Pullback Calculation

A more general equation for how to compute pullbacks is as follows: If $f: M \to N$ is a smooth map of smooth manifolds and $\omega$ is an $n$-form on $N$, written in local coordinates as $$ \omega = \sum_I \omega_I dy^{i_1} \wedge \cdots \wedge dy^{i_n}$$ then $f^*\omega$ can be written in induced coordinates as $$ f^*\omega = \sum_I (\omega_I \circ f) d(y^{i_1} \circ f) \wedge \cdots \wedge d(y^{i_n} \circ f)$$ where you think of $y^i$ as the function which picks out the $i^{th}$ coordinate of $f$.

Your $\omega$ is given by $$ \omega = \omega_{23} dx_2 \wedge dx_3 + \omega_{31} dx_3 \wedge dx_1 + \omega_{12} dx_1 \wedge dx_2$$ where $\omega_{23} = \frac{x_1}{r^3}, \omega_{31} = \frac{x_2}{r^3}$ andd $\omega_{12} = \frac{x_3}{r^3}$. So now you just use the formula: I will do the $\omega_{23} dx_2 \wedge dx_3$ term as an example:

$$ f^*(\omega_{23} dx_2 \wedge dx_3) = (\omega_{23} \circ f) d(x_2 \circ f) \wedge d(x_3 \circ f) $$

Now \begin{align*} \omega_{23} \circ f &= \frac{\sin\theta \cos\phi}{\sqrt{\sin^2\theta\cos^2\phi + \sin^2\theta \sin^2\phi + \cos^2\theta}^3} = \sin\theta\cos\phi \\ d(x_2\circ f) &= d(\sin\theta\sin\phi) = \cos\theta \sin\phi d\theta + \sin\theta \cos\phi d\phi \\ d(x_3\circ f) &= d(\cos\theta) = -\sin\theta d\theta. \end{align*} Putting it all together, we get \begin{align*} (\omega_{23} \circ f) d(x_2 \circ f) \wedge d(x_3 \circ f) &= (\sin\theta\cos\phi)(\cos\theta\sin\phi d\theta + \sin\theta\cos\phi d\phi) \wedge (-\sin\theta d\theta ) \\ &= \sin^3\theta\cos^2\phi d\theta \wedge d\phi \end{align*}

(Of course, modulo mistakes in my computations!)