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 ${\displaystyle x^{a}}$, the coordinate-induced basis ${\displaystyle e_{a}}$ of the tangent space is given by

${\displaystyle e_{a}={\frac {\partial }{\partial x^{a}}}}$

and the dual basis ${\displaystyle \omega ^{a}}$ of the cotangent space is

${\displaystyle \omega ^{a}=dx^{a}.\,}$

References

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