Unit radial vector field

Question

Lee's book defines the unit radial vector field in normal coordinates as

$$ \partial_r:= \frac{x^i}{r(x)} \partial_i$$

and $r(x):=\sqrt{\sum_i (x^i)^2}$

Now this is a unit vector field iff $$g(\partial_r,\partial_r)=1.$$

By linearity this is equivalent to $$1= \frac{x^iy^j}{r^2} g(\partial_i,\partial_j).$$

Is there now any reason that this should hold?

Answer 1

On the very next page, Proposition 5.11(e) shows that it is always a unit vector field.