Definition:Local Coordinates

For neighborhood $$U \ $$ of a point $$p \ $$, $$U \subset X^n \ $$ in an $$n \ $$-dimensional manifold $$X^n \ $$, local coordinates are a set of functions $$\left\{{x_i}\right\}_{i=1}^n \ $$ defined $$x_i:X^n \to \R \ $$ defined so that if $$a,b \in U \ $$ and $$\forall i, x_i(a)=x_i(b), \ $$ then $$ p=q \ $$.