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 [math]\displaystyle{ x^a }[/math], the coordinate-induced basis [math]\displaystyle{ e_a }[/math] of the tangent space is given by

[math]\displaystyle{ e_a = \frac{\partial}{\partial x^a} }[/math]

and the dual basis [math]\displaystyle{ \omega^a }[/math] of the cotangent space is

[math]\displaystyle{ \omega^a=dx^a. \, }[/math]

References

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