Cotangent space
In differential geometry, the cotangent space is a vector space associated with a point x {\displaystyle x} on a smooth (or differentiable) manifold M {\displaystyle {\mathcal {M}}} ; one can define a cotangent space for every point on a smooth manifold. Typically, the cotangent space, T x ∗ M {\displaystyle T_{x}^{*}\! {\mathcal {M}}} is defined as the dual space of the tangent space at x {\displaystyle x} , T x M {\displaystyle T_{x}{\mathcal {M}}} , although there are more direct definitions (see below).