0x024 Vector Analysis

Multivariable Functions

Definition (multivariable function) A multivariable function is a function $f: D \subset R^n \to R^m$ assigns a vector $f(x) \in R^m$ to each $x \in D$

$$f(x) = (f_1 (x), f_2(x), …, f_m(x))$$

Definition (level set) Let $f: D \subset R^n \to R$ be a real valued function and $c$ a number. The set of all points $(x_1, x_2, …, x_n)$ in the domain $D$ where $f(x_1, x_2, …, x_n) = c$ is called the $c$ level set of $f$

Definition (continuity) A function $f: D \subset R^n \to R^m$ is continuous at $X$ if for every tolerance $\epsilon > 0$, there exists a precision $\delta > 0$ such that

$$ (\forall y:|| x-y|| < \delta) \implies ||f(x) – f(y)|| < \epsilon$$

A function is continuous on $D$ if it is continuous at every point in $D$

Proposition (elementwise continuity implies continuity) A function $F: D \subset R^n \to R^m$, denoted

$$F(X) = (f_1(X), f_2(X), …, f_m(X))$$

is continuous on $D$ if and only if each component $f_j: D \subset R^n \to R$ is continuous

Differentiation

Differentiable Functions

Definition (differentiability) A function from $R^2 \to R$ defined on an open disk centered at $A$ is differentiable at $A$ if $f(A+H) -f(A)$ can be well approximated by a linear function $l$ in the following sense:

$$\lim_{||H|| \to 0} \frac{f(A+H) – f(A) – l(H)}{||H||} = 0$$

Reference

[1] Tao, Terence. Analysis. Vol. 1. Hindustan Book Agency, 2006.

[2] Tao, Terence. Analysis. Vol. 2. Hindustan Book Agency, 2006.

[3] Abbott, Stephen. Understanding analysis. Vol. 2. New York: Springer, 2001.

[4] Lax, Peter D., and Maria Shea Terrell. Multivariable Calculus with Applications. Springer, 2017.

[5] 杉浦光夫. “解析入門 I.” 東京大学出版会

[6] 杉浦光夫. “解析入門 II.” 東京大学出版会