Caculus: Early Transcendentals

這是關於書籍《Caculus》的讀書筆記。

1 書籍信息

• 書籍名稱：Calculus: Early Transcendentals
• 作者：James Stewart
• 版次：6th

2 簡介

主要內容：

• Diagnostic Tests： 四個測試，分別關於基礎代數、解析幾何、函數以及三角學。
• A Preview of Calculus：一個關於微積分的簡介以及一些可以引發興趣的問題。

1. Functions and Models：函數及數學模型，首先使用語言、數字、可視化、代數的方法來表示函數，之後利用數學模型使用上述四個方法複習包括指數函數、對數函數等在內的標準函數。
2. Limits and Derivatives：極限及導數，通過前面關於切線及速度的例子來說明極限。極限將通過四種方式來說明：描述、圖形、數字、代數。之後將介紹導數及其例子。
3. Differentiation Rules：求導法則，所有的基本函數，包括指數函數，對數函數，反三角函數的求導規則都在這裡介紹。
4. Applications of Differentiation：求導的應用。使用中值定理對極限和曲線的形狀的判定進行推理。
5. Integrals：The area problem and the distance problem serve to motivate the definite integral。
6. Applications of Integration：Here I present the applications of integration—area, volume, work, average value—that can reasonably be done without specialized techniques of integration. General methods are emphasized.不使用積分方法而使用常規的方法來求積分區域、體積、work、平均值。
7. Techniques of Integration：標準積分方法的介紹。
8. Further Applications of Integration：積分的進階應用，關於弧長度及表面積，以及關於生物學、經濟學、物理學方面的一些應用。
9. Differential Equations：Modeling is the theme that unifies this introductory treatment of differential equations. Direction fields and Euler』s method are studied before separable and linear equations are solved explicitly, so that qualitative, numerical, and analytic approaches are given equal consideration. These methods are applied to the exponential, logistic, and other models for population growth. The first four or five sections of this chapter serve as a good intro-duction to first-order differential equations. An optional final section uses predator-prey models to illustrate systems of differential equations.
10. Parametric Equations and Polar Coordinates: This chapter introduces parametric and polar curves and applies the methods of calculus to them.
11. Infinite Sequences and Series: The convergence tests have intuitive justifications (see page 697) as well as formal proofs. Numerical estimates of sums of series are based on which test was used to prove conver-gence. The emphasis is on Taylor series and polynomials and their applications to physics. Error estimates include those from graphing devices.
12. Vectors and The Geometry of Space: 三維分析幾何和向量被分為兩章，這一章介紹向量、點、線、面和表面。The material on three-dimensional analytic geometry and vectors is divided into two chapters. Chapter 12 deals with vectors, the dot and cross products, lines, planes, and surfaces.
13. This chapter covers vector-valued functions, their derivatives and integrals, the length and curvature of space curves, and velocity and acceleration along space curves, culminating in Kepler』s laws.
14. Partial Derivatives： Functions of two or more variables are studied from verbal, numerical, visual, and algebraic points of view. In particular, I introduce partial derivatives by looking at a specific column in a table of values of the heat index (perceived air temperature) as a function of the actual temperature and the relative humidity. Directional derivatives are estimated from contour maps of temperature, pressure, and snowfall.
15. Multiple Integrals: 多重積分。Contour maps and the Midpoint Rule are used to estimate the average snowfall and average temperature in given regions. Double and triple integrals are used to compute probabilities, surface areas, and (in projects) volumes of hyperspheres and volumes of intersections of three cylinders. Cylindrical and spherical coordinates are introduced in the context of evaluating triple integrals.
16. Vector Calculus：Vector fields are introduced through pictures of velocity fields showing San Francisco Bay wind patterns. The similarities among the Fundamental Theorem for line integrals, Green』s Theorem, Stokes』 Theorem, and the Divergence Theorem are emphasized.
17. Second-Order Differential Equations：Since first-order differential equations are covered in Chapter 9, this final chapter deals with second-order linear differential equations, their application to vibrating springs and electric circuits, and series solutions.

3 關鍵詞

• algebra: 代數
• Caculus: 微積分
• Derivative：導數
• Differentiation：求導
• Integral：積分
• Integration：積分(名詞)
• Mean Value Theorem：中值定理
• trigonometry: 三角學