Lecture 0
This is a series of Advanced Calculus notes of class Fall 2009 and Spring 2010 lectured by Prof. Jin-Tzu Chen 陳金次.
Calculus is the application of fundamental theorem of calculus.
Advanced calculus is a step stone toward modern mathematics.
Basic properties of continuous functions:
- Intermediate value theorem
- It has min/max value on closed interval
Starting from these properties, we may derive two mean value theorems.
From there we may derive the fundamental theorem of calculus.
Let \(f\) be a continuous function on \([a, b]\),
- If \(G(x) = \int_a^x \! f(t) \, \mathrm{d}t\), then \(\frac{\mathrm{d}}{\mathrm{d}x} \! G(x) = f(x)\)
- If \(\frac{\mathrm{d}}{\mathrm{d}x} \! F(x) = f(x)\), then \(\int_a^b \! f(x) \, \mathrm{d}x = F(b) - F(a)\)
高等微積分探討連續函數的基本性質、實數的性質、完備性。由公設建構實數
實數完備性 \(\rightarrow\) 微積分基本定理 \(\rightarrow\) 連續函數、黎曼可積
畢達哥拉斯的基本信條：給定線段 a, b 必定存在線段 c 同時量盡 a, b，即無理數不存在
歐氏幾何五大公設：
- 任意兩點可連一直線
- 線段可任意延長
- 所有直角皆相等
- 給定 O, A 兩點，存在有一圓 C 以 O 為圓心過 A 點
- 過線外一點，可作一直線平行已知直線（平行公設）
球面是非歐幾何（球面上過北極的線一定與赤道相交，牴觸平行公設）
Why cannot you compare imaginary numbers?
- Suppose \(i < 2i\)
- Subtract \(i\) on both sides, you get \(0 < i\)
- Multiply \(i\) on both sides, you get \(-1 < -2\), which is a contradiction
