Vector Field

Statements

Definition. A vector field $X$ on an open subset $U$ of ${\mathbb{R}}^n$ is a vector-valued function $X:U\to {\mathbb{R}}^m$ that assigns to each point $p\in U$ a vector ${\mathbb{R}}^m$.

Remark. From the corollary that tangent vectors are derivations, the image of vector field $X:U\to T_p({\mathbb{R}}^n)$ at $p\in U$, denoted as $X_p$, has basis $\{\partial /\partial x^i{|}_p\}$. Thus, the vector $X_p$ is a linear combination $X_p={\sum }_{i=1}^n{a}^i(p){\frac{\partial }{\partial x^i}|}_p,p\in U,a^i(p)\in \mathbb{R}.$

References

@Tu11, pp. 14 and @Boo75 Section 2.5