About the book
This handbook is a continuation of the Handbook of Elementary Mathematics by the same author and includes material usually studied in mathematics courses of higher educational institutions.
The designation of this handbook is two fold.
Firstly, it is a reference work in which the reader can find definitions (what is a vector product?) and factual information, such as how to find the surface of a solid of revolution or how to expand a function in a trigonometric series, and so on. Definitions, theorems, rules and formulas (accompanied by examples and practical hints) are readily found by reference to the comprehensive index or table of contents.
Secondly, the handbook is intended for systematic reading. It does not take the place of a textbook and so full proofs are only given in exceptional cases. However, it can well serve as material for a first acquaintance with the subject. For this purpose, detailed explanations are given of basic concepts, such as that of a scalar product (Sec. 104), limit (Secs. 203~206), the differential (Secs. 228-235), or infinite series (Secs. 270, 366-370). All rules are abundantly illustrated with examples, which form an integral part of the handbook (see Secs. 50-62, 134, 149, 264-266, 369, 422, 498, and others). Explanations indicate how to proceed when a rule ceases to be valid; they also point out errors to be avoided (see Secs. 290, 339, 340, 379, and others).
The theorems and rules are also accompanied by a wide range of explanatory material. In some cases, emphasis is placed on bringing out the content of a theorem to facilitate a grasp of the proof. At other times, special examples are illustrated and the reasoning is such as to provide a complete proof of the theorem if applied to the general case (see Secs. 148, 149, 369, 374). Occasionally, the explanation simply refers the reader to the sections on which the proof is based. Material given in small print may be omitted in a first reading however, this does not mean it is not important.
Considerable attention has been paid to the historical background of mathematical entities, their origin and development. This very often helps the user to place the subject matter in its proper perspective. Of particular interest in this respect are Secs. 270, 366 together with Secs. 271, 383, 399, and 400, which, it is hoped, will give the reader a clearer understanding of Taylor’s series than is usually obtainable in a formal exposition. Also, biographical information from the lives of mathematicians has been included where deemed advisable.
The book was translated from Russian by George Yankovsky was published in 1987 (fifth reprint) by Mir Publishers.
Author(s): M.Vygodsky
Edition: 5
Publisher: Mir
Year: 1980
Language: English
Pages: 935
City: Moscow
Tags: Higher mathematics, Calculus, Analytical Geometry, Differential Equations
Contents
PLANE ANALYTIC GEOMETRY
1. The Subject of Analytic Geometry 19
2. Coordinates 20
3. Rectangular Coordinate System 20
4. Rectangular Coordinates 21
5. Quadrants 21
6. Oblique Coordinate System 22
7. The Equation of a Line 23
8. The Mutual Positions of a Line and a Point 24
9. The Mutual Positions of Two Lines 25
10. The Distance Between Two Points 25
11. Dividing a Line-Segment in a Given Ratio 26
1la. Midpoint of a Line-Segment
12. Second-Order Determinant
13. The Area of a Triangle
14. The Straight Line. An Equation Solved for the Ordinate (Slope-
Intercept Form) 28
15. A Straight Line Parallel to an Axis 30
16. The General Equation of the Straight Line 31
17. Constructing a Straight Line on the Basis of ItsEquation 32
18. The Parallelism Condition of Straight Lines 32
19. The Intersection of Straight Lines 34
20. The Perpendicularity Condition of Two StraightLines 35
21. The Angle Between Two Straight Lines 36
22. The Condition for Three Points Lying on OneStraight Line 38
23. The Equation of a Straight Line Through Two Points (Two-Point Form) 39
24. A Pencil of Straight Lines 40
