18.013A Calculus with Applications, Spring 2005
譯者：姚凱
編輯：劉慕華、洪曉慧
一個正方體示意圖的特寫。（圖片由麻省理工學院開放式課程提供。）
課程重點
本課程的特點是彙集了一些學習工具，包括一套互動的Java Applets微積分術語辭彙表以及一整套以線上教科書形式出現的課堂講稿。
課程描述
這是一門大學部的一維和多維微積分課程。對於在高中曾學過微積分的同學，該課程需要一個半學期的學習時間。本課程的設計是完整而獨立的，因此對於沒有任何微積分基礎的同學也可以學習。
技術要求
MathML版本的教科書的執行需要安裝有MathPlayer plugin的Internet 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®plugin software來觀看MathML和HTML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。
先修課程
一年的高中微積分或等同的課程，且微積分課程AB或微積分課程BC有關AB部分的分數需有4或5，微積分測驗，或等同的標準國際考試分數，或是在18.01先修學分測驗中為成績為前百分之五十。
本課程在第一學期的前半部分進行。但對於只選18.013A的同學，需要參加整個學期的課程。
課程概況和安排
課程18.013A是一門供已經在高中時學習過微積分的學生學習的一個半學期的微積分課。這門課希望能涵蓋完整的內容，所以對勇於嘗試的學生，即便沒有任何微積分基礎也可能跟上課程的。課程中我們用了一些相當新的工具，像是applets，目的要讓這個主題學起來更加簡單且更為有趣。但我們的目的並不是僅僅把它變成一門簡單的微積分課，舊有課程中所具有的教材仍然包含在內，只是因為有著applets和電子資料表的利用，因而使其更易理解。
現代節省人力的裝置並未使生活變的簡單和容易。相反地，它節省我們的時間讓我們能夠在生活中做更多的事，使得生活更為複雜和忙碌。同樣的，我們希望這些新的工具能夠讓在學生花相同或更多的精力的前提下，學得更多更透徹。因此課程總體上所包含的是比一般微積分課程更深、更多的內容。事實上，我們已嘗試儘量在課本的每一章節中加入新的材料，正如在網站上可以看到的。我們為何要這樣做呢？某種程度上僅僅是人性的弱點：避免在在創造時失去理智。同時也是為了讓已經學習過這些主題的學生能夠保持對於它的興趣。最後我們所要證明的是：這些在不久之前還很少教、即使教了也很難懂的內容，是能夠簡單地被學生吸收和利用的。
某些額外的教材僅僅會讓學生感到困惑，某些內容非常粗糙，有許多改進的空間，某些部分則是讓你想到可以加入一些更好的內容。如果你能夠利用或學習這些材料的話，作者會感到很欣慰。如果對內容有任何意見的話，我們非常感謝並希望您利用電子郵件通知我們，特別對於發現課程不足的人。這個部份的教材仍然在改進的階段。
關於內容：有何更新？
這門微積分課有何更新呢？第0章中介紹電子資料表的部份是全新的。第一章中對於一些標準函數的討論是新的。第二章中利用圖示對於三角函數的幾何定義是新的，還有第三章中(3.8)討論不同測度的部分也是新的（可能弊大於利）。第四和第五章中主要的更新為applets，儘管在這個階段中對於本征向量和本征值的介紹或許是不常見的。第六章中，對於在各維度中微分的定義是一較為新穎的觀念；但我想它能夠使學生瞭解為何有微分的規則、這些規則是些什麼以及為何微積分是有用的。第七章中的數值微分至少對我來說是新的。第八章中除了將其應用在所有的維度外，沒有太多的更新。
除了applets的部分之外，課程中最為創新的部份可能為數值分析，對於單變數以及多變數微積分的同時探討，複數平面上的積分介紹和物理上的應用。雖然這些應用在物理中很常見，但卻很少在微積分課程中討論。
我們的目的是讓課程中包含足夠進階領域的數學，使得學生瞭解它們並激勵學生更深入地學習它。
如何使用本網站？
相信學生可以自我學習這些資料，但這很少奏效。學生很可能在某些部分被卡住，沒有動力繼續學習。他們試著克服它但熱情卻越來越少，最終產生一種對其抗拒的心態。選修一門組織良好的課程可以藉由要求他們面對一些障礙，比如考試和作業來避免這種事情發生。
如果你在學習中遇到困難，你可以給我們發電子郵件，我們會試著解答。
教學時程
本課程被設計為自學微積分。內容被劃分為以下的“章節”。
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 Antiderivative 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 
相關閱讀資料
使用須知：本部分的內容均以教科書的形式給出。課程分為不同章節，每章節包括很多網頁，使用網頁上的導航按紐可以進入下一章或回到上一層網頁。
18.013A 線上教科書Online Textbook (HTML®)
作業
複習習題1  5
習題1
給定以下五個向量：A = (1, 2, 3); B = (2, 3,
5); C = (x, y, z); D = (cos t, sin t, t 2); E = (2, 1,
0).
