Coordinate-induced basis

In mathematics, a coordinate-induced basis is a basis for the tangent space or cotangent space of a manifold that is induced by a certain coordinate system. Given the coordinate system $$ x^a $$, the coordinate-induced basis $$e_a$$ of the tangent space is given by
 * $$ e_a = \frac{\partial}{\partial x^a} $$

and the dual basis $$ \omega^a $$ of the cotangent space is
 * $$ \omega^a=dx^a. \, $$