25. The Equation of a Straight Line Through a Given Point and Parallel to a Given Straight Line (Point-Slope Form) 42
26. The Equation of a Straight Line Through a Given Point and Perpendicular to a Given Straight Line 43
27. The Mutual Positions of a Straight Line and aPair of Points 44
28. The Distance from a Point to a Straight Line 44
29. The Polar Parameters (Coordinates) of a Straight Line 45
30. The Normal Equation of a Straight Line 47
31. Reducing the Equation of a Straight Line to the Normal Form 48
32. Intercepts 49
33. Intercept Form of the Equation of a Straight Line 50
34. Transformation of Coordinates (Statement of theProblem) 51
35. Translation of the Origin 52
36. Rotation of the Axes 53
37. Algebraic Curves and Their Order 54
38. The Circle 56
39. Finding the Centre and Radius of a Circle 57
40. The Ellipse as a Compressed Circle 58
41. An Alternative Definition of the Ellipse 60
42. Construction of an Ellipse from the Axes 62
43. The Hyperbola 63
44. The Shape of the Hyperbola, Its Vertices andAxes 65
45. Construction of a Hyperbola from Its Axes 67
46. The Asymptotes of a Hyperbola 67
47. Conjugate Hyperbolas 68
48. The Parabola 69
49 Construction of a Parabola from a Given Parameter p 70
50. The Parabola as the Graph of the Equation y = ax^{2} + bx + c 70
51. The Directrices of the Ellipse and of the Hyperbola 73
52. A General Definition of the Ellipse, Hyperbola and Parabola 75
53. Conic Sections 77
54. The Diameters of a Conic Section 78
55. The Diameters of an Ellipse 79
56. The Diameters of a Hyperbola 80
57. The Diameters of a Parabola 82
58. Second-Order Curves (Quadric Curves) 83
59. General Second-Degree Equation 85
60. Simplifying a Second-Degree Equation. General Remarks 86
61. Preliminary Transformation of a Second-Degree Equation 86
62. Final Transformation of a Second-Degree Equation 88
63. Techniques to Facilitate Simplification of a Second-Degree Equation 95
64. Test for Decomposition of Second-Order Curves 95
65 Finding Straight Lines that Constitute a Decomposable Second-Order Curve 97
66. Invariants of a Second-Degree Equation 99
67. Three Types of Second-Order Curves 102
68. Central and Noncentral Second-Order Curves (Conics) 104
69. Finding the Centre of a Central Conic 105
70. Simplifying the Equation of a Central Conic 107
71. The Equilateral Hyperbola as the Graph of the Equation y= k/x 109
72. The Equilateral Hyperbola as the Graph of the Equation
y = (mx + n)/(px + q) 110
73. Polar Coordinates 112
74. Relationship Between Polar and Rectangular Coordinates 114
75. The Spiral of Archimedes 116
76. The Polar Equation of a Straight Line 118
77. The Polar Equation of a Conic Section 119
SOLID ANALYTIC GEOMETRY
78. Vectors and Scalars. Fundamentals 120
79. The Vector in Geometry 120
80. Vector Algebra 121
81. Collinear Vectors 121
82. The Null Vector 122
83. Equality of Vectors 122
84. Reduction of Vectors to a Common Origin 123
85. Opposite Vectors 123
86. Addition of Vectors 123
87. The Sum of Several Vectors 125
88. Subtraction of Vectors 126
89. Multiplication and Division of a Vector by a Number 127
90. Mutual Relationship of Collinear Vectors (Division of a Vector
by a Vector) 128
91. The Projection of a Point on an Axis 129
92. The Projection of a Vector on an Axis 130
93. Principal Theorems on Projections of Vectors 132
94. The Rectangular Coordinate System in Space 133
95. The Coordinates of a Point 134
96. The Coordinates of a Vector 135
97. Expressing a Vector in Terms of Components and in Terms of
Coordinates 137
98. Operations Involving Vectors Specified by their Coordinates 137 99. Expressing a Vector in Terms of the Radius Vectors of Its Origin and Terminus 137
100. The Length of a Vector. The Distance Between Two Points 138
101 The Angle Between a Coordinate Axis and aVector 139
102. Criterion of Collinearity (Parallelism) of Vectors 139
103. Division of a Segment in a Given Ratio 140
104. Scalar Product of Two Vectors 141
104a. The Physical Meaning of a Scalar Product 142
105. Properties of a Scalar Product 142
