I Ex. 19 Gauge Fields, Knots & Gravity: Is Rotation Correct?

ergospherical
Science Advisor
Homework Helper
Education Advisor
Insights Author
Messages
1,097
Reaction score
1,384
Wanted to check with you guys that I'm not going crazy...

Exercise 19: Let ##\phi : \mathbf{R}^2 \rightarrow \mathbf{R}^2## be a counterclockwise rotation by angle ##\theta##. Let ##\partial_x, \partial_y## be the coordinate vector fields on ##\mathbf{R}^2##. Show, at any point of ##\mathbf{R}^2##, that ##\phi_*\partial_x = (\cos{\theta})\partial_x - (\sin{\theta}) \partial_y## and also ##\phi_*\partial_y = (\sin{\theta})\partial_x + (\cos{\theta})\partial_y##

The effect of the pushforward is just a rotation of the vectors, so presumably one would instead expect ##\phi_* \partial_x = (\cos{\theta})\partial_x + (\sin{\theta})\partial_y##, right?

The rotation is ##(x,y) \mapsto \phi(x,y) = (x\cos{\theta} - y\sin{\theta}, \ x\sin{\theta} + y\cos{\theta})##. Let ##f \in C^{\infty}(\mathbf{R}^2)## be a test function, then the pushforward of ##\partial_x## is\begin{align*}
((\phi_* \partial_x)(f))(\phi(x),\phi(y)) &= (\partial_x(\phi^* f))(x,y) \\
&= (\partial_x (f \circ \phi))(x,y) \\
\end{align*}One can determine the ##x##-component of ##\phi_* \partial_x## by letting ##f=x##,\begin{align*}
((\phi_* \partial_x)(x))(\phi(x),\phi(y)) &= (\partial_x(x \circ \phi))(x,y) \\
&= (\partial_x(x\cos{\theta} - y\sin{\theta}))(x,y) \\
&= \cos{\theta}
\end{align*}Similarly, put ##f=y## to obtain ##((\phi_* \partial_x)(y))(\phi(x),\phi(y)) = \sin{\theta}##. Then\begin{align*}
\phi_* \partial_x = (\cos{\theta})\partial_x + (\sin{\theta})\partial_y
\end{align*}as before. I haven't misread something?
 
Physics news on Phys.org
I agree with your conclusion. Alternatively, consider curve ##\gamma: t \mapsto (t,a)##, which has ##\partial_x## as its tangent vector. You will find that ##\phi \circ \gamma (t) = (t \cos\theta - a \sin\theta, t \sin\theta + a \cos\theta)##, which clearly has the tangent vector ##\cos\theta \partial_x + \sin\theta \partial_y##.
 
  • Like
Likes ergospherical
Great, thanks!
 
Is there an errata for the errata? :wink:
1641022076628.png
 
I started reading a National Geographic article related to the Big Bang. It starts these statements: Gazing up at the stars at night, it’s easy to imagine that space goes on forever. But cosmologists know that the universe actually has limits. First, their best models indicate that space and time had a beginning, a subatomic point called a singularity. This point of intense heat and density rapidly ballooned outward. My first reaction was that this is a layman's approximation to...
In this video I can see a person walking around lines of curvature on a sphere with an arrow strapped to his waist. His task is to keep the arrow pointed in the same direction How does he do this ? Does he use a reference point like the stars? (that only move very slowly) If that is how he keeps the arrow pointing in the same direction, is that equivalent to saying that he orients the arrow wrt the 3d space that the sphere is embedded in? So ,although one refers to intrinsic curvature...
Thread 'Dirac's integral for the energy-momentum of the gravitational field'
See Dirac's brief treatment of the energy-momentum pseudo-tensor in the attached picture. Dirac is presumably integrating eq. (31.2) over the 4D "hypercylinder" defined by ##T_1 \le x^0 \le T_2## and ##\mathbf{|x|} \le R##, where ##R## is sufficiently large to include all the matter-energy fields in the system. Then \begin{align} 0 &= \int_V \left[ ({t_\mu}^\nu + T_\mu^\nu)\sqrt{-g}\, \right]_{,\nu} d^4 x = \int_{\partial V} ({t_\mu}^\nu + T_\mu^\nu)\sqrt{-g} \, dS_\nu \nonumber\\ &= \left(...

Similar threads

Replies
1
Views
890
Replies
8
Views
2K
Replies
0
Views
1K
Replies
4
Views
952
Replies
2
Views
714
Back
Top