## CALCULUS (4)

#### Department of Mathematics Kwok-Wing Tsoi (蔡國榮)

### 課程簡介

**中文課程名稱：**微積分4

**英文課程名稱：**CALCULUS (4)

**授課教師：**Kwok-Wing Tsoi (蔡國榮)

**學分數：**2 學分

**開課單位：**Department of Mathematics

**建立日期：**2023 年 2 月

### 課程概述

In this final module of Calculus (MATH4009), we will develop calculus on “vector fields”. Vector fields are vector-valued functions arise naturally from Physics and we will discuss how to make sense of integrals of them over curves and surfaces. Topics to be discussed include:

- Line integrals and Green’s Theorem

- Conservative of vector fields

- Surface integrals and Flux

- Stokes’ and Divergence Theorem

In particular, Green’s, Stokes' and Divergence Theorem can be regarded as a vast generalisation of the Fundamental Theorem of Calculus, for line and surface integrals. As an application, we will derive the Gauss' Law that describes the flux of an inverse square field across a closed surface.

Finally, to complete the discussion on limits of a function or a (infinite) sum of functions in the course of the study of Calculus, the definitions of limits of sequences and series are also introduced, which provide the theoretical basis of the introduction of a “power series”. “Power series” is a generalization of polynomials and can be used to represent elementary as well as more general functions, which paves the way for more advanced analysis of functions, necessary in practical applications.

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. This course also provides discussion sections in which studentsareable to make their skills in handling calculations in Calculus more proficient under the guidance of our teaching assistants.

### 課程目標

On successful completion of this module students should be able to:

(1) Parametrise curves and surfaces in Cartesian and other coordinates, including polar, cylindrical and spherical coordinates

(2) Understand and be able to calculate line, surface integrals with respect to various coordinate systems.

(3) Understand and prove properties of a conservative vector field

(4) State the Green's, Divergence and Stokes' Theorems and use them to aid calculations

(5) Apply these techniques to problems in mechanics (work done, circulation and flux)

(6) Analyse convergence and divergence of sequences and series

(7) Apply basic properties and calculus of a power series

(8) State and apply the Taylor's Theorem to resolve problems about smooth functions

(9) Approximate an infinite series by a partial sum and be able to estimate the error incurred

### 課程要求

Assumed knowledge :

- MATH4006, 4007, 4008,

- Basic trigonometry, vector geometry,

- Determinants of 2x2 and 3x3 matrices (knowledge in linear algebra will be useful but not necessary)