106. The Scalar Products of Base Vectors 144
107. Expressing a Scalar Product in Terms of the Coordinates of the Factors 145
108. The Perpendicularity Condition of Vectors 146
109. The Angle Between Vectors 146
110. Right-Handed and Left-Handed Systems ofThree Vectors 147
111. The Vector Product of Two Vectors 148
112. The Properties of a Vector Product 150
113. The Vector Products of the Base Vectors 152
114. Expressing a Vector Product in Terms of the Coordinates of
the Factors 152
115. Coplanar Vectors 154
116. Scalar Triple Product 154
117 Properties of a Scalar Triple Product 155
118. Third-Order Determinant 156
119. Expressing a Triple Product in Terms of the Coordinates of the
Factors 169
120. Coplanarity Criterion in Coordinate Form 159
121. Volume of a Parallelepiped 160
122. Vector Triple Product 161
123. The Equation of a Plane 161
124. Special Cases of the Position of a Plane Relative to a Coordinate System 162
125. Condition of Parallelism of Planes 163
126. Condition of Perpendicularity of Planes 164
127. Angle Between Two PlaneS 164
128. A Plane Passing Through a Given Point Parallel to a Given Plane 165
129. A Plane Passing Through Three Points 165
130. Intercepts on tne Axes 166
131. Intercept Form of the Equation of a Plane 166
132. A Plane Passing Through Two Points Perpendicular to a Given Plane 167
133. A Plane Passing Through a Given Point Perpendicular to Two Planes 167
134. The Point of Intersection of Three Planes 168
135. The Mutual Positions of a Plane and a Pair of Points 169
136. The Distance from a Point to a Plane 170
137. The Polar Parameters (Coordinates) of a Plane 170
138. The Normal Equation of a Plane 172
139. Reducing the Equation of a Plane to the Normal Form 173
140. Equations of a Straight Line in Space 174
141. Condition Under Which Two First-Degree Equations Represent a Straight Line 176
142. The Intersection of a Straight Line and a Plane 177
143. The Direction Vector 179
144. Angles Between a Straight Line and the Coordinate Axes 179 145. Angle Between Two Straight Lines 180 146. Angle Between a Straight Line and a Plane 181
147. Conditions of Parallelism and Perpendicularity of a Straight Line and a Plane 181
148. A Pencil of Planes 182
149. Projections of a Straight Line on the Coordinate Planes 184
150. Symmetric Form of the Equation of a Straight Line 185
151. Reducing the Equations of a Straight Line to Symmetric Form 187
152. Parametric Equations of a Straight Line 188
153. The Intersection of a Plane with a Straight Line Represented Parametrically 189
154. The Two-Point Form of the Equations of a Straight Line 190
155. The Equation of a Plane Passing Through a Given Point Perpendicular to a Given Straight Line 190
156. The Equations of a Straight Line Passing Through a Given Point Perpendicular to a Given Plane 190
157. The Equation of a Plane Passing Through a Given Point and a Given Straight Line 191
158. The Equation of a Plane Passing Through a Given Point Parallel to Two Given Straight Lines 192
159. The Equation of a Plane Passing Through a Given Straight Line and Parallel to Another Given Straight Line 192
160. The Equation of a Plane Passing Through a Given Straight Line and Perpendicular to a Given Plane 193
161. The Equations of a Perpendicular Dropped from a Given Point onto a Given Straight Line 193
162. The Length of a Perpendicular Dropped from a Given Point onto a Given Straight Line 195
163. The Condition for Two Straight Lines Intersecting or Lying in a Single Plane 196
164. The Equations of a Line Perpendicular to Two Given Straight Lines 197
165. The Shortest Distance Between Two Straight Lines 199
165a. Right-Handed and Left-Handed Pairs of Straight Lines 201
166. Transformation of Coordinates 202
167. The Equation of a Surface 203 168. Cylindrical Surfaces Whose Generatrices Are Parallel to One of the Coordinate Axes 204
169. The Equations of a Line 205
170. The Projection of a Line on a Coordinate Plane 206
171. Algebraic Surfaces and Their Order 209
