---- Let $M$ be a [[smooth manifold]] of dimension $n$. Let $\boldsymbol p \in M$. > [!definition] Definition. ([[tangent space at a point of a smooth manifold]]) > The **tangent space** to $M$ at $\boldsymbol p$ is the [[vector space]] of all [[tangent vector to a smooth manifold|tangent vectors]] at $\boldsymbol p$, denoted $T_{\boldsymbol p}M$. > > $T_{\boldsymbol p}M$ has dimension $n$, [[the rank theorem for free modules|hence]] $T_{\boldsymbol p}M \cong \mathbb{R}^{n}$, but there is no canonical [[linear isomorphism]]. A choice of [[coordinate chart]] $(U, \varphi)=\big( U, (x^{1}, \dots, x^{n}) \big)$ about $\boldsymbol p$ induces a choice of [[linear isomorphism|isomorphism]] $T_{\boldsymbol p}M \to \mathbb{R}^{n}$. > > Indeed, if we view $\mathscr{U}:=(U_{\boldsymbol p}^{\alpha}, \varphi_{\boldsymbol p}^{\alpha})_{\alpha \in I}$ as the collection of all charts on $M$ about $\boldsymbol p$, an element of $T_{\boldsymbol p}M$ takes the form of a map $\mathscr{U} \to \mathbb{R}^{n},$ so that $T_{\boldsymbol p}M = \{ \mathscr{U} \xrightarrow{f} \mathbb{R}^{n} \text{ obeying the transformation law} \}.$ If we choose our 'preferred chart' $(U, \varphi)$ about $\boldsymbol p$, we can define $\begin{align} T_{\boldsymbol p} M& \to \mathbb{R}^{n} \\ f & \mapsto f(U, \varphi) \end{align}$ and this is a (non-canonical) [[linear isomorphism]]. > The [[basis]] for $T_{p}M$ corresponding to to the [[free module|standard basis]] of $\mathbb{R}^{n}$ under the above identification is denoted $\frac{ \partial }{ \partial x^{i} } |_{\boldsymbol p}$ and called the **coordinate basis for $T_{\boldsymbol p}M$**. Any vector $\boldsymbol v \in T_{\boldsymbol p}M$ may be written uniquely "in local coordinates" as $\boldsymbol v = \sum_{i} v^{i} \frac{ \partial }{ \partial x^{i} } |_{\boldsymbol p}= v^{i} \frac{ \partial }{ \partial x^{i} }|_{\boldsymbol p} $ where Einstein notation has been employed. The $v^{i}$ are called the **components of $\boldsymbol v$** with respect to the coordinate basis. ^definition > [!equivalence] The tangent space in terms of derivations. > Note that the standard basis vectors of $T_{\boldsymbol p}M$ given by the local coordinates $(x_{i})$ and $(x_{i}')$ are related by [^1] $\frac{ \partial }{ \partial x_{i} } |_{\boldsymbol p} = \frac{ \partial x_{j}' }{ \partial x_{i} } |_{\boldsymbol p} \ \frac{ \partial }{ \partial x_{j}' } |_{\boldsymbol p}.$ > This means that a first-order derivation $\begin{align} a: C^{\infty}(M) & \to \mathbb{R} \\ f & \mapsto \sum_{i} a_{i} \frac{ \partial f }{ \partial x_{i} } (p) \end{align}$ >is well-defined independent of choice of [[coordinate chart]] $(x_{i})$. ^equivalence [^1]: Indeed, in the $(x_{i})$ coordinate system, $\frac{ \partial }{ \partial x_{i} } |_{\boldsymbol p}$ has components $\boldsymbol a = (0,\dots, \overbrace{1}^{i^{th} \text{ slot}},\dots, 0)$, which transform to components $\boldsymbol a'= \frac{ \partial \boldsymbol x' }{ \partial x_{j} } a_{j} = \frac{ \partial (x_{1}',\dots, x_{n}') }{ \partial x_{j} } a_{j}= \frac{ \partial (x_{1}',\dots,x_{n}') }{ \partial x_{i} } \cancel{a_{i}}^{=1}$ in the $(x_{i}')$ system, i.e., $\boldsymbol a'=\left( \frac{ \partial x_{1}' }{ \partial x_{i} },\dots, \frac{ \partial x_{n}' }{ \partial x_{i} } \right)$. Thus, we have $\frac{ \partial }{ \partial x_{i} } |_{\boldsymbol p}= \sum_{j} \frac{ \partial x_{j}' }{ \partial x_{i} } |_{\boldsymbol p}\ \frac{ \partial }{ \partial x_{j}' } |_{\boldsymbol p} .$ Note that this formula looks a lot like the multivariate chain rule. > [!note] Remark. > Recall that we also have an 'undergraduate' definition for [[tangent space to an embedded Euclidean submanifold]]. Given a [[coordinate chart]] $\varphi$, this definition defines $T_{\boldsymbol p}M$ to be the space spanned by the (partial) derivatives of $\varphi ^{-1}: \Omega \subset \mathbb{R}^{n} \to U \subset M$ with respect to each coordinate $x_{1},\dots,x_{n}$ of $\mathbb{R}^{n}$; each derivative is a column vector in a [[Jacobian]]. In this case, the coordinate basis notation above is no longer just notation... I think there are other reasons why the notation makes sense too, but we haven't seen them yet... ^note > [!justification] > The [[linear map|linearity]] of the transformation law is what ensures this is a [[vector space]]. Indeed, let $(U_{\alpha}, \varphi_{\alpha})_{\alpha \in I}$ denote all of the [[coordinate chart|coordinate charts]] about $\boldsymbol p$. Consider two tangent vectors $\boldsymbol v, \boldsymbol w \in T_{\boldsymbol p}M$, characterized by assigning coordinate vectors $(v_{1},\dots,v_{n})_{\alpha},(w_{1},\dots,w_{n})_{\alpha} \in \mathbb{R}^{n}$ to each chart $(U_{\alpha}, \varphi_{\alpha})$. > We may readily define $\boldsymbol v + \boldsymbol w$ to be the assignment of $(v_{1} + w_{1}, \dots, v_{n} + w_{n})_{\alpha} \text{ to } (U_{\alpha}, \varphi_{\alpha})$ for each $\alpha \in I$, for considering $\alpha, \alpha'$ and the [[transition map]] [[Jacobian]] $\boldsymbol J_{\boldsymbol p}$ we have $\boldsymbol J_{p}(\boldsymbol v + \boldsymbol w)= \boldsymbol J_{p}\boldsymbol v + \boldsymbol J_{p} \boldsymbol w$ which verifies that $\boldsymbol v+ \boldsymbol w$ obeys the transformation law. Similar reasoning goes for scalar multiplication. The rest of the [[vector space]] axioms are straightforward to check. ^justification ---- #### ---- #### 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 > ```