[Coursera] Vector Calculus for Engineers
믿고보는 제프리 채스노프 교수의 강좌다. 이전에 강좌를 보면 다음과 같다.
1. [커세라]이과생을 위한 행렬 대수(Matrix Algebra for Engineers)
2. [커세라] 이과생을 위한 미분 방정식(Differential Equations for Engineers)
벡터 미적분은 상대론, 전자기학, 유체역학 등 모든 이공계의 기초다.
이 강좌는 벡터 미적분학(Vector Calculus)으로 공학도라면 반드시 알아야 할 내용을 다루고 있다. 벡터 미적분학 혹은 다변수(한 객체를 복수의 변수로 표현하니까) 미적분학 혹은 미적분-3이라고 부르기도 한다.
5주짜리 강좌로 기본 이론과 응용을 다룬다. 첫주는 스칼라와 벡터 장(fields), 둘째주는 장의 미분(differentiating fields), 셋째주는 장의 중적분(multiple integrals)과 좌표계(Polar/Cylindrical/Spherical Coordinates), 넷째주는 선적분(Line Integrals)과 면적분(Surface Integrals), 다섯째주는 그래디언트 정리(gradient theorem), 발산정리(the divergence theorem), 스토크스 정리(Stokes’ theorem) 등을 포함한 벡터 미적분의 기본 이론을 다룬다. 이 적분 정리들은 전자기학, 유체역학등 공학 분야에 적용되는 기본정리들 이다.
아울러 다음과 같은 벡터 미적분의 응용을 다룰 것이다.
- 최소 자승법(The method of least sguares)
- 3중적 법칙과 이상기체 법칙(Ideal Gas law)
- 전자기 파(Electromagnetic Waves)
- 원심력: 뉴튼의 운동법칙(Newton's equations for central force)
- 호의 길이 및 표면적(Arc lengths and surface areas)
- 일과 에너지(Work and energy theorem)
- 질량과 전자기 플럭스(Mass and electromagnetic flux)
- 에너지 보존 법칙(Law of energy conservation)
- 연속 방정식(The continuity equations)
- 맥스웰 방정식의 미분형(Differential form of Maxwell's equations)
강의록은 아래의 링크를 통해 얻을 수 있다.
http://www.math.ust.hk/~machas/vector-calculus-for-engineers.pdf
----------------------------------------
[강의목차]
----------------------------------------
[강의목차]
------------------------
W1-2: 1강. 벡터(Vectors)
- 벡터연산 결합법칙(Associative Law)
- 삼각형 중점 원리(Triangle Midpoint Theorem)
W1-3: 2강. 직교좌표계(Cartesian Coordinates)
- 뉴튼 중력 힘(Newton's equation for the force between two masses)
W1-4: 3강. 스칼라 곱(Dot Product)
- 벡터 내적의 교환 및 분배(Commutative and Distributive Properties)
- 표준 단위 벡터 사이의 내적 특성(Dot Product between Standard Unit Vectors)
- 코사인 법칙(Law of Cosines)
- 행렬에 대하여(Do you know matrices?)
W1-5: 4강. 벡터 곱(Cross Product)
- 벡터 외적의 교환 및 분배(Commutative and Distributive Properties)
- 표준 단위 벡터 사이의 외적 특성(Cross Product Between Standard Unit Vectors)
- 결합 법칙(Associative Property)
W1-6: 연습문제:벡터(Practice Quiz: Vectors)
W1-7: 5강. 해석기하:직선(Analytic Geometry of Lines)
- 선의 매개 변수 식(Parametric Equation for a Line)
W1-8: 6강. 해석기하: 평면(Analytic Geometry of Planes)
- 면의 방정식(Equation for a Plane)
W1-9: 연습문제:해석기하(Practice Quiz: Analytic Geometry)
W1-10: 7강. 크로네커 델타와 리바이-씨비타 기호(Kronecker Delta & Levi-Civita symbol)
- 외적(Cross Product)
- 크로네커 델타 δ 항등식(Kronecker Delta δ Identities)
- 벡터연산 결합법칙(Associative Law)
- 삼각형 중점 원리(Triangle Midpoint Theorem)
W1-3: 2강. 직교좌표계(Cartesian Coordinates)
- 뉴튼 중력 힘(Newton's equation for the force between two masses)
W1-4: 3강. 스칼라 곱(Dot Product)
- 벡터 내적의 교환 및 분배(Commutative and Distributive Properties)
- 표준 단위 벡터 사이의 내적 특성(Dot Product between Standard Unit Vectors)
- 코사인 법칙(Law of Cosines)
- 행렬에 대하여(Do you know matrices?)
