教學時程
微積分及應用
因舊版課程無指定課堂作業與考試,因此統整所有作業、講義、考試內容合併列出。
微積分及應用
因舊版課程無指定課堂作業與考試,因此統整所有作業、講義、考試內容合併列出。
本課程被設計為自學微積分。內容被劃分為以下的“章節”。
章 節 |
課程單元 |
|
前言Preface |
0 |
電子資料表The Spreadsheet |
1 |
哲學,數字和函數 Philosophy, Numbers and Functions |
2 |
指數函數與三角函數 The Exponential Function and Trigonometric Functions |
3 |
向量,內積,矩陣乘法和距離 Vectors, Dot Products, Matrix Multiplication and Distance |
4 |
平行四邊形的面積,行列式,體積,超體積以及向量積 Area of a Parallelogram, Determinants, Volume and Hypervolume, the Vector Product |
5 |
二維和三維中的向量與幾何 Vectors and Geometry in Two and Three Dimensions |
6 |
微分函數,積分和微分 Differentiable Functions, the Derivative and Differentials |
7 |
根據定義的積分計算 Computation of Derivatives from their Definition |
8 |
根據法則的積分演算 Calculation of Derivatives by Rule |
9 |
向量場的積分和極座標系中的梯度 Derivatives of Vector Fields and the Gradient in Polar Coordinates |
10 |
高等積分,泰勒級數,二次逼近和逼近精確度 Higher Derivatives, Taylor Series, Quadratic Approximations and Accuracy of Approximations |
11 |
多維空間中的二次逼近 Quadratic Approximations in Several Dimensions |
12 |
微分的應用:直接用於線性逼近 Applications of Differentiation: Direct Use of Linear Approximation |
13 |
解方程Solving Equations |
14 |
極值Extrema |
15 |
曲線Curves |
16 |
幾個重要的例子和一個物理學中的表述 Some Important Examples and a Formulation in Physics |
17 |
積的規則和向量的微分 The Product Rule and Differentiating Vectors |
18 |
複數和複數方程 Complex Numbers and Functions of Them |
19 |
反導數或不定積分 The Anti-derivative or Indefinite Integral |
20 |
曲線下方的面積及其多種推廣 The Area under a Curve and its Many Generalizations |
21 |
微積分在一維中的基本定理 The Fundamental Theorem of Calculus in One Dimension |
22 |
微積分在多維中的基本定理:加性測度,斯托克斯定理以及散度定理 The Fundamental Theorem of Calculus in Higher Dimensions; Additive Measures, Stokes Theorem and the Divergence Theorem |
23 |
線積分化簡為通常積分和相對簡化形式 Reducing a Line Integral to an Ordinary Integral and Related Reductions |
24 |
面積分化簡為多重積分和雅可比矩陣 Reducing a Surface Integral to a Multiple Integral and the Jacobian |
25 |
數值積分法Numerical Integration |
26 |
微分方程的數值解 Numerical Solution of Differential Equations |
27 |
積分練習Doing Integrals |
28 |
介紹電場和磁場 Introduction to Electric and Magnetic Fields |
29 |
磁場,電磁感應和電動力學 Magnetic Fields, Magnetic Induction and Electrodynamics |
30 |
級數Series |
31 |
平面,曲面,體積的積分練習 Doing Area, Surface and Volume Integrals |
32 |
一些線性代數Some Linear Algebra |
33 |
二階微分方程Second Order Differential Equations |