我先说好，这个是用来理解Stoke’s 定理，而且很多是平时的想法和笔记想到啥写啥，尽量严谨，但是肯定还有很多不严谨的地方
$§0$ Euclidean space
基本定义的构建
$§1$ Differentiation
Definition(Limit): Let $f:A ⊂ \mathbb{R}^{n} → \mathbb{R}^{m}$ ,where A is an open set, $x_{0}$ be in A or a boundary point of A, N be a neighborhood of $b ∈ \mathbb{R}^{m}$ . We say \textbf{f is eventually in N as x approaches x_{0}} if exists a neighborhood U of $x_{0}$ such that $x ≠ x_{0}, x ∈ U, and x ∈ A$ imply $f(x) \in N$. We say $f(x) approaches b$ as x approaches $x_{0}$ or in symbols, $$ lim_{x → x_{0}} f(x) = b $$
($ϵ & δ$)
Definition(Continuity) Let $f: A ⊂ \mathbb{R}^{n} → \mathbb{R}^{m}$ be a given function with domain A. We say f is continuous at $x_{0}$ if and only if $$ lim_{x → x_{0}} f(x) = f(x_{0}) $$