W1-5: 4강. 벡터 곱(Cross Product)
- 벡터 외적의 교환 및 분배(Commutative and Distributive Properties)
- 표준 단위 벡터 사이의 외적 특성(Cross Product Between Standard Unit Vectors)
- 결합 법칙(Associative Property)
W1-6: 연습문제:벡터(Practice Quiz: Vectors)
W1-7: 5강. 해석기하:직선(Analytic Geometry of Lines)
- 선의 매개 변수 식(Parametric Equation for a Line)
W1-8: 6강. 해석기하: 평면(Analytic Geometry of Planes)
- 면의 방정식(Equation for a Plane)
W1-9: 연습문제:해석기하(Practice Quiz: Analytic Geometry)
W1-10: 7강. 크로네커 델타와 리바이-씨비타 기호(Kronecker Delta & Levi-Civita symbol)
- 외적(Cross Product)
- 크로네커 델타 δ 항등식(Kronecker Delta δ Identities)
- 레바이-씨비타 ε 항등식(Levi-Civita ε Identities)
W1-11: 8강. 벡터 곱의 등가 연산공식(Vector Identities)
- 자코비 항등식(Jacobi Identity)
- 3차원의 라그랑지 항등식(Lagrange's Identity in Three Dimensions)
W1-12: 연습문제:벡터 대수(Practice Quiz: Vector Algebra)
- 자코비 항등식(Jacobi Identity)
- 3차원의 라그랑지 항등식(Lagrange's Identity in Three Dimensions)
W1-12: 연습문제:벡터 대수(Practice Quiz: Vector Algebra)
W1-13: 9강. 스칼라장과 벡터장(Scalar and Vector Field)
- 스칼라 장과 벡터 장의 예(Examples of Scalar and Vector Fields)
W1-14: 1주차 평가문제(Week One Assessment)
2주: 다변수 미분(Differentiation)
------------------------------------------
장(fields)의 최대최소값(곡률)을 찾는 문제를 풀기 위해 스칼라와 벡터장의 미분한다. 특히 다변수 벡터장은 편미분 한다. 다변수 미분의 연쇄법칙(chain rule)을 다룬다. 나블라(nabla 혹은 델 del) 𝛁 연산자와 함께 그래디언트(gradient, 𝛁), 다이버젼스(divergence, 𝛁·), 컬(curl, 𝛁✕), 라플라시언(Laplacian)이 어떻게 정의되는지 살펴볼 것이다.
벡터의 미분 특성(identities)를 배우고 크로네커 델타와 리바이-시비타 기호로 표현하는 방법을 알아본다. 벡터 연산자들이 맥스웰 방정식이라고 하는 자유공간에서 전자기파 방정식에 어떻게 활용되는지도 살펴본다.
W2-1: 2주차 공부할 내용 소개(Introduction)
W2-2: 10강. 편미분(Partial Derivatives)
- 편미분 계산법(Computing Partial Derivatives)
- 테일러 급수 확장(Taylor Series Expansions)
W2-3: 11강. 최소자승법(The Method of Least Squares)
- 최소자승법(Least-squares Method)
W2-4: 12강. 연쇄법칙(Chain Rule)
- 연쇄 법칙(Chain Rule)
W2-5: 13강. 3중 곱 법칙(Triple Product Rule)
W2-6: 14강. 3중 곱 법칙: 예제(Triple Product Rule: Example)
- 선형 함수의 3중곱(Triple Product Rule for a Linear Function)
- 4중곱 규칙(Quadruple Product Rule)
W2-7: 연습문제: 편미분(Practice Quiz: Partial Derivatives)
15강, Gradient
- Computing the Gradient
16강, Divergence
- Computing the Divergence
17강, Curl
- Computing the Curl
18강, Laplacian
W2-2: 10강. 편미분(Partial Derivatives)
- 편미분 계산법(Computing Partial Derivatives)
- 테일러 급수 확장(Taylor Series Expansions)
W2-3: 11강. 최소자승법(The Method of Least Squares)
- 최소자승법(Least-squares Method)
W2-4: 12강. 연쇄법칙(Chain Rule)
- 연쇄 법칙(Chain Rule)
W2-5: 13강. 3중 곱 법칙(Triple Product Rule)
W2-6: 14강. 3중 곱 법칙: 예제(Triple Product Rule: Example)
- 선형 함수의 3중곱(Triple Product Rule for a Linear Function)
- 4중곱 규칙(Quadruple Product Rule)
W2-7: 연습문제: 편미분(Practice Quiz: Partial Derivatives)
15강, Gradient
- Computing the Gradient
16강, Divergence
- Computing the Divergence
17강, Curl
- Computing the Curl
18강, Laplacian
- Computing the Laplacian
연습문제: The Del Operator
연습문제: The Del Operator
19강, Vector Derivative Identities
20강, Vector Derivative Identities (Proof)
- Vector Derivative Identities
- The Material Acceleration
- The Material Acceleration
21강, Electromagnetic Waves
- Wave Equation for the Magnetic Field
- Wave Equation for the Magnetic Field
연습문제: Vector Calculus Algebra
2주차 평가문제
WEEK 3: Integration and Curvilinear Coordinates
Scalar and vector fields can be integrated. We learn about double and triple integrals, and line integrals and surface integrals. Curvilinear coordinates, namely polar coordinates in two dimensions, and cylindrical and spherical coordinates in three dimensions, are used to simplify problems with cylindrical or spherical symmetry. We learn how to change variables in multidimensional integrals using the Jacobian of the transformation.