做以下題目：
寫出A + B + C之總合。
計算 A•B.
計算A×(B+C).
求滿足C•A = 0和 C•B = 0時的 x,，y ，z的值。
求A 和B. B 和 D 所成角的餘弦值（ 用t的函數表示）。
求 E 在 B上的投影。
求列式為A, B 和 E的行列式；再求出列式為 A, B 和 C的行列式。
假定點P的座標為x = 1, y = 2, z = 3，它的球面座標 ρ, θ 和 Φ為多少?
邊長為A，B，E的平形六面體的體積為多少？
求 D 在平面xy 上的投影。它的長度為多少？
習題2
已知直線上有點A和B
寫出該直線上兩點的參數表示形式。
寫出該直線上兩點處一單位長度的“法向量”。
求與該向量垂直的兩個方向。
已知平面上有點 A, B 和E：
寫出該平面的參數表示形式（兩個參數）。
求該平面的垂線。
求滿足該平面所有點的方程。
假定有一種新的不同與以往的向量內積V@W ，它具有以下性質：對於所有的V 和 @都有V@V = 0，且它們為線性變化，因此可以使用分配律
藉由(V + W)@(V + W)推導出與V@W + W@V相關的內容。
習題3
區分以下指定變數的函數：
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
求(sin y)ex的梯度
求該函數在單位向量為 (cos t, sin t)的方向上的方向導數
sin (ex) 對於x=0的線性逼近
(r×v)對於t求導數， v 為dr/dt；假定 dv/dt 和 r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?
求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 ）。
習題4
求函數r = (x2 + y2)1/2 和 ρ = (x2 + y2 + z2)1/2的梯度。
求1/ρ的梯度。
求 cos θ和θ的梯度。
求 (y, z, x)的旋度。
求ρ/ρ3 的散度(記住ρ = (x, y, z))。
求它的旋度。
習題5
求sin xy在x = 1, y = 2 （弧度）時的二次線性逼近。
該函數在何處具有臨界點（兩側偏倒數均為 0）。
至少找出一個鞍點。
計算出(a×b)•(a×b)交換點和內積後的結果,用同樣的方法寫出三元內積的形式， 得到一個與內積形式等價的交錯式。
以下哪個函數在x = 0出有意義？ (1cos x)/x2, x2/sinx, (sin x cos x)/x2？
使用工具或軟體
.需要Java® plugin software來執行本部分的Java®檔案
預備積分Precalculus
函數的運算 Operations on Functions
直線的斜率Slope of a Line
單變數的微積分Single Variable Calculus
導數與切線Derivative and Tangent Line
常數、一次、二次、三次線性逼近 Constant, Linear, Quadratic and Cubic Approximations
牛頓的方法 Newton's Method
雙變數的拉格朗日乘數Lagrange Multipliers with Two Variables
向量與代數Vectors and Algebra
轉動的座標系Rotating Coordinates
行列式與向量的積 Determinant and Vector Products
利用矩陣計算向量乘法Multiplication of a Vector by a Matrix
三維中的線性逼近 Linear Transformations in Three Dimensions
三維線性幾何中的應用Application to 3D Linear Geometry
空間中的線Lines in Space
空間中的面Planes in Space
曲線 Curves
極座標圖Polar Plotter
二維中的曲線 Curves in Two Dimensions
三維中的曲線Curves in Three Dimensions
場和曲面Fields and Surfaces
等高線，梯度與方向函數Contour Lines, Gradients and Directional Derivatives
曲線與曲面Curves and Surfaces
雙變數方程Functions of Two Variables
雙變數雙方程的牛頓的方法Newton's Method with Two Equations and Two Variables
複數與複數函數Complex Numbers and Functions
曲線的積分Integration on Curves
曲線與向量場Curves and Vector Fields
線積分 Line Integrals
面的積分Integration on Surfaces
流量積分Flux Integrals
積分的界限Integration Bounds
微分方程 Differential Equations
一階微分方程First Order ODE
二階微分方程Second Order ODE
一階微分方程組 System of First Order ODE
應用 Applications
二維中的靜電場Static Electric Fields in Two Dimensions
三維中的靜電場 Static Electric Fields in Three Dimensions
二維中的平穩磁場 Stationary Magnetic Fields in Two Dimensions
RLC（電阻－電感－電容）串聯電路 Series RLC Circuit
研習資料
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 
<v 
列向量，或1乘k的矩陣。 A vector written as a row, or 1 by k matrix 
dx 
變數x的無窮小的變化; dy, dz, dr 等同理 An "infinitesimal" or very small change in the variable x; also similarly dy, dz, dr etc... 
