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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 {\displaystyle x^{a}} , the coordinate-induced basis e a {\displaystyle e_{a}} of the tangent space is given by
e a = ∂ ∂ x a {\displaystyle e_{a}={\frac {\partial }{\partial x^{a}}}}and the dual basis ω a {\displaystyle \omega ^{a}} of the cotangent space is
ω a = d x a . {\displaystyle \omega ^{a}=dx^{a}.\,}- D.J. Hurley, M.A. Vandyck Topics in Differential Geometry: a New Approach Using D-Differentiation (2002 Springer) p. 5