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微積分4
CALCULUS (4)
余正道   109下

課程概述
這是一門半學期的課程,分為「向量微積分」與「泰勒展式」兩大主題。
「向量微積分」討論的主體是定義域、值域皆屬於 r^n的函數(又稱為「向量場」)。我們將定義如何在曲線或曲面上積分向量場,並介紹作用在向量場上的兩種微分運算,「散度」與「旋度」。課程將解釋green定理、stokes定理、散度定理如何結合向量場的微分與積分運算,而被理解為高維度的「微積分基本定理」。應用上,我們將推導電磁學中的 gauss 定律,計算封閉曲面的電通量。
「泰勒展式」這主題推廣「極限」的概念,探討如何以多項式逼近複雜的函數。為了達到這個目的,我們將介紹無窮級數與冪級數的收斂性,利用泰勒定理估計餘項,進而推導出常見函數的泰勒展式。最後我們將示範多項式逼近的實際應用。
課堂上將講解定義並推導重要定理,以培養學生邏輯推理與分析能力;同時會示範微積分在各領域的應用,幫助學生將微積分與其他專業科目結合。本課程還設有習題課,學生將在助教的帶領下熟練微積分的計算。

this half-semester course contains two main topics which are vector calculus and taylor series.
vector calculus deals with functions whose domain and range are both in r^n which are also called vector fields. we will make sense of integrating vector fields over curves and surfaces and introduce two differential operators acting on them, the divergence and curl. we will explain how green’s theorem, stokes’ theorem, and the divergence theorem connect the integration and differentiation of vector fields and are regarded as higher-dimensional fundamental theorem of calculus. as an application, we will derive gauss law in electromagnetism that describes the flux of an inverse square field across a closed surface.
the topic of taylor series extends the concept of limit to approximating complicated functions by polynomials. we will introduce the convergence of series and power series, use taylor’s theorem to estimate remainder terms, and derive taylor series for common functions. finally, applications of approximating functions by polynomials are illustrated.
definitions are discussed and the most important theorems are derived in the lectures with a view to help students to develop their abilities in logical deduction and analysis. practical applications of calculus in various fields are illustrated in order to promote a more organic interaction between the theory of calculus and students own fields of study. this course also provides discussion sections in which students are able to make their skills in handling calculations in calculus more proficient under the guidance of our teaching assistants.

課程目標
修完本課程學生能熟悉微積分工具,並應用在各學科。「微積分1, 2, 3, 4」將奠定學生修讀工程數學、分析、微分方程等進階課程的基礎。

students would be familiar with calculus as a tool and be able to apply it in various subjects after finishing this course. "calculus 1, 2, 3, 4" provide the basis for the study of various advanced courses like engineering mathematics, analysis and differential

課程要求
修這門課以前,學生要熟練高中數學。學生應積極參與課堂和習題課的活動與討論。

students participating in the course should be already skilled in high school mathematics. they are expected to attend and participate actively in lectures as well as discussion sessions.

指定閱讀
textbook: james stewart, daniel clegg, and saleem watson, calculus early transcendentals, 9th edition.

參考書目
其他相關資訊
微積分統一教學網站: http://www.math.ntu.edu.tw/~calc/default.html
台大微積分考古題: u+a0http://www.math.ntu.edu.tw/~calc/cl_n_34455.html
數學知識網站:u+a0http://episte.math.ntu.edu.tw/cgi/mathfield.pl?fld=cal
免費線上數學繪圖軟體desmos calculator:u+a0https://www.desmos.com/calculator
免費知識型計算引擎:u+a0https://www.wolframalpha.com

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