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

In summary, the pushforward of a vector field created by a counterclockwise rotation by angle ##\theta## is equivalent to the rotation of the vector field.
  • #1
ergospherical
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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?
 
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  • #2
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##.
 
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Likes ergospherical
  • #3
Great, thanks!
 
  • #5
Is there an errata for the errata? :wink:
1641022076628.png
 

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

1. What are gauge fields?

Gauge fields are mathematical constructs used in theoretical physics to describe fundamental forces. They are vector fields that represent the strength and direction of a particular force at every point in space.

2. What is the significance of knots in gauge fields?

Knots in gauge fields play a crucial role in understanding the topology of space. They are used to describe the behavior of gauge fields in three-dimensional space and are essential in theories of quantum gravity.

3. How does rotation factor into gauge fields?

Rotation is a fundamental aspect of gauge fields. In fact, gauge fields are defined by their transformation properties under rotations. This allows for a unified description of both rotational and non-rotational forces.

4. Can gauge fields be used to describe gravity?

Yes, gauge fields can be used to describe gravity through the theory of general relativity. In this theory, gravity is described as the curvature of spacetime caused by the presence of matter and energy. This curvature is represented by a gauge field known as the metric tensor.

5. Are there any practical applications of gauge fields?

Yes, gauge fields have many practical applications in fields such as particle physics, cosmology, and condensed matter physics. They are also used in technologies such as MRI machines and particle accelerators.

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