Group velocity vector in spherical coordinates

I am trying to understand a derivation from the textbook Radiation Processes in Plasmas by G. Bekefi (p. 14).

Start with the group velocity vector $$\mathbf{w}=\frac{\partial \omega}{\partial \mathbf{k}}=\hat{\textbf{e}}_{x} \frac{\partial \omega}{\partial k_{x}}+\hat{\textbf{e}}_{y} \frac{\partial \omega}{\partial k_{y}}+\hat{\textbf{e}}_{z} \frac{\partial \omega}{\partial k_{z}}.$$

The propagation vector k is of a wave of frequency $$\omega$$, where both k and $$\omega$$ which have spread (d$$\omega, d\textbf{k}$$) and can be expressed in spherical coordinates $$(k, \theta, \phi): \omega=\omega(k, \theta, \phi) \text{ or } k=k(\omega, \theta, \phi).$$

I understand that

$$w_{k}=\frac{\partial \omega}{\partial k}$$, $$w_{\theta}=\frac{1}{k}\frac{\partial \omega}{\partial \theta}$$, and $$w_{\phi}=\frac{1}{k \sin \theta}\frac{\partial \omega}{\partial \phi}.$$

But the text also says that

$$w_{k}=\left ( \frac{\partial k}{\partial \omega} \right )^{-1},$$ $$w_{\theta}=-\frac{1}{k}\frac{\partial k}{\partial \theta}\left ( \frac{\partial k}{\partial \omega} \right )^{-1},$$ and $$w_{\phi}=-\frac{1}{k \sin \theta}\frac{\partial k}{\partial \phi}\left ( \frac{\partial k}{\partial \omega} \right )^{-1}$$

and I don't understand how those expressions are derived. Could someone explain?

I didn't read the book and I am not expect in this field, but if $$w_{k}=\frac{\partial \omega}{\partial k}=\left(\frac{\partial k}{\partial \omega}\right)^{-1}$$, then the other equalities are obvious :
$$w_{\theta}=\frac{1}{k}\frac{\partial \omega}{\partial \theta}= \frac{1}{k}\frac{\partial \omega}{\partial k} \frac{\partial k}{\partial \theta} = \frac{1}{k}\frac{\partial k}{\partial \theta}\left ( \frac{\partial k}{\partial \omega} \right )^{-1}$$
$$w_{\phi}=\frac{1}{k sin \theta}\frac{\partial \omega}{\partial \phi} = \frac{1}{k sin \theta}\frac{\partial \omega}{\partial k}\frac{\partial k}{\partial \phi} = \frac{1}{k sin \theta}\frac{\partial k}{\partial \phi}\left ( \frac{\partial k}{\partial \omega} \right )^{-1}$$