172. The Sphere 209
173. The Ellipsoid 210
174. Hyperboloid of One Sheet 213
175. Hyperboloid of Two Sheets 215
176. Quadric Conical Surface 217
177. Elliptic Paraboloid 218
178. Hyperbolic Paraboloid 220
179. Quadric Surfaces Classified 221
180. Straight-Line Generatrices of Quadric Surfaces 224
181. Surfaces of Revolution 225
182. Determinants of Second and Third Order 226
183. Determinants of Higher Order 229
184. Properties of Determinants 231 185. A Practical Technique for Computing Determinants 233
186. Using Determinants to Investigate and Solve Systems of Equations 236
187. Two Equations in Two Unknowns 236
188. Two Equations in Three Unknowns 238
189. A Homogeneous System of Two Equations in Three Unknowns 240
190 Three Equations in Three Unknowns 241
190a. A System of n Equations in n Unknowns 246
FUNDAMENTALS OF MATHEMATICAL ANALYSIS
191. Introductory Remarks 247
192. Rational Numbers 248
193. Real Numbers 248
194. The Number Line 249
195. Variable and Constant Quantities 250
196. Function 250
197. Ways of Representing Functions 252
198. The Domain of Definition of a Function 254
199. Intervals 257
200. Classification of Functions 258
201. Basic Elementary Functions 259
202. Functional Notation 259
203. The Limit of a Sequence 261
204. The Limit of a Function 262
205. The Limit of a Function Defined 264
206. The Limit of a Constant 265
207. Infinitesimals 265
208. Infinities 266
209. The Relationship Between Infinities and Infinitesimals 267
210. Bounded Quantities 267
211. An Extension of the Limit Concept 267
212. Basic Properties of Infinitesimals 269
213. Basic Limit Theorems 270
214. The Number e 271
215. The Limit of sin x / x as x → 0 273
216. Equivalent Infinitesimals 273
217. Comparison of Infinitesimals 274
217a. The Increment of a Variable Quantity 276
218. The Continuity of a Function at a Point 277
219. The Properties of Functions Continuous at a Point 278
219a. One-Sided (Unilateral) Limits. The Jump of a Function 278
220. The Continuity of a Function on a Closed Interval 279
221. The Properties of Functions Continuous on a Closed Interval 280
DIFFERENTIAL CALCULUS
222. Introductory Remarks 282
223. Velocity 282
224. The Derivative Defined 284
225. Tangent Line 285
226. The Derivatives of Some Elementary Functions 287
227. Properties of a Derivative 288
228. The Differential 289
229. The Mechanical Interpretation of a Differential 290
230. The Geometrical Interpretation of a Differential 291
231. Differentiable Functions 291
232. The Differentials of Some Elementary Functions 294
233. Properties of a Differential 294
234. The Invariance of the Expression f'(x) dx 294
235. Expressing a Derivative in Terms of Differentials 295
236. The Function of a Function (Composite Function) 296
237. The Differential of a Composite Function 296
238. The Derivative of a Composite Function 297
239. Differentiation of a Product 298
240. Differentiation of a Quotient (Fraction) 299
241. Inverse Function 300
242. Natural Logarithms 302
243. Differentiation of a Logarithmic Function 303
244. Logarithmic Differentiation 304
245. Differentiating an Exponential Function 306
246. Differentiating Trigonometrie Functions 307
247. Differentiating Inverse Trigonometrie Functions 308
247a. Some Instructive Examples 309
248. The Differential in Approximate Calculations 311
249. Using the Differential to Estimate Errors in Formulas 318
250. Differentiation of Implicit Functions 315
251. Parametric Representation of a Curve 316
252. Parametric Representation of a Function 318
253. The Cycloid 320
254. The Equation of a Tangent Line to a Plane Curve 321
254a. Tangent Lines to Quadric Curves 323
255. The Equation of a Normal 323
256. Higher-Order Derivatives 324
257. Mechanical Meaning of the Second Derivative 325
258. Higher-Order Differentials 326
259. Expressing Higher Derivatives in Terms of Differentials 329
260. Higher Derivatives of Functions Represented Parametrically 330 261. Higher Derivatives of Implicit Functions 331
262. Leibniz Rule 332
