Cotangent space
= Cotangent space
{title2=$T_p^*M$}
The cotangent space at $p$ is the <dual vector space> $T_p^*M=(T_pM)^*$. Its elements are <covectors> on the <tangent space>. In a <manifold chart> $x$, the basis $dx^i|_p$ is dual to $\partial_i|_p$. Under a coordinate change to $y$, the coefficients of $\alpha=a_i dx^i=b_jdy^j$ obey $b_j=\sum_i a_i\partial x^i/\partial y^j$. The disjoint union of these spaces is the <cotangent bundle>.