Tangent Space

linear-logic

Definition

Def

Let $M$ be a smooth manifold (a surface) inside ${\mathbb{R}}^n$, and let $p$ be a point on $M$.

Consider all possible smooth curves $\gamma :(-\epsilon ,\epsilon )\to M$ such that $\gamma (0)=p$. The tangent space $T_{p}M$ is the set of all possible velocity vectors of these curves at $t=0$:

$$T_{p}M=\{v\in {\mathbb{R}}^n\mid \exists \text{curve }\gamma \text{ on }M\text{ s.t. }\gamma (0)=p\wedge {\gamma }^{\prime }(0)=v\}$$

Tangent Space [ Gradient Definition]

Let a surface be defined by the level set of a function $F(x)=c$.

$$S=\{x\in {\mathbb{R}}^n\mid F(x)=c\}$$

The tangent space at point $p\in S$ is the null space (kernel) of the derivative (gradient) of $F$ at $p$:

$$T_{p}S=\{v\in {\mathbb{R}}^n\mid ⟨\nabla F(p),v⟩=0\}$$