Let \(M\) be a [smooth manifold](smooth manifold.md). The cotangent space at a point \(p\in M\), is the dual space of the tangent space at \(p\), i.e.
\begin{equation*} T^*_pM=(T_pM)^*. \end{equation*}The elements of \(T^*_pM\) are called (tangent) covectors at \(p\).
Remarks
- A cotangent vector is also called covariant vector .
- the derivative of \(f\colon M\to \mathbb{R}\) at a certain point can be seen as a covector