 MyOOPS開放式課程 18.013A微積分及應用  2005春季課程

18.013A Calculus with Applications, Spring 2005 MathML版本的教科書的執行需要安裝有MathPlayer plug-inInternet Explorer 6或更新的瀏覽器，或者Netscape® 7.x / Mozilla 1.0瀏覽器。另外需要安裝Java®plug-in software來觀看MathMLHTML版本教科書中的applet 檔。還需要安裝Flash™Player software以執行本課程網頁中的Flash™多媒體。

 章    節 課程單元 前言Preface 0 電子資料表The Spreadsheet 1 哲學，數字和函數 Philosophy, Numbers and Functions 2 指數函數與三角函數 The Exponential Function and Trigonometric Functions 3 向量，內積，矩陣乘法和距離 Vectors, Dot Products, Matrix Multiplication and Distance 4 平行四邊形的面積，行列式，體積，超體積以及向量積 Area of a Parallelogram, Determinants, Volume and Hypervolume, the Vector Product 5 二維和三維中的向量與幾何 Vectors and Geometry in Two and Three Dimensions 6 微分函數，積分和微分 Differentiable Functions, the Derivative and Differentials 7 根據定義的積分計算 Computation of Derivatives from their Definition 8 根據法則的積分演算 Calculation of Derivatives by Rule 9 向量場的積分和極座標系中的梯度 Derivatives of Vector Fields and the Gradient in Polar Coordinates 10 高等積分，泰勒級數，二次逼近和逼近精確度 Higher Derivatives, Taylor Series, Quadratic Approximations and Accuracy of Approximations 11 多維空間中的二次逼近 Quadratic Approximations in Several Dimensions 12 微分的應用：直接用於線性逼近 Applications of Differentiation: Direct Use of Linear Approximation 13 解方程Solving Equations 14 極值Extrema 15 曲線Curves 16 幾個重要的例子和一個物理學中的表述 Some Important Examples and a Formulation in Physics 17 積的規則和向量的微分 The Product Rule and Differentiating Vectors 18 複數和複數方程 Complex Numbers and Functions of Them 19 反導數或不定積分 The Anti-derivative or Indefinite Integral 20 曲線下方的面積及其多種推廣 The Area under a Curve and its Many Generalizations 21 微積分在一維中的基本定理 The Fundamental Theorem of Calculus in One Dimension 22 微積分在多維中的基本定理：加性測度，斯托克斯定理以及散度定理 The Fundamental Theorem of Calculus in Higher Dimensions; Additive Measures, Stokes Theorem and the Divergence Theorem 23 線積分化簡為通常積分和相對簡化形式 Reducing a Line Integral to an Ordinary Integral and Related Reductions 24 面積分化簡為多重積分和雅可比矩陣 Reducing a Surface Integral to a Multiple Integral and the Jacobian 25 數值積分法Numerical Integration 26 微分方程的數值解 Numerical Solution of Differential Equations 27 積分練習Doing Integrals 28 介紹電場和磁場 Introduction to Electric and Magnetic Fields 29 磁場，電磁感應和電動力學 Magnetic Fields, Magnetic Induction and Electrodynamics 30 級數Series 31 平面，曲面，體積的積分練習 Doing Area, Surface and Volume Integrals 32 一些線性代數Some Linear Algebra 33 二階微分方程Second Order Differential Equations

18.013A 線上教科書Online Textbook (HTML®)

AB BD 所成角的餘弦值（ 用t的函數表示）。

EB上的投影

D 在平面xy 上的投影。它的長度為多少？

sin (2x)

(sin xy)ex+y固定 y，關於x微分

x2 + y2 - 3xy固定 y.關於x微分

(sin (y + s sin t))e-(x+s cos t).關於s微分，其他變數固定

(sin y)e-x的梯度

sin (ex) 對於x=0的線性逼近

(r×v)對於t求導數，  vdr/dt；假定 dv/dtr的方向一致，那麼答案為多少？

1/x 為何不可微? tan x 為何不可微? |x| 為何不可微?

