Calculus 3 (微積分3) Calculus of multivariable functions together with its profound applications in various subject areas are introduced in this half-semester course. Especially, topics about differentiation include partial derivatives, tangent planes, linear approximations, directional derivatives, and the chain rule. Also, applications such as finding extreme values and methods of Lagrange multipliers are discussed. Topics about integration involve definitions of multiple integrals and iterated integrals, Fubini’s theorem, change of variables, and how multiple integrals are used in probability. 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 sessions in which students are able to improve their skills in handling calculations and explore applications of Calculus under the guidance of teaching assistants. 這是一門半學期的課程，主要介紹多變數函數的微積分運算，和其在各領域豐富的應用。 微分主題包含多變數函數的偏微分、切平面、線性逼近、方向導數，和連鎖律；並討論求函數極值，Lagrange乘子法等應用問題。 積分部分涵蓋多重積分與逐次積分的定義，Fubini定理，和變數變換；並探究機率如何使用重積分。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計算並探索微積分的應用。

Upon completing this course, students are expected to be able to : 1. describe curves in plane or space using parametric equations and compute geometric quantities of curves by using differentiation and integration. 2. compute partial derivatives and understand their geometric meaning. 3. apply the chain rule to compute derivatives of composed functions in multivariables & directional derivatives. 4. determine local extrema of a given two-variable function. 5. use Lagrange multiplier to resolve constrained optimization problems. 6. compute double integrals by Fubini's Theorem and/or change of variables, understand the geometric meanings of double integrals. 學生修習本課程後，應具備下列能力： 1. 使用參數方程描述平面或空間中的曲線，並通過微分和積分計算曲線的幾何量 2. 計算偏導數並理解其幾何意義 3. 應用 chain rule 計算多變數組合函數的導數及方向導數 4. 判斷給定的二變數函數的局部極值 5. 使用 Lagrange multipliers解決受限優化問題 6. 通過Fubini定理和/或變數代換計算二重積分，並理解二重積分的幾何意義

Before taking this course, students should be skilled in high school mathematics and finish the online Precalculus Self Diagnostic Test https://cool.ntu.edu.tw/courses/50879 which is designed for NTU freshmen. Students are expected to attend and participate actively in lectures as well as discussion sessions. 學生應熟練高中數學，並完成為台大新生預備的線上「微積分學前自我檢測」https://cool.ntu.edu.tw/courses/50879。 學生應出席並積極參與課堂與習題課的討論。

To ensure maximum engagement and the highest learning outcomes, students are advised to allocate a minimum of 8 hours per week to independent study after class to the following tasks :  1. Assimilate and organize the course materials, including the definitions, theorems, and formulae introduced during lectures. 2. Review and reproduce the examples and problem-solving methodologies demonstrated by the instructor or teaching assistants. 3. Ensure the timely and thorough completion of all assigned work, including WeBWorK exercises, written homework, and Worksheets. 4. Reflect critically on any challenging areas or ambiguities encountered. Proactively pinpoint concepts needing clarification and immediately seek guidance from the instructor or teaching assistants. Students are strongly encouraged to utilize all designated office hours and support sessions. 為了達到最好的學習效果，鼓勵同學每周花 8 小時課後時間，依序完成以下任務 Step 1. 理解、整理並背下課堂中介紹的定義、定理與公式 Step 2. 複習課堂上的重要例題 Step 3. 寫 WeBWorK作業、紙本作業、學習單 Step 4. 回顧寫作業中遇到的瓶頸，如果有不完全理解的內容，盡快尋求助教和老師的協助。 強烈鼓勵同學參加 office hours 和助教習題課。

Textbook: James Stewart, Daniel Clegg, and Saleem Watson, Calculus Early Transcendentals, 9th edition, Metric Version. ISBN: 978-626-7533-06-2 Other useful websites 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:  http://www.math.ntu.edu.tw/~calc/cl_n_34455.html 數學知識網站: http://episte.math.ntu.edu.tw/cgi/mathfield.pl?fld=cal  免費線上數學繪圖軟體Desmos Calculator: https://www.desmos.com/calculator  免費知識型計算引擎: https://www.wolframalpha.com