ds 
非常小的距離變化A small change in distance 
ρ 
變數(x2 + y2 + z2)1/2 或球體座標系中到原點的距離 The variable (x2 + y2 + z2)1/2 or distance to the origin in spherical coordinates 
r 
變數(x2 + y2)1/2或在三維座標或極座標中到Z軸的距離 The variable (x2 + y2)1/2 or distance to the z axis in three dimensions or in polar coordinates 
M 
矩陣M的行列式（ 由他的行或列確定面積或側面平行的區域的體積） The determinant of a matrix M (whose magnitude is the area or volume of the parallel sided region determined by its columns or rows) 
M 
矩陣M的行列式的量，是體積，面積或超體積 The magnitude of the determinant of the matrix M, which is a volume or area or hypervolume 
det M 
M的行列式The determinant of M 
M1 
矩陣M的逆The inverse of the matrix M 
v×w 
向量v和向量w的內積或叉積。 The vector product or cross product of two vectors, v and w 
θvw 
向量v和w的夾角。The angle made by vectors v and w 
A•B×C 
標量三重積，或是由列A,B,C所構成矩陣的行列式值。 The scalar triple product, the determinant of the matrix formed by columns A, B, C 
uw 
向量w方向上的單位向量，它的含義與w/w相同 A unit vector in the direction of the vector w; it means the same as w/w 
df 
函數f的無窮小變化，小至對於所有函數皆可利用線性近似表示。 A very small change in the function f, sufficiently small that the linear approximation to all relevant functions holds for such changes 
df/dx 
f對x的微分，是對於函數f的線性逼近。 The derivative of f with respect to x, which is the slope of the linear approximation to f 
f ' 
f對變數的微分，通常對x。 The derivative of f with respect to the relevant variable, usually x 
∂f/∂x 
函數f對x的偏微分，固定y和z。一般來說函數f對某一指定變數q的偏微分 所代表的是在其餘變數固定的情形之下，df對dq的比值。 由於對於哪個變數保持固定可能產生誤解，所以應該明確標注出來。 The partial derivative of f with respect to x, keeping y, and z fixed. In general a partial derivative of f with respect to a variable q is the ratio of df to dq when certain other variables are held fixed. Where there is possible misunderstanding over which variables are to be fixed that information should be made explicit 
(∂f/∂x)r,z 
固定r和z，函數f對x的偏微分。 The partial derivative of f with respect to x keeping r and z fixed 
grad f 
各分量分別為f對x,y和z的偏微分的向量場： [(∂f/∂x), (∂f/∂y), (∂f/∂z)] or (∂f/∂x)i + (∂f/∂y)j + (∂f/∂z)k；稱作f的梯度 The vector field whose components are the partial derivatives of the function f with respect to x, y and z: [(∂f/∂x), (∂f/∂y), (∂f/∂z)] or (∂f/∂x)i + (∂f/∂y)j + (∂f/∂z)k; called the gradient of f 
∇ 
向量運算(∂/∂x)i + (∂/∂x)j + (∂/∂x)k, 讀作 "del" The vector operator (∂/∂x)i + (∂/∂x)j + (∂/∂x)k, called "del" 
∇f 
f的梯度；它與uw的內積為f在w方向上的方向導數 The gradient of f; its dot product with uw is the directional derivative of f in the direction of w 
∇•w 
向量場w的散度；為向量算符∇ 與向量w的內積， 或(∂wx /∂x) + (∂wy /∂y) + (∂wz /∂z) The divergence of the vector field w; it is the dot product of the vector operator ∇ with the vector w, or (∂wx /∂x) + (∂wy /∂y) + (∂wz /∂z) 
curl w 
向量算符∇與向量w的叉積。 The cross product of the vector operator ∇ with the vector w 
∇×w 
w的旋度，分量為[(∂fz /∂y)  (∂fy /∂z), (∂fx /∂z)  (∂fz /∂x), (∂fy /∂x)  (∂fx /∂y)] The curl of w, with components [(∂fz /∂y)  (∂fy /∂z), (∂fx /∂z)  (∂fz /∂x), (∂fy /∂x)  (∂fx /∂y)] 
∇•∇ 
