---- Let $M$ be a [[smooth manifold]] of dimension $n$, and $\mathscr{A}$ a [[atlas of a manifold|smooth atlas]]. > [!definition] Definition. ([[tangent vector to a smooth manifold]] — coordinate-laden definition) > In order to specify a given tangent vector using coordinates, one specifies how it is represented in all possible coordinate systems and checks that it '[[transformation law|transforms like a tangent vector should]]' under a change of coordinates. Specifically: > > A **tangent vector** to $M$ at $\boldsymbol p \in M$ is an assignment to each [[coordinate chart|chart]] $(U, \varphi) \ni \boldsymbol p$ an $n$-tuple of Euclidean coordinates, in a manner that is '[[tensor of type (p,q)|tensorially]] compatible'. That is, if said assignment maps $\begin{align} > (U, \varphi)& \mapsto \boldsymbol a=(a_{1},\dots,a_{n}) \in \mathbb{R}^{n} \\ > (U', \varphi') & \mapsto \boldsymbol a' = (a_{1}',\dots, a_{n}') \in \mathbb{R}^{n} > \end{align}$ then we must have $\boldsymbol a'= D |_{\boldsymbol p} (\varphi '\circ \varphi ^{-1}) \boldsymbol a$, where $D |_{\boldsymbol p}(\varphi' \circ \varphi ^{-1})$ is the [[transition map]] [[derivative]] at $\boldsymbol p$. > > If we impose local coordinates $\varphi=(x_{1},\dots,x_{n})$, $\varphi'=(x_{1}',\dots,x_{n}')$ on $U$ and $U'$ and understand the notation $\frac{ \partial x_{i}' }{ \partial x_{j} } |_{\boldsymbol p}$ to mean the $ij$th entry of the [[transition map]] [[Jacobian]] $\boldsymbol J_{p}$, this [[transformation law]] is the assertion that $a_{i}'= \sum_{j=1}^{n} \frac{ \partial x_{i}' }{ \partial x_{j} } |_{\boldsymbol p} \ a_{j} $ > holds for all $i \in [n]$. > > Note that the standard basis vectors of $T_{\boldsymbol p}M$ given by the local coordinates $(x_{i})$ and $(x_{i}')$ are related by[^5] $\frac{ \partial }{ \partial x_{i} } |_{\boldsymbol p} = \frac{ \partial x_{j}' }{ \partial x_{i} } |_{\boldsymbol p} \ \frac{ \partial }{ \partial x_{j}' } |_{\boldsymbol p}$ > which is what one would expect in view of the [[chain rule]] from calculus. [^5]: One way to see this is to set up the equation $\boldsymbol a= \boldsymbol J \boldsymbol a'$ for $\boldsymbol J=\left( \frac{ \partial x_{j} }{ \partial x_{i}' } \right)$. Then multiply both sides by $\boldsymbol J ^{-1} =\left( \frac{ \partial x_{j}' }{ \partial x_{i} } \right)$. $\frac{ \partial }{ \partial x_{i} }$ will be the $i$th column of $\boldsymbol J ^{-1}$, which has the required form. > [!definition] Definition. ([[tangent vector to a smooth manifold]] — coordinate-free definition) > Our coordinate-free definition of a tangent vector will be phrased in terms of [[derivation|derivations]]. > > Note that the set $C^{\infty}(M)$ of [[smooth maps between manifolds|smooth functions]] $f:M \to \mathbb{R}$ is an associative [[algebra]]: smooth functions can be added, scaled, and pointwise-multiplied. One defines the tangent space $T_{\boldsymbol p}M$ as the space of "[[derivation|derivations]] *at $\boldsymbol p$*": [[linear map|linear maps]] $D:C^{\infty}(M) \to \mathbb{R}$ satisfying the Leibniz rule $\fa f, g \in C^{\infty}(M) :D(fg)=D(f) \cdot g(\boldsymbol p) + f(\boldsymbol p)D(g).