📂Geometry

Tangent Vectors on Simple Surfaces

Definition¹

Consider a point $p = \mathbf{x}(a,b)$ on a coordinate patch $\mathbf{x} : U \to \mathbb{R}^{3}$. If a vector $\mathbf{X}$ is the velocity vector at $p$ of some curve $\mathbf{x}(U)$ on the curve passing through $p$, then $\mathbf{X}$ is defined as the tangent vector to the simple surface $\mathbf{x}$.

In other words, if for any arbitrary $\epsilon > 0$, there exists a suitably short curve $\boldsymbol{\alpha} : (-\epsilon, \epsilon) \to \mathbf{x}(U) \subset \mathbb{R}^{3}$ that satisfies the following condition

$$ \boldsymbol{\alpha}(0) = p \quad \text{and} \quad \boldsymbol{\alpha}^{\prime}(0) = \left. \dfrac{d \boldsymbol{\alpha}}{d t}\right|_{t=0}= \mathbf{X} \quad \text{and} \quad \boldsymbol{\alpha} (t) = \mathbf{x}\left( \alpha_{1}(t), \alpha_{2}(t) \right) $$

then $\mathbf{X}$ is called the tangent vector to the simple surface $\mathbf{x}$.

Description

The set of tangent vectors defined as above becomes a vector space by the theorem below and is actually the same as the tangent plane. Therefore, the tangent plane is called the tangent space.

• The set of all tangent vectors to $M$ at point $p \in M$ on the surface $M$ is denoted as $T_{p}M$ and called the tangent space.

$$ T_{p}M = \left\{ \text{all vectors tangent to } M \text{ at } p \right\} $$

This way of definition is also used when defining tangent vectors on a differential manifold. At first glance, the reason for defining it through the consideration of such curves $\boldsymbol{\alpha}$ may not be readily accepted, but as you continue to study differential geometry or encounter generalizations to manifolds, it will naturally be understood.