高等微積分

內(nèi)容概要

　　本書是本科生的微積分教學(xué)用書，主要內(nèi)容為：牛頓運(yùn)動學(xué)基本定律（開篇），向量代數(shù)，天體力學(xué)簡介，線性變換，微分形式和微分演算，隱函數(shù)反函數(shù)定理，重積分演算，曲線曲面積分，微積分基本定理，經(jīng)典場論基本定理，愛因斯坦狹義相對論簡介。本書特別注意數(shù)學(xué)與物理、力學(xué)等自然科學(xué)的內(nèi)在聯(lián)系和應(yīng)用。作者在理念導(dǎo)引、內(nèi)容選擇、程度深淺、適用范圍等方面都有相當(dāng)周密的考慮。從我們國內(nèi)重點(diǎn)大學(xué)的教學(xué)角度看，本書的難易程度與物理、力學(xué)和電類專業(yè)數(shù)學(xué)課的微積分相當(dāng)，而思想內(nèi)容則要深刻和生動些，因此適于用作這些專業(yè)本科生的教科書或?qū)W習(xí)參考書。
　　

書籍目錄

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

章節(jié)摘錄

　　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.　　……

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