拉普拉斯運算元，為微分運算元：(∂2/∂x2) + (∂/∂y2) + (∂/∂z2) The Laplacian, the differential operator: (∂2/∂x2) + (∂/∂y2) + (∂/∂z2) 
f "(x) 
f對x的二階導數；亦為f '(x)的導數 The second derivative of f with respect to x; the derivative of f '(x) 
d2f/dx2 
f對x的二階導數The second derivative of f with respect to x 
f(2)(x) 
f對x的二階導數的另一種形式 Still another form for the second derivative of f with respect to x 
f(k)(x) 
f對x的k階導數；亦為f(k1) (x)的導數 The kth derivative of f with respect to x; the derivative of f(k1) (x) 
T 
沿著曲線的單位切向量；若曲線為r(t)， T = (dr/dt)/dr/dt Unit tangent vector along a curve; if curve is described by r(t), T = (dr/dt)/dr/dt 
ds 
沿著曲線的長度的微分A differential of distance along a curve 
κ 
曲線的曲率；為單位切向量對距離取導數所得向量的大小： dT/ds The curvature of a curve; the magnitude of the derivative of its unit tangent vector with respect to distance on the curve: dT/ds 
N 
沿著dT/ds方向上的單位向量，與T垂直。 A unit vector in the direction of the projection of dT/ds normal to T 
B 
垂直與平面T 和 N的單位向量, 即面的曲率 A unit vector normal to the plane of T and N, which is the plane of curvature 
τ 
曲線的擾率; dB/ds The torsion of a curve; dB/ds 
g 
引力常數The gravitational constant 
F 
力學中力的標準符號The standard symbol for force in mechanics 
k 
彈簧的彈性係數The spring constant of a spring 
pi 
第I個質點的動量The momentum of the ith particle 
H 
物理系統的哈密頓量，它的能量用{ri} 和 {pi}即 位置和動量表示 The Hamiltonian of a physical system, which is its energy expressed in terms of {ri} and {pi}, position and momentum 
{Q, H} 
Q 和 H的泊松括弧The Poisson bracket of Q and H 
f(x)的反導數， 用x的函數表示出來An antiderivative of f(x) expressed as a function of x 

f從a到b的定積分。當 f 為正值且 a < b , 那它代表 直線 y = a, y = b與x軸和函數曲線所圍成的面積 The definite integral of f from a to b. When f is positive and a < b holds, then this is the area between the x axis the lines y = a, y = b and the curve that represents the function f between these lines 

L(d) 
在等區間大小d下，f在每一區間中取左端點所得的黎曼和。 A Riemann sum with uniform interval size d and f evaluated at the left end of each subinterval 
R(d) 
在等區間大小d下，f在每一區間中取右端點所得的黎曼和。 A Riemann sum with uniform interval size d and f evaluated at the right end of each subinterval 
M(d) 
在等區間大小d下，f在每一區間中取極大值點所對應的點所得的黎曼和。 A Riemann sum with uniform interval size d and f evaluated at the maximum point of f in each subinterval 
m(d) 
在等區間大小d下，f在每一區間中取極小值點所對應的點所得的黎曼和。 A Riemann sum with uniform interval size d and f evaluated at the minimum point of f in each subinterval 
以下為系統擷取之英文原文
18.013A Calculus with Applications
Spring 2005
Closeup 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 selfstudy program in differential calculus. The content is organized into "chapters" below.
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 Antiderivative 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 
Readings
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:
Compute A•B.
Compute A×(B+C).
Find values for x, y and z for which C•A = 0 and C•B = 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)ex.
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 1/ρ.