sin (ex)反函數的導數（完整地定義一個反函數需要給出它的值域；在此處可以忽略 ）。

1/ρ的梯度。

cos θθ的梯度。

(y, z, x)的旋度。

ρ/ρ3 的散度(記住ρ = (x, y, z))

sin xyx = 1, y = 2 （弧度）時的二次線性逼近。

.需要Java® plug-in software來執行本部分的Java®檔案

RLC（電阻－電感－電容）串聯電路 Series RLC Circuit

 符號辭彙表 符號 含義 i -1的平方根The square root of minus one f(x) 自變數為x的函數f的值The value of the function f at argument x sin(x) 自變數為x的正弦函數值 The value of the sine function at argument x exp(x) 自變數為x的指數函數值，也記作ex The value of the exponential function at argument x. This is often written as ex a^x a的x次方；有理數x定義為反函數 The number a raised to the power x; for rational x is defined by inverse functions ln x exp x的反函數The inverse function to exp x ax 與a^x相同Same as a^x logba 為了得出a ，b; blogba = a The power you must raise b to in order to get a; blogba = a cos x 自變數為x的餘弦函數的值（正弦的餘） The value of the cosine function (complement of the sine) at argument x tan x 由 sin x/cos x得出Works out to be sin x/cos x cot x 正切函數的餘的值或 cos x/sin x The value of the complement of the tangent function or cos x/sin x sec x 正割函數的值，即 1/cos x Value of the secant function, which turns out to be 1/cos x csc x 正割函數餘的值，即 1/sin x Value of the complement of the secant, called the cosecant. It is 1/sin x asin x 自變數為x的正弦的反函數的值. 即 x = sin y The value, y, of the inverse function to the sine at argument x. Means x = sin y acos x 自變數為x的餘弦的反函數的值. 即 x = cos y The value, y, of the inverse function to cosine at argument x. Means x = cos y atan x 自變數為x的正切的反函數的值. 即  x = tan y The value, y, of the inverse function to tangent at argument x. Means x = tan y acot x 自變數為x的餘切反函數的值. 即  x = cot y The value, y, of the inverse function to cotangent at argument x. Means x = cot y asec x 自變數為x的正割的反函數的值. 即  x = sec y The value, y, of the inverse function to secant at argument x. Means x = sec y acsc x 自變數為x的反割的反函數的值. 即  x = csc y The value, y, of the inverse function to cosecant at argument x. Means x = csc y θ 角的標準符號. 除注明外一般用弧度制，特別用於atan x/y 當 x, y, z 是描述三維空間裏的點的時候 A standard symbol for angle. Measured in radians unless stated otherwise. Used especially for atan x/y when x, y, and z are variables used to describe point in three dimensional space i, j, k 分別是x 、y、 z方向上的單位向量 Unit vectors in the x y and z directions respectively (a, b, c) x 分量為a，y分量為 b，z分量為c的向量 A vector with x component a, y component b and z component c (a, b) x 分量為a，y分量為 b的向量 A vector with x component a, y component b (a, b) 向量a和b的內積The dot product of vectors a and b a•b 向量a和b的內積The dot product of vectors a and b (a•b) 向量a和b的內積The dot product of vectors a and b |v| 向量v的量The magnitude of the vector v |x| X的絕對值The absolute value of the number x Σ 表示和，它的參數和起始標號常標示在此符號的下方而終止標號則標示於其上方。 例如從j=1到n的和可寫為 ， 意為1 + 2 + … + n Used to denote a summation, usually the index and often their end values are written under it with upper end value above it. For example the sum of j for j=1 to n is written as . This signifies 1 + 2 + … + n M 用於表示數值或其他物件的矩陣或陣列。 Used to represent a matrix or array of numbers or other entities |v> 行向量，其物件排列成行，可視為k乘1的矩陣。 A column vector, that is one whose components are written as a column and treated as a k by 1 matrix

18.013A Calculus with Applications

Spring 2005 Close-up of a diagram of a cube. (Image by MIT OCW.)

Course Highlights

This course features a collection of learning tools, including a set of interactive Java Applets, a glossary of calculus terminology, and a full set of lecture notes in the form of an online textbook.

