---- > [!definition] Definition. ([[cotangent space]]) > Let $M$ be a [[smooth manifold]]. For each $p \in M$, we define the **cotangent space at $p$**, denoted by $T_{p}^{*}M$, to be the [[dual vector space|dual space]] of the [[tangent space at a point of a smooth manifold|tangent space]] at $p$: $T_{p}^{*}M = (T_{p}M)^{*}.$ Elements of $T_{p}^{*}M$ are called **tangent covectors at $p$**, **cotangent vectors at $p$**, or just **covectors at $p$**. > > Given smooth local [[coordinate chart|coordinates]] $(x^{i})$ on an open subset $U \subset M$, for each $p \in U$ the [[tangent space at a point of a smooth manifold|coordinate basis]] $(\frac{ \partial }{ \partial x^{i} } |_{p})$ gives rise to a [[dual basis]] for $T_{p}^{*}M$, which we denote by $(dx^{i} |_{p})$. Any covector $\omega \in T_{p}^{*}M$ can thus be written uniquely as $\omega=\omega_{i} \ dx^{i} |_{p}$, where $\omega_{i}=\omega\left( \frac{ \partial }{ \partial x^{i} } |_{p} \right).$ > The components $\omega_{i}$ obey the [[transformation law]] $\omega_{i}= \frac{ \partial \tilde{x}^{j} }{ \partial x^{i} } \tilde{\omega}_{j}.$ > The basis vectors $dx^{i}$ obey the [[transformation law]] $d\tilde{x}^{i}=\frac{ \partial \tilde{x}^{i} }{ \partial x^{j} } |_{p} dx^{j}.$ > Note that while the choice of notation $\frac{ \partial }{ \partial x^{i} } |_{p}$ for the [[tangent space at a point of a smooth manifold|coordinate basis]] of $T_{p}M$ was merely suggestive, the notation $dx^{i} |_{p}$ is not: viewing $x^{i}: U \to \mathbb{R}$ as a [[smooth maps between manifolds|smooth map between manifolds]], its [[differential of a smooth map between smooth manifolds|differential]] at $p$ $dx^{i} |_{p}: T_{p}M \to T_{x^{i}(p)}\mathbb{R}=\mathbb{R}$ > is the element of $T_{p}^{*}M$ given by definition as the [[matrix]] $\begin{bmatrix} > \overbrace{ \frac{ \partial }{ \partial x^{1} } |_{p} x^{i} }^{ =0 } & \cdots & \overbrace{ \frac{ \partial }{ \partial x^{i} } |_{p} x^{i} }^{ =1 } & \cdots & \overbrace{ \frac{ \partial }{ \partial x^{n} } |_{p} x^{i} }^{ =0 } > \end{bmatrix},$ > i.e., is the dual basis vector to $\frac{ \partial }{ \partial x^{i} } |_{p}$. > > > [!justification] The transformation law. > Suppose that $(\widetilde{x}^{i})$ is another set of smooth [[coordinate chart|coordinates]] around $p$, and let $(d\tilde{x}^{i})$ denote the [[basis]] for $T_{p}^{*}M$ [[dual basis|dual]] to the [[tangent space at a point of a smooth manifold|coordinate basis]] $(\frac{ \partial }{ \partial \widetilde{x}^{j} } |_{p})$ of $T_{p}M$. How do the components transform? > > Writing $d\tilde{x}^{i}$ in terms of $dx^{i}$ mostly comes down to recalling the [[transformation law]] for [[tangent vector to a smooth manifold|tangent vectors]]. In particular, recall that $\frac{ \partial }{ \partial x^{i} }= \frac{ \partial \tilde{x}^{j} }{ \partial x^{i} } \frac{ \partial }{ \partial \tilde{x}^{j} }$. Hence $\begin{align} > \omega_{i} &= \omega\left( \frac{ \partial }{ \partial x^{i} } |_{p} \right) \\ > &= \omega \left( \frac{ \partial \tilde{x}^{j} }{ \partial x^{i} } |_{p} \ \frac{ \partial }{ \partial \tilde{x}^{j} } |_{p} \right) \\ > &= \frac{ \partial \tilde{x}^{j} }{ \partial x^{i} } \tilde{\omega}_{j}. > \end{align}$ > > > As for basis vectors, have $d\tilde{x}^{i}=\frac{ \partial \tilde{x}^{i} }{ \partial x^{j} } |_{p} dx^{j}.$ ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```