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

[ergospherical]

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?

 

[Orodruin]

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 $\cos\theta \partial_x + \sin\theta \partial_y$.

 

[ergospherical]

Great, thanks!

 

[Demystifier]

Check also errata for the book:
https://math.ucr.edu/home/baez/errata.html

 

[Orodruin]

Is there an errata for the errata?