Course Description

This is an undergraduate course on differential calculus in one and several dimensions. It is intended as a one and a half term course in calculus for students who have studied calculus in high school. The format allows it to be entirely self contained, so that it is possible to follow it without any background in calculus.

Technical Requirements

Special software is required to use some of the files in this course: .jar.

*Some translations represent previous versions of courses.

Syllabus

Prerequisites

A year of high school calculus or the equivalent, with a score of 4 or 5 on the AB, or the AB portion of the BC, Calculus test, or an equivalent score on a standard international exam, or a passing grade on the first half of the 18.01 Advanced Standing exam.

This course is given in the first half of the first term. However, those wishing credit for 18.013A only, must attend the entire semester.

Overview and Format

The course 18.013A as it appears here is intended as a one and a half term course in calculus for students who have studied calculus in high school. It is intended to be self contained, so that it is possible to follow it without any background in calculus, for the adventurous. It makes use of some tools that are relatively new, such as applets, which are intended to make the subject easier to learn and more fun. However it was not our intention to make this course merely an easy calculus course, covering all the same material as the traditional course, but easier to assimilate because of the applets and use of spreadsheets.

Modern labor saving devices in practice do not make life easier and simpler. Instead they save us time that allows us to do more in our lives and to make life more complicated and busier than ever. In the same sense it is our hope that these new tools allow the student to learn more and learn more thoroughly with the same amount of, and even perhaps more, effort than before. Thus, the material covered is substantially greater in depth and variety than is normally attempted in a calculus course. In fact, we have attempted to inject new material into almost every chapter of the text as it appears on this site. Why have we done so? In part it is mere human frailty: to keep from going crazy while creating this material. Also, though, it is in order to maintain interest in the subject among those who have already been exposed to it. Finally, it is to show how easy it now is to absorb and use material that not long ago was utterly inaccessible to students, rarely taught, and when taught rarely mastered.

Obviously some of the extra material contributes mainly to confuse students, some is very badly done though it could be improved, some might suggest to you even better things that might be included. The authors would be delighted if you find a way to use this material or to learn from it. We would be extremely grateful if you would send us via email any comments you have about it, especially from those who don't like something they see. It is still at a stage in which it can be changed.

Content Notes: What is new here?

What then is new for a calculus course here? Chapter 0 which introduces the spreadsheet is entirely new. The discussion of standard functions in Chapter 1 is new. The geometric definitions of the trigonometric functions by the illustration is new in Chapter 2, and the section on various metrics (3.8) is new in Chapter 3. (It probably does more harm than good there) The main innovations in Chapters 4 and 5 are the applets, though introduction of the concepts of eigenvectors and eigenvalues at this stage is perhaps unusual. Chapter 6, which involves definition of differentiation in all dimensions is novel; but I think it helps make it possible for students to see why the rules of differentiation are what they are, and why calculus is useful. Chapter 7 on numerical differentiation is new at least to me. There is not much new in Chapter 8 except for it applying in all dimensions.

Probably the major innovations in this course apart from the applets, are the sections on numerical analysis, the study of single and multiple variable calculus together, introduction of integration in the complex plane, and the applications to physics, which while commonplace in physics, rarely are discussed in a calculus course.

Our intent here is to cover enough about more advanced areas of mathematics to make students aware of them and to encourage their wanting to learn more about them.

How To Use This Web Site

This material could conceivably be studied by a student on his or her own, but this seldom works out. Students tend to get stuck on something, and, having no goad to keep them going, they try to get past it with decreasing energy, and ultimately develop mental blocks against going on. Having an organized course prevents this by forcing them to face obstacles like exams and assignments.

If you attempt this and get stuck, as is almost inevitable, you could try emailing us and we can try to unstick you.

Calendar

This course is designed as a self-study program in differential calculus. The content is organized into "chapters" below.

