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}.\,

References

D.J. Hurley, M.A. Vandyck Topics in Differential Geometry: a New Approach Using D-Differentiation (2002 Springer) p. 5