> [!definition] > > Let $X$ be a [[Manifold|manifold]] and $p \in X$, then the **cotangent space** at $p$, > $ > T^*_pX = (T_pX)^* > $ > is the [[Topological Dual|dual]] of the [[Tangent Space|tangent space]]. > > Let $(U, \varphi)$ be a chart at $p$, then the isomorphism between $T_p^*X$ and $T_{(p, U, \varphi)}X$ induces an isomorphism between $T_p^*X$ and $(T_{(p, U, \varphi)}X)^*$. Let $\omega \in T_p^*X$, then $\widehat \omega \in (T_{(p, U, \varphi)}X)^*$ defined $\angles{\wh \omega, \hat v} = \angles{\omega, v}$ is its representation in the concrete tangent space. > > In particular, if $X$ is a [[n-Manifold|n-manifold]], then $T_{(p, U, \varphi)}X$ comes with the standard [[Basis|basis]] $\seq{e_i}$, and $(T_{(p, U, \varphi)}X)^*$ comes with the corresponding [[Dual Space|dual basis]] $\seqf{\varepsilon^i}$. So for any $\wh\omega \in (T_{(p, U, \varphi)}X)^*$, > $ > \wh\omega = \wh w_i \varepsilon^i \quad \text{where} \quad \wh\omega_i = \angles{\wh \omega , e_i} > $ > Identifying $\seqf{e_i}$ with the coordinate tangent vectors $\seqf{\ppi}$, and $\seqf{\varepsilon^i}$ with $\seqf{dx^i|_p}$ yields the coordinate representation > $ > \omega = \omega_idx^i|_p \quad \text{where} \quad \omega_i = \angles{\omega, \ppip} > $ > [!theorem] > > Let $X$ be a $n$-manifold, $p \in X$, and $(U, \varphi)$ and $(V, \psi)$ be two charts at $p$. Then the translation between the concrete tangent spaces > $ > D(\psi \circ \varphi^{-1})(p): T_{(p, U, \varphi)}X \to T_{(p, V, \psi)} > $ > induces a [[Dual Map|dual map]] > $ > D(\psi \circ \varphi^{-1})(p)^*: (T_{(p, V, \psi)}X)^* \to (T_{(p, U, \varphi)}X)^* > $ > Let $\omega \in T_p^*X$ with representations $\wh \omega \in (T_{(p, U, \varphi)}X)^*$ and $\td \omega \in (T_{(p, V, \psi)}X)^*$, and $\psi \circ \phi^{-1} = (\td x^1, \cdots, \td x^n)$ be the transition map, then > $ > \wh \omega_i = \td\omega_j\frac{\partial \td x^j}{\partial x^i}(p) > $ > *Proof*. The dual to the transition map allows translating between the concrete cotangent spaces as > $ > \wh \omega = D(\psi \circ \varphi^{-1})(p)^* \cdot \td \omega > $ > Let $\seqf{a_i}$ be the standard basis in $T_{(p, U, \varphi)}X$ and $\seqf{\alpha_i}$ be the corresponding dual basis, $\seqf{e_i}$ be the standard basis in $T_{(p, V, \psi)}X$ and $\seqf{\varepsilon_i}$ be the corresponding dual basis. Then the dual map with respect to the dual bases is just the transpose of $D(\psi \circ \varphi^{-1})(p)$. Hence > $ > \td \omega = \td \omega_i\varepsilon^i \mapsto \td\omega_i\frac{\partial \td x^i}{\partial x^j}(p)\alpha_j > $ > under the dual map, meaning that > $ > \wh\omega_j = \td\omega_i\frac{\partial \td x^i}{\partial x^j}(p) > $