Vector Fields and Scalar Functions
Definition 1. Let $D \subset \mathbb{R}^m$ for $m \in \mathbb{N}$. Then
(1) A scalar function on a domain set $D$ is a function $f: D \rightarrow \mathbb{R}$.
(2) A vector-valued function, or vector function, or vector field on $D$ is a function $\textbf{f}: D \rightarrow \mathbb{R}^n$ defined by $\textbf{f}(\textbf{x}) = (f_1(\textbf{x}), f_2(\textbf{x}), \cdots, f_n(\textbf{x}))$ for scalar functions $f_1, ..., f_n$ on $D$ for $\textbf{x} \in D$.
함수값이 스칼라, 여기서는 $\mathbb{R}$의 원소이면 scalar function, 함수값이 벡터, 여기서는 $\mathbb{R}^n (n > 1)$이면 vector function이라고 한다.
Level Curve, Level Surface
Definition 2. The set of points in the plane where a function $f(x, y)$ has a constant value $f(x, y) = c$ is called a level curve or level surface of $f$.
Limit and Continuity of Two-Variables Scalar Fields
Definition 3. We say that a function $f(x, y)$ approaches the limit $L$ as $(x, y)$ approaches $(x_0, y_0)$, and write $$\lim_{(x, y) \to (x_0, y_0)} f(x, y) = L$$ if, for every number $\varepsilon > 0$, there exists a corresponding number $\delta > 0$ such that for all $(x, y)$ in the domain of $f$, $$|f(x, y) - L| < \varepsilon \text{ whenever } 0 < \sqrt{(x - x_0)^2 + (y - y_0)^2} < \delta.$$
Note. $$\lim_{(x, y) \to (x_0, y_0)} x = x_0, \\ \lim_{(x, y) \to (x_0, y_0)} y = y_0, \\ \lim_{(x, y) \to (x_0, y_0)} k = k \\ (\text{for any number } k).$$
Definition 4. A function $f(x, y)$ is continuous at the point $(x_0, y_0)$ if
(1) $f$ is defined at $(x_0, y_0)$,
(2) $\lim_{(x, y) \to (x_0, y_0)} f(x, y)$ exists,
(3) $\lim_{(x, y) \to (x_0, y_0)} f(x, y) = f(x_0, y_0)$.
A function is continuous if it is continuous at every point of its domain.
Two-variables 이상의 multi variable scalar function도 마찬가지의 방법으로 극한과 연속을 정의한다.
Two-Path Test for Nonexistence of a Limit
If a function $f(x, y)$ has different limits along two different paths in the domain of $f$ as $(x, y)$ approaches $(x_0, y_0)$, then $\lim_{(x, y) \to (x_0, y_0)} f(x, y)$ does not exist.
single variable에서는 방향이 좌, 우밖에 없었으므로 좌극한, 우극한만 고려하면 됐었다. 그러나 two-variable, 나아가 multi variable에서는 주어진 점에 여러 방향으로 접근할 수 있음을 유의해야 한다. 거꾸로 말하면 극한이 존재한다면 어떤 방향에서 접근해도 동일한 극한이 존재해야 한다는 뜻이다. 즉 서로 다른 두 방향에서 접근할 때 극한값이 다르다면 주어진 점에서 극한이 존재하지 않음을 증명할 수 있다.
Theorem 1
Theorem 1. If $f$ is continuous at $(x_0, y_0)$ and $g$ is a single-variable function continuous at $f(x_0 , y_0)$, then the composite function $h = g ∘ f$ defined by $h(x, y) = g(f(x, y))$ is continuous at $(x_0, y_0)$.