$ > > Every tangent vector $v=a_{i} \frac{ \partial }{ \partial x_{i} } |_{\boldsymbol p}$ in the 'coordinate-laden' definition above induces a well-defined derivation $v: C^{\infty}(M) \to \mathbb{R}$ at $\boldsymbol p$ by evaluation the row vector $Df |_{p}$ at $\boldsymbol a$: $\begin{align} > v=a_{i} \frac{ \partial }{ \partial x_{i} } |_{\boldsymbol p} : C^{\infty}(M) & \to \mathbb{R} \\ > (f:M \to \mathbb{R}) & \mapsto D_{v} f |_{p}:=a_{i} \frac{ \partial f }{ \partial x_{i} } (\boldsymbol p). > \end{align}$ > The converse is true (meaning the two definitions indeed carry the same data), although we shall not prove it in this course. > Sometimes the space of derivations at a point $p$ is denoted $C^{\infty}_{p}(M)$. A third perspective on tangent vectors comes from considering velocity vectors of curves, as defined [[velocity vector of a parameterized curve|here]]. $d(\exp_{p})_{v=0}=\left( X_{p} \mapsto \frac{d}{dt}\exp_{p}(tX) |_{t=0} \right)$ # Old [[tangent space at a point of a smooth manifold#^definition]] > [!justification] > It is worth checking in Euclidean space why this definition is well-motivated. > Consider a local [[parameterized curve|curve]] $\boldsymbol x(t)=\big( x_{1}(t), \dots, x_{n}(t) \big)$ in $\mathbb{R}^{n}$, put $\boldsymbol x(\boldsymbol 0):= \boldsymbol p$. Its [[velocity vector of a parameterized curve|tangent vector]] at $\boldsymbol p$ is $\boldsymbol x'(\boldsymbol 0)=\boldsymbol a \in \mathbb{R}^{n}$. > Now suppose $\boldsymbol y=\boldsymbol y(\boldsymbol x)$ is a local change of coordinates of the ambient space (i.e., a [[Euclidean diffeomorphism|diffeomorphism]] of neighborhood of $\boldsymbol p$ in $\mathbb{R}^{n}$ onto its image). Then, using the [[chain rule]], $\frac{d}{dt} |_{t=0} \boldsymbol y \circ \boldsymbol x(t)= \boldsymbol y' \big( \boldsymbol x(\boldsymbol 0) \big) \boldsymbol x'(\boldsymbol 0)=\boldsymbol y'(\boldsymbol p) \boldsymbol a=\sum_{i=1}^{n} \frac{ \partial \boldsymbol y }{ \partial x_{i} }(\boldsymbol p) \ a_{i} ,$ where $\frac{ \partial \boldsymbol y }{ \partial x_{i} }(\boldsymbol p)$ is the $i$th column of the [[Jacobian]] $\boldsymbol J_{p} \in \mathbb{R}^{n \times n}$ of $\boldsymbol y$ at $\boldsymbol p$. > The idea is to use this '(contravariant) tensoriality' to motivate an abstract definition — a tangent vector will be one that [[transformation law|transforms]] 'like a tangent vector should' under a change of coordinates. ^justification # Old definition A **tangent vector** to $M$ at $\boldsymbol p \in M$ is an assignment to each [[coordinate chart|chart]] $(U,\varphi) \ni \boldsymbol p$ an $n$-tuple of "coordinates" $\boldsymbol a =(a_{1},\dots,a_{n}) \in \mathbb{R}^{n}$ so that if we take another [[coordinate chart|chart]] $(U', \varphi') \ni \boldsymbol p$ — so that there are [[coordinate chart|local coordinates]] $(x_{i})_{i =1}^{n}, (x_{i}')_{i=1}^{n}$ on $U,U'$ respectively — then $\boldsymbol a'= \boldsymbol J_{p} \boldsymbol a$, where $\boldsymbol J_{p}= \left[ \frac{ \partial x_{i}' }{ \partial x_{j} } |_{\boldsymbol p} \right]_{i,j=1}^{n}$ is the [[Jacobian]] of the [[transition map]] $\varphi' \circ \varphi ^{-1}$. That is, the transformation law $a_{i}' = \sum_{j=1}^{n} \frac{ \partial x_{i}' }{ \partial x_{j} } |_{\boldsymbol p} \ a_{j}$is obeyed for all $i \in [n]$. ---- #### ---- #### 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 > ``` Recall the notion of a [[tangent vector to a regular surface|tangent vector to smooth embedded Euclidean submanifold]] $M$ at a point $\boldsymbol p$: it is merely the [[velocity vector of a parameterized curve|velocity (tangent) vector]] $\gamma'(0)$ to a [[parameterized curve|local curve]] $\gamma:(-\varepsilon, \varepsilon) \to M$ with $\gamma(0)=\boldsymbol p$. ![[Pasted image 20250402143522.png|300]]