Course calendar. chapter # Topics
 Preface 0 The Spreadsheet 1 Philosophy, Numbers and Functions 2 The Exponential Function and Trigonometric Functions 3 Vectors, Dot Products, Matrix Multiplication and Distance 4 Area of a Parallelogram, Determinants, Volume and Hypervolume, the Vector Product 5 Vectors and Geometry in Two and Three Dimensions 6 Differentiable Functions, the Derivative and Differentials 7 Computation of Derivatives from their Definition 8 Calculation of Derivatives by Rule 9 Derivatives of Vector Fields and the Gradient in Polar Coordinates 10 Higher Derivatives, Taylor Series, Quadratic Approximations and Accuracy of Approximations 11 Quadratic Approximations in Several Dimensions 12 Applications of Differentiation: Direct Use of Linear Approximation 13 Solving Equations 14 Extrema 15 Curves 16 Some Important Examples and a Formulation in Physics 17 The Product Rule and Differentiating Vectors 18 Complex Numbers and Functions of Them 19 The Anti-derivative or Indefinite Integral 20 The Area under a Curve and its Many Generalizations 21 The Fundamental Theorem of Calculus in One Dimension 22 The Fundamental Theorem of Calculus in Higher Dimensions; Additive Measures, Stokes Theorem and the Divergence Theorem 23 Reducing a Line Integral to an Ordinary Integral and Related Reductions 24 Reducing a Surface Integral to a Multiple Integral and the Jacobian 25 Numerical Integration 26 Numerical Solution of Differential Equations 27 Doing Integrals 28 Introduction to Electric and Magnetic Fields 29 Magnetic Fields, Magnetic Induction and Electrodynamics 30 Series 31 Doing Area, Surface and Volume Integrals 32 Some Linear Algebra 33 Second Order Differential Equations

Special software is required to use some of the files in this section: .jar.

A note to the user: The contents in this section are presented in the format of a textbook. This section is divided into chapters, and each chapter contains a number of pages. Navigation on each page allows the user to move to the next or previous page.

18.013A Online Textbook (HTML®) 18.013A Online Textbook (MATHML)

Assignments

Review Exercises 1 - 5

Exercise 1

Given the following five vectors: A = (1, 2, 3); B = (2, -3, 5); C = (x, y, z); D = (cos t, sin t, t 2); E = (-2, 1, 0).
Do each of the following:

Form the sum: A + B + C.

Compute AB.

Compute A×(B+C).

Find values for x, y and z for which CA = 0 and CB = 0.

Find the cosine of the angle between A and B. Between B and D (the answer will be a function of t).

Find the projection of E on B.

Find the determinant whose columns are A, B and E; also find the determinant whose columns are A, B and C.

Suppose the point P has coordinates x = 1, y = 2, z = 3. What are its spherical coordinates ρ, θ and Φ?

What is the volume of the parallelepiped with edges A, B and E?

Find the projection of D into the xy plane. What is its length?

Exercise 2

Consider the line containing the points A and B above.

Give a parametric representation of the points on that line.

Find a unit length "tangent vector" that points in the direction of the line.

Find two directions normal to that vector.

Consider the plane containing the points A, B and E:

Find a (two parameter) parametric representation of the plane.

Find a normal to the plane.

Find an equation that points on the plane all obey.

Suppose we have a new and different product of vectors V@W that has the property V@V = 0 for all V and @ is linear in each argument so that you can apply the distributive law.

Deduce something about V@W + W@V by applying same to (V + W)@(V + W).

Exercise 3

Differentiate the following functions with respect to the indicated variables:

sin (2x).

(sin xy)ex+y with respect to x for fixed y.

x2 + y2 - 3xy with respect to y for fixed x.

(sin (y + s sin t))e-(x+s cos t) with respect to s everything else fixed.

Find the gradient of (sin y)e-x.

Find the directional derivative of this function in the direction whose unit vector is (cos t, sin t).

Find the linear approximation to sin (ex) at x = 0.

Evaluate the derivative with respect to t of (r×v) where v is dr/dt; suppose that dv/dt is in the direction of r. What then is the answer?

Where is 1/x not differentiable? Where is tan x not differentiable? Where is |x| not differentiable?

