For what point p is $df_p$ injective if $ f $ is a parametrization of a 3-dim sphere

Question

A 3-dimensional sphere $\mathbb{S} \subseteq \mathbb{R}^4$ can be parameterized by $$ f(\varphi, \psi, \theta) = (\cos \varphi \cos\psi \cos\theta,\sin \varphi \cos\psi \cos\theta, \sin \psi \cos\theta, \sin \theta ) $$ For what point p is $ df_p$ injective?

I considered $$ df_p = \begin{pmatrix} -sin \varphi \cos\psi \cos\theta & -cos \varphi \sin\psi \cos\theta & -cos \varphi \cos\psi \sin\theta \\ \cos \varphi \cos\psi \cos\theta & -sin \varphi \sin\psi \cos\theta & -sin \varphi \cos\psi \sin \theta \\ 0 & \cos\psi \cos\theta & -sin \ \psi \sin\theta \\ 0 & 0 & \cos \theta \end{pmatrix} $$

for $p = (\varphi, \psi, \theta) $ .

Now $df_p$ is injective if and only if $rank(df_p) = 3$.

So I need to prove for what p all columns of $ df_p$ are linearly independent? And how would I do that?

Answer 1

Hint $df_p$ has rank $3$ iff at least one of its $3 \times 3$ minors has rank $3$.