---- - Let $M$ be a [[smooth]] $k$-[[differentiable Euclidean submanifold (with or without boundary)|manifold]] in $\rrn$; - Let $p \in M$; - Let $\alpha: U \to V$ be a [[coordinate patch]] on $M$ about $p$. > [!definition] Definition. ([[tangent space to an embedded Euclidean submanifold]]) > Define the **tangent space to $M$ at $p$** as $T_{p}(M) = \{ (p; D\alpha(x_{0}))v : v \in \rrk \},$ > where $x_{0}=\alpha ^{-1}(p)$. > [!generalization] > [[tangent space at a point of a smooth manifold]] ^generalization > [!intuition] > ![[CleanShot 2023-02-10 at 16.33.02.jpg]] > ![[CleanShot 2023-02-10 at 16.33.52.jpg]] **Remark.** This definition does not depend on the choice of [[coordinate patch]] — see below.![[CleanShot 2023-02-10 at 16.32.07.jpg]] > [!note] Discussion. > The equation of the tangent *plane* to a [[differentiable Euclidean submanifold|regular surface]] $S$ which is the [[graph]] of a $C^{r}$ function $z=f(x,y)$, at the point $p_{0}=(x_{0},y_{0})$, is given by the solution set of the equation $z=f(x_{0},y_{0}) + \frac{ \partial f }{ \partial x }|_{(x_{0},y_{0})}(x-x_{0}) + \frac{ \partial f }{ \partial y } |_{(x_{0},y_{0})}(y-y_{0}).$ To see this, start with the tangent space to $S$ at $(x_{0},y_{0})$. Since $S$ is the [[graphs of smooth functions are regular surfaces|graph of a]] $C^{r}$ function, we can cover it with a single [[coordinate patch]]: namely, the patch $\alpha(u,v):=(u,v, f(u,v))$. So then the tangent space to $S$ at our point is >$P=\span \left( \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ \frac{ \partial f }{ \partial x }|_{(x_{0},y_{0})} & \frac{ \partial f }{ \partial y } |_{(x_{0},y_{0})} \end{bmatrix} \right).$ This a two-dimensional [[linear subspace]] of $\mathbb{R}^{2}$, i.e., a [[plane]] passing through the origin. A normal vector to $P$ can be obtained by negating the [[cross product]] of the [[basis]] vectors formed by the columns of $P$ $\begin{align} (A,B,C)= & ( \frac{ \partial f }{ \partial x } |_{(x_{0},y_{0})}, \frac{ \partial f }{ \partial y }|_{(x_{0},y_{0})} ,-1) \end{align}$ and now the [[plane#Point-Normal Equation]] tells us that the plane with normal vector $(A,B,C)$ that passes through $(x_{0},y_{0},\underbrace{f(x_{0},y_{0}))}_{z_{0}}$ — i.e., the plane tangent to $S$ at our point — is $\begin{align} \{(x,y,z) \in& \mathbb{R}^3 : \vec n \cdot (\vec p - \vec p_0) = 0\} \\ =& \{ (x,y,z) \in \mathbb{R}^3 : A(x - x_0) + B(y-y_0) + C(z - z_0)=0\} \\ = & \{ (x,y,z) \in \mathbb{R}^{3} : \frac{ \partial f }{ \partial x } |_{(x_{0},y_{0})},(x-x_{0}) + \frac{ \partial f }{ \partial y }|_{(x_{0},y_{0})}(y-y_{0} ) \\ & -(z-z_{0}) = 0\} \\ = & \{ (x,y,z) \in \mathbb{R}^{3} : \frac{ \partial f }{ \partial x } |_{(x_{0},y_{0})},(x-x_{0}) + \frac{ \partial f }{ \partial y }|_{(x_{0},y_{0})}(y-y_{0} ) \\ & + f(x_{0},y_{0}) = z\} \end{align}$ which is exactly the equation we were looking for. ^603a37 ---- #### ---- #### 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 > ```