics book proposes the following statement for an isentropic proccess: $$dH=vdp \Rightarrow \nabla H=v\nabla p$$ What's the rigorous way to get this result (...
Pablo1571's user avatar
0 votes
1 answer
87 views

How do I reconcile these two definitions of acceleration?

How do I reconcile these two definitions of acceleration? $$a=\frac{d\bar{v}}{dt}=(\frac{dv^k}{dt}+v^i v^j \Gamma^k_{ij})\bar{e}_k \tag{1}$$ and $$a^k=v^{\small\beta} \nabla_{\small\beta} v^k.\tag{2}$$...
jelly ears's user avatar
0 votes
1 answer
81 views

Volume of a two-dimensional sphere in a fixed three sphere geometry

I'm just starting to read Hartle's Gravity and he gives the following equation for the volume of a 2D sphere of radius $r$ if space was a fixed 3-sphere geometry on page 20. $$V = 4\pi a^3\left(\frac{...
nourbaki's user avatar
0 votes
2 answers
186 views

What's the physical meaning of $\vec A \times \vec \nabla$? [closed]

What's the physical meaning of $\vec A \times \vec \nabla$? Gradient represents a field. What if we try we write "opposite" of that? I know that if direction of $\vec \nabla \times \vec A$ ...
Unknown's user avatar
  • 91
0 votes
2 answers
204 views

Physical meaning of the exterior derivative of the first law of thermodynamics

We know, $$ dU = d \overline{q} - d \overline{W}.$$ suppose we took the exterior derivative on both sides, then: $$ 0= d( d \overline{q}) - d( d \overline{W})$$ This means, $$ d^2 \overline{q} = d^2 \...
Brian's user avatar
  • 8,040
1 vote
1 answer
79 views

Alternative formula for the affine connection in a new coordinate basis

In Hobsons's General Relativity: An Introduction for Physicists, pg. 64, he gave two different expressions for the affine connection $\Gamma'^a_{bc}$ in a transformed coordinate basis $x'^a$ (the ...
TaeNyFan's user avatar
  • 4,276
-1 votes
2 answers
203 views

Divergence of vector fields intuitive understanding

I am troubled by the idea of positive and negative divergence of a vector field. I understand that the idea of e.g. positive div is that a gas expands everywhere (for the velocity field of a gas ...
SheppLogan's user avatar
1 vote
1 answer
50 views

Path Coordinates: direction problem (doubt) in derivative of tangential vector

Why is the direction of derivative of tangential vector perpendicular to the direction of the tangential vector?
Prakul Virdi's user avatar
4 votes
2 answers
5k views

How is dot or cross product possible using the del operator?

Yesterday in class my teacher told me that the del operator has a direction but no value of its own (as its an operator). So it can't be called exactly a vector. But in vector calculus we see that div ...
Theoretical's user avatar
  • 1,430
5 votes
1 answer
1k views

Is the inverse of the deformation gradient simply the deformation gradient of the inverse transformation?

If we have a continuum where the initial positions are denoted $X$ and the positions after some deformation are denoted $x$, the deformation gradient is defined: $$ F = \frac{\partial x}{\partial X} $...
nnn's user avatar
  • 103
3 votes
2 answers
969 views

Differential operators in curvilinear coordinates

In the appendix A of Griffith's Electrodynamics text, he cites Spivak's Calculus on Manifolds as a reference more a more complete treatment of taking the gradient, curl, divergence, and Laplacian in ...
15 votes
3 answers
44k views

Derive vector gradient in spherical coordinates from first principles

Trying to understand where the $\frac{1}{r sin(\theta)}$ and $1/r$ bits come in the definition of gradient. I've derived the spherical unit vectors but now I don't understand how to transform ...
Lucidnonsense's user avatar