Video: IntroductionVideo: Double and Triple Integrals | Lecture 22
Reading: Computing the Mass of a Cube
Video: Example: Double Integral with Triangle Base | Lecture 23
Reading: Volume of a surface above a parallelogram
Practice Quiz: Multidimensional Integration
Video: Polar Coordinates | Lecture 24
Reading: Inverse Formula
Reading: Some Common Two-Dimensional Vectors
Video: Central Force | Lecture 25
Reading: Angular Momentum
Video: Change of Variables (single integral) | Lecture 26
Video: Change of Variables (double integral) | Lecture 27
Reading: Mass of a Disk
Reading: Gaussian Integral
Practice Quiz: Polar Coordinates
Video: Cylindrical Coordinates | Lecture 28
Reading: Del in Cylindrical Coordinates
Reading: Divergence of a Unit Vector
Reading: Divergence and Curl of the Unit Vectors
Video: Spherical Coordinates (Part A) | Lecture 29
Reading: Spherical and Cartesian Unit Vectors
Reading: Change-of-variables formula
Reading: Integrating a function that only depends on distance from the origin
Reading: Mass of a Sphere
Video: Spherical Coordinates (Part B) | Lecture 30
Reading: Derivatives of the Unit Vectors
Reading: Divergence and Curl of the Unit Vectors
Reading: Laplacian of 1/r
Practice Quiz: Cylindrical and Spherical Coordinates
Video: Line Integral of a Vector Field | Lecture 31
Reading: Line Integral around a Square
Reading: Line Integral around a Circle
Video: Surface Integral of a Vector Field | Lecture 32
Reading: Surface Integral over a Sphere
Practice Quiz: Vector Integration
Discussion Prompt: Week Three Discussion
Graded: Week Three Assessment
WEEK 4: Fundamental Theorems
The fundamental theorem of calculus links integration with differentiation. Here, we learn the related fundamental theorems of vector calculus. These include the gradient theorem, the divergence theorem, and Stokes' theorem. We show how these theorems are used to derive continuity equations, define the divergence and curl in coordinate-free form, and convert the integral version of Maxwell's equations into their more famous differential form.
Video: Introduction
Video: Gradient Theorem | Lecture 33
Reading: Gradient Theorem
Video: Conservative Vector Fields | Lecture 34
Reading: Conservative Vector Fields
Practice Quiz: Gradient Theorem
Video: Divergence Theorem | Lecture 35
Reading: Divergence Theorem for a Sphere
Video: Divergence Theorem: Example I | Lecture 36
Reading: Test the Divergence Theorem for a Cube
Reading: Divergence Theorem for a Cube
Video: Divergence Theorem: Example II | Lecture 37
Reading: Test the Divergence Theorem for a Sphere
Reading: Divergence Theorem for a Sphere
Video: Continuity Equation | Lecture 38
Reading: Continuity Equation
Reading: Electrodynamics Continuity Equation
Practice Quiz: Divergence Theorem
Video: Green's Theorem | Lecture 39
Reading: Test Green's Theorem for a Square
Reading: Test Green's Theorem for a Circle
Video: Stokes' Theorem | Lecture 40
Reading: Stokes' Theorem in Two Dimensions
Reading: Test Stokes' Theorem
Practice Quiz: Stokes' Theorem
Video: Meaning of the Divergence and the Curl | Lecture 41
Reading: The Navier-Stokes Equation
Video: Maxwell's Equations | Lecture 42
Reading: Electric Field of a Point Charge
Reading: Magnetic Field of a Wire
Discussion Prompt: Week Four Discussion
Video: Concluding Remarks
Reading: Please Rate this Course
Reading: Acknowledgments
Graded: Week Four Assessment
댓글 없음:
댓글 쓰기