高等微积分

出版时间：2009-11  出版社：清华大学出版社  作者：David M. Bressoud  页数：386  
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内容概要

　　本书是本科生的微积分教学用书，主要内容为：牛顿运动学基本定律（开篇），向量代数，天体力学简介，线性变换，微分形式和微分演算，隐函数反函数定理，重积分演算，曲线曲面积分，微积分基本定理，经典场论基本定理，爱因斯坦狭义相对论简介。本书特别注意数学与物理、力学等自然科学的内在联系和应用。作者在理念导引、内容选择、程度深浅、适用范围等方面都有相当周密的考虑。从我们国内重点大学的教学角度看，本书的难易程度与物理、力学和电类专业数学课的微积分相当，而思想内容则要深刻和生动些，因此适于用作这些专业本科生的教科书或学习参考书。
　　

书籍目录

preface xi
1 f = ma1
　1.1 prelude to newton's principia 1
　1.2 equal area in equal time 5
　1.3 the law of gravity 9
　1.4 exercises16
　1.5 reprise with calculus 18
　1.6 exercises26
2 vector algebra 29
　2.1 basic notions29
　2.2 the dot product 34
　2.3 the cross product39
　2.4 using vector algebra 46
　2.5 exercises 50
3 celestial mechanics 53
　3.1 the calculus of curves 53
　3.2 exercises05
　3.3 orbital mechanics 06
　3.4 exercises75
4 differential forms 77
　4.1 some history77
　4.2 differential 1-forms 79
　4.3 exercises 86
　4.4 constant differential 2-forms 89
　4.5 exercises 96
　4.6 constant differential k-forms 99
　4.7 prospects 105
　4.8 exercises 107
5 line integrals, multiple integrals 111
　5.1 the riemann integral 111
　5.2 linelntegrals.113
　5.3 exercises llo
　5.4 multiple- -integrals 120
　5.5 using multiple integrals 131
　5.6 exercises
6 linear transformations 139
　6.1 basicnotions.139
　0.2 determinants 146
　6.3 history and comments 157
　6.4 exercises 158
　6.5 invertibility 165
　6.6 exercises
7 differential calculus 171
　7.1 limits 171
　7.2 exercises 178
　7.3 directional derivatives 181
　7.4 the derivative 187
　7.5 exercises 197
　7.6 the chain rule._a201
　7.7 usingthegradient.205
　7.8 exercises 207
8 integration by pullback 211
　8.1 change of variables 211
　8.2 interlude with'lagrange 213
　8.4 thesurfacelntegral 221
　8.5 heatflow228
　8.6 exercises 230
9 techniques of differential calculus 233
　9.1 implicitdifferentiation 233
　9.2 invertibility 238
　9 3 exercises 244
　9.4 locating extrema 248
　9.5 taylor's formula in several variables 254
　9.6 exercises 262
　9.7 lagrangemultipliers266
　9 8 exercises277
10 the fundamental theorem of calculus 279
　10.1 overview 279
　10.2 independence of path 286
　10.3 exercises 294
　10.4 the divergence theorems 297
　10.5 exercises 310
　10.6 stokes' theorem 314
　10.7 summary for r3 321
　10.8 exercises 323
　10.9 potential theory 326
11 e = mc2 333
　11.2 flow in space-time 338
　11.3 electromagnetic potential 345
　11.4 exercises 349
　11.5 specialrelativity 352
　11.6 exercises 360
appendices
　a an opportunity missed 361
　b bibliography365
　c clues and solutions367
index 382

章节摘录

　　1.1  Prelude to Newton's Principia　　Popular mathematical history attributes to Isaac Newton (1642-1727) andGottfried Wilhelm Leibniz (1646-1716) the distinction of having invented calculus. Of course, it is not nearly so simple as that. Techniques for evaluating areas and volumes as limits of computable quantities go back to theGreeks of the classical era. The rules for differentiating polynomials and theuses of these derivatives were current before Newton or Leibniz were born.Even the fundamental theorem of calculus, relating integral and differentialcalculus, was known to Isaac Barrow (1630-1677), Newton's teacher. Yetit is not inappropriate to date calculus from these two men for they werethe first to grasp the power and universal applicability of the fundamentaltheorem of calculus. They were the first to see an inchoate collection ofresults as the body of a single unified theory.　　Newton's preeminent application of calculus is his account of celestialmechanics in Philosophive Naturalis Principia Mathematica or Mathematical Principles of Natural Philosophy. Ironically, he makes very little specificmention of calculus in it. This may, in part, be due to the fact that calculuswas still sufficiently new that he felt it would be suspect. In part, it is areflection of an earlier age in which mathematicians jealously guarded powerful new techniques and only revealed the fruits of their labors.　　……

用户评论 (总计1条)

•   　　 作者在序言中说这本书受两本书的启发：Tom Apostol的Calculus--作者念本科时的课本，和H. Edward的 Advanced Calculus: A Differential Forms Approach。
  　　 我感觉这本书可称得上是“小说型”的课本，认真读它，做好习题，你会进入与Newton,Maxwell,Poincare,E.Cartan同呼吸的境界。看看这书中行列式的引入用的篇幅就知道是“用户友好”之作。
  　　 90年代初，在天元基金赞助下，世图出了影印版“第二学年微积分”，清华社这次影印的跟世图是一个版本，窃以为叫“高等微积分”并不妥当，基本可视为“数学分析“的同义词，会吓跑本来应有的读者--理工科学生。而且清华的影印本太贵了点儿。
  　　 陈天权教授的评价：“（该书）是作者在Pennsylvania州立大学的讲义．作者在Freeman Dyson的鼓励下写成了这本多元微积分．它的数学内容并不深，但是它与力学，电动力学及狭义相对论结合在一起讲．使得数学与物理的相互影响历历在目．”