mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a coordinate-induced basis is a basis for the

tangent space In mathematics, the tangent space of a manifold is a generalization of to curves in two-dimensional space and to surfaces in three-dimensional space in higher dimensions. In the context of physics the tangent space to a manifold at a point can be ...

or cotangent space of a

manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...

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

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 Differential geometry {{differential-geometry-stub