Critical points and immersion map

Question

I was dealing with the following problem from differential geometry yesterday.

Let $ x_1, \ldots, x_4 $ be points in general position in $ \mathbb{R} ^3 $ (that is they don't lie in a plane). Let $ q_1, \ldots, q_4 \in \mathbb{R} $ be electric charges placed at these points. The potential function of the resulting electric field is given by $$ V_q= \frac{q_1}{r_1} + \ldots + \frac{q_4}{r_4} $$ where $ r_i=|x-x_i| $. The critical points of $ V_q $ are called equilibrum points of the electric field and an equilibrum point is non-degenerate if the critical point is. Prove that for almost all $ q $ the equilibrum points of $ V_q $ are non-degenerate and finite in number. Show that the map $ \mathbb{R}^3 - \{ x_1,\ldots , x_4 \} \rightarrow \mathbb{R}^4 $ with coordinates $ r_1, \ldots, r_4 $ is an immersion."

So far I was able to find the critical points and for showing that the given map is an immersion on the second part I was considering to find the rank of derivative and check if it is equal to the dimension of $ \mathbb{R}^3 - \{ x_1, \ldots, x_4 \} $ but not sure if this is the right approach or if there is some easier way to show it is an immersion. Any help would be appreciated.

Answer 1

Denote by $f:\mathbb{R}^3-\{x_1,\dots,x_4\}\to \mathbb{R}^4$ your second map, so we have $$f(y)=\left( \vert y-x_i\vert\right)_{1\leq i\leq 4}.$$ By differentiating at $y\in \mathbb{R}^3-\{x_1,\dots,x_4\}$, we have $$d_yf\cdot h=\left( \frac{\langle y-x_i,h\rangle}{\vert y-x_i\vert}\right)_{1\leq i\leq 4}.$$ So $h\in Ker(d_yf)\iff h\in Vect(y-x_1,\dots,y-x_4)^\perp$.

Can you conclude from here? Have you finished the first part of the question?