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
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? (1cos 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
Single Variable Calculus
Vectors and Algebra
Application to 3D Linear
Geometry
Curves
Fields and Surfaces
Complex Numbers and
Functions
Integration on Curves
Integration on Surfaces
Differential Equations
Applications
Study Materials
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 
<v  A vector written as a row, or 1 by k matrix 
dx  An "infinitesimal" or very small change in the variable x; also similarly dy, dz, dr etc... 
ds  A small change in distance 
ρ  The variable (x2 + y2 + z2)1/2 or distance to the origin in spherical coordinates 
r  The variable (x2 + y2)1/2 or distance to the z axis in three dimensions or in polar coordinates 
M  The determinant of a matrix M (whose magnitude is the area or volume of the parallel sided region determined by its columns or rows) 
M  The magnitude of the determinant of the matrix M, which is a volume or area or hypervolume 
det M  The determinant of M 
M1  The inverse of the matrix M 
v×w  The vector product or cross product of two vectors, v and w 
θvw  The angle made by vectors v and w 
A•B×C  The scalar triple product, the determinant of the matrix formed by columns A, B, C 
uw  A unit vector in the direction of the vector w; it means the same as w/w 
df  A very small change in the function f, sufficiently small that the linear approximation to all relevant functions holds for such changes 
df/dx  The derivative of f with respect to x, which is the slope of the linear approximation to f 
f '  The derivative of f with respect to the relevant variable, usually x 
∂f/∂x  The partial derivative of f with respect to x, keeping y, and z fixed. In general a partial derivative of f with respect to a variable q is the ratio of df to dq when certain other variables are held fixed. Where there is possible misunderstanding over which variables are to be fixed that information should be made explicit 
(∂f/∂x)r,z  The partial derivative of f with respect to x keeping r and z fixed 
grad f  The vector field whose components are the partial derivatives of the function f with respect to x, y and z: [(∂f/∂x), (∂f/∂y), (∂f/∂z)] or (∂f/∂x)i + (∂f/∂y)j + (∂f/∂z)k; called the gradient of f 
∇  The vector operator (∂/∂x)i + (∂/∂x)j + (∂/∂x)k, called "del" 
∇f  The gradient of f; its dot product with uw is the directional derivative of f in the direction of w 
∇•w  The divergence of the vector field w; it is the dot product of the vector operator ∇ with the vector w, or (∂wx /∂x) + (∂wy /∂y) + (∂wz /∂z) 
curl w  The cross product of the vector operator ∇ with the vector w 
∇×w  The curl of w, with components [(∂fz /∂y)  (∂fy /∂z), (∂fx /∂z)  (∂fz /∂x), (∂fy /∂x)  (∂fx /∂y)] 
∇•∇  The Laplacian, the differential operator: (∂2/∂x2) + (∂/∂y2) + (∂/∂z2) 
f "(x)  The second derivative of f with respect to x; the derivative of f '(x) 
d2f/dx2  The second derivative of f with respect to x 
f(2)(x)  Still another form for the second derivative of f with respect to x 
f(k)(x)  The kth derivative of f with respect to x; the derivative of f(k1) (x) 
T  Unit tangent vector along a curve; if curve is described by r(t), T = (dr/dt)/dr/dt 
ds  A differential of distance along a curve 
κ  The curvature of a curve; the magnitude of the derivative of its unit tangent vector with respect to distance on the curve: dT/ds 
N  A unit vector in the direction of the projection of dT/ds normal to T 
B  A unit vector normal to the plane of T and N, which is the plane of curvature 
τ  The torsion of a curve; dB/ds 
g  The gravitational constant 
F  The standard symbol for force in mechanics 
k  The spring constant of a spring 
pi  The momentum of the ith particle 
H  The Hamiltonian of a physical system, which is its energy expressed in terms of {ri} and {pi}, position and momentum 
{Q, H}  The Poisson bracket of Q and H 
An antiderivative of f(x) expressed as a function of x  
The definite integral of f from a to b. When f is positive and a < b holds, then this is the area between the x axis the lines y = a, y = b and the curve that represents the function f between these lines  
L(d)  A Riemann sum with uniform interval size d and f evaluated at the left end of each subinterval 
R(d)  A Riemann sum with uniform interval size d and f evaluated at the right end of each subinterval 
M(d)  A Riemann sum with uniform interval size d and f evaluated at the maximum point of f in each subinterval 
m(d)  A Riemann sum with uniform interval size d and f evaluated at the minimum point of f in each subinterval 
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