Find the derivative of an inverse function to sin (ex) (to define an inverse function completely you have to specify a range; ignore that here).

Exercise 4

Find the gradient of the function r = (x2 + y2)1/2 and ρ = (x2 + y2 + z2)1/2.

Find the gradients of cos θ and of θ.

Find the curl of (y, z, x).

Find the divergence of ρ/ρ3 (remember that ρ = (x, y, z)).

Find the curl of same.

Exercise 5

Find the quadratic approximation to sin xy at x = 1, y = 2 (radians).

Where does this function have critical points (both partial derivatives are 0).

Find at least one saddle point.

Evaluate (a×b)•(a×b) by switching a dot and cross product and expressing the triple cross product according the rule for doing same, to get an alternate expression for the same thing entirely in terms of dot products.

Which of the following functions can be defined at x = 0? (1-cos x)/x2, x2/sinx, (sin x cos x)/x2?

Tools

Special software is required to use some of the files in this section: .jar.

Precalculus

Operations on Functions Trigonometric Functions Slope of a Line

Single Variable Calculus

Derivative and Tangent Line Constant, Linear, Quadratic and Cubic Approximations Newton's Method Numerical Integration Lagrange Multipliers with Two Variables

Vectors and Algebra

Rotating Coordinates Operations on Vectors Determinant and Vector Products Multiplication of a Vector by a Matrix Linear Transformations in Three Dimensions

Application to 3D Linear Geometry

Lines in Space Planes in Space

Curves

Polar Plotter Curves in Two Dimensions Curves in Three Dimensions

Fields and Surfaces

Directional Derivatives Contour Lines, Gradients and Directional Derivatives Curves and Surfaces Functions of Two Variables Newton's Method with Two Equations and Two Variables

Complex Numbers and Functions

Complex Numbers Complex Functions

Integration on Curves

Curves and Vector Fields Line Integrals

Integration on Surfaces

Flux Integrals Integration Bounds

Differential Equations

First Order ODE Second Order ODE System of First Order ODE

Applications

Static Electric Fields in Two Dimensions Static Electric Fields in Three Dimensions Stationary Magnetic Fields in Two Dimensions Series RLC Circuit

Study Materials

Glossary of Notations notation Meaning
 i The square root of minus one f(x) The value of the function f at argument x sin(x) The value of the sine function at argument x exp(x) The value of the exponential function at argument x. This is often written as ex a^x The number a raised to the power x; for rational x is defined by inverse functions ln x The inverse function to exp x ax Same as a^x logba The power you must raise b to in order to get a; blogba = a cos x The value of the cosine function (complement of the sine) at argument x tan x Works out to be sin x/cos x cot x The value of the complement of the tangent function or cos x/sin x sec x Value of the secant function, which turns out to be 1/cos x csc x Value of the complement of the secant, called the cosecant. It is 1/sin x asin x The value, y, of the inverse function to the sine at argument x. Means x = sin y acos x The value, y, of the inverse function to cosine at argument x. Means x = cos y atan x The value, y, of the inverse function to tangent at argument x. Means x = tan y acot x The value, y, of the inverse function to cotangent at argument x. Means x = cot y asec x The value, y, of the inverse function to secant at argument x. Means x = sec y acsc x The value, y, of the inverse function to cosecant at argument x. Means x = csc y θ A standard symbol for angle. Measured in radians unless stated otherwise. Used especially for atan x/y when x, y, and z are variables used to describe point in three dimensional space i, j, k Unit vectors in the x y and z directions respectively (a, b, c) A vector with x component a, y component b and z component c (a, b) A vector with x component a, y component b (a, b) The dot product of vectors a and b a•b The dot product of vectors a and b (a•b) The dot product of vectors a and b |v| The magnitude of the vector v |x| The absolute value of the number x Σ Used to denote a summation, usually the index and often their end values are written under it with upper end value above it. For example the sum of j for j=1 to n is written as . This signifies 1 + 2 + … + n M Used to represent a matrix or array of numbers or other entities |v> A column vector, that is one whose components are written as a column and treated as a k by 1 matrix

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