263. Rolle’s Theorem 334
264. Lagrange’s Mean-Value Theorem 335
265. Formula of Finite Increments 337
266. Generalized Mean-Value Theorem (Cauchy) 339
267. Evaluating the Indeterminate Form 0/0 341
268. Evaluating the Indeterminate Form ∞/∞ 344
269. Other indeterminate Expressions 345
270. Taylor’s Formula (Historical Background) 347
271. Taylor’s Formula 351
272. Taylor’s Formula for Computing the Values of a Function 353
273. Increase and Decrease of a Function 360
274. Tests for the Increase and Decrease of a Function at a Point 362 274a. Tests for the Increase and Decrease of a Function in an Interval 363
275. Maxima and Minima 364
276. Necessary Condition for a Maximum and a Minimum 365
277. The First Sufficient Condition for a Maximum and a Minimum 366 278. Rule for Finding Maxima and Minima 366
279. The Second Sufficient Condition for a Maximum and a Minimum 372 280. Finding Greatest and Least Values of a Function 372
281. The Convexity of Plane Curves. Point of Inflection 379
282. Direction of Concavity 380
283. Rule for Finding Points of Inflection 381
284. Asymptotes 383
285. Finding Asymptotes Parallel to the CoordinateAxes 383
286. Finding Asymptotes Not Parallel to the Axis ofOrdinates 386
287. Construction of Graphs (Examples) 388
288. Solution of Equations. General Remarks 392
289. Solution of Equations. Method of Chords 394
290. Solution of Equations. Method of Tangents 396
291. Combined Chord and Tangent Method 398
INTEGRAL CALCULUS
292. Introductory Remarks 401
293. Antiderivative 403
294. Indefinite Integral 404
295. Geometrical Interpretation of Integration 406
296. Computing the Integration Constant from Initial Data 409
297. Properties of the Indefinite Integral 410
298. Table of Integrais 411
299. Direct integration 413
300. Integration by Substitution (Change of Variable) 414
301. Integration by Parts 418
302. Integration of Some Trigonometric Expressions 421
303. Trigonometrie Substitutions 426
304. Rational Functions 426
304a. Taking out the Integral Part 426
305. Techniques for Integrating Rational Fractions 427
306. Integration of Partial Rational Fractions 428
307. Integration of Rational Functions (General Method) 431
308. Factoring a Polynomial 438
309. On the Integrability of Elementary Functions 439
310. Some Integrais Dependent on Radicals 439
311. The Integral of a Binomial Differential 441
312. Integrais of the Form ∫ R (x, √(ax^{2} + bx + c) dx 443
313. Integrais of the Form ∫ R (sin x, cos x) dx 445
314. The Definite Integral 446
315. Properties of the Definite Integral 450
316. Geometrical Interpretation of the Definite Integral 452
317. Mechanical Interpretation of the Definite Integral 453
318. Evaluating a Definite Integral 455
318a. The Bunyakovsky Inequality 456
319. The Mean-Value Theorem of Integral Calculus 456
320. The Definite Integral as a Function of the Upper Limit 458
321. The Differential of an Integral 460
322. The Integral of a Differential. The Newton-Leibniz Formula 462 323. Computing a Definite Integral by Means of the Indefinite
Integral 464
324. Definite Integration by Parts 465
325. The Method of Substitution in a Definite Integral 466
326. On Improper Integrais 471
327. Integrais with Infinite Limits 472
328. The Integral of a Function with a Discontinuity 476
329. Approximate Integration 480
330. Rectangle Formulas 483
331. Trapezoid Rule 485
332. Simpson’s Rule (for Parabolic Trapezoids) 486
333. Areas of Figures Referred to Rectangular Coordinates 488
334. Scheme for Employing the Definite Integral 490
335. Areas of Figures Referred to Polar Coordinates 492
336. The Volume of a Solid Computed by the Shell Method 494
337. The Volume of a Solid of Revolution 496
338. The Arc Length of a Plane Curve 497
339. Differential of Arc Length 499
340. The Arc Length and Its Differential inPolarCoordinates 499
341. The Area of a Surface of Revolution 501
PLANE AND SPACE CURVES (FUNDAMENTALS)
342. Curvature 503
343. The Centre, Radius and Circle of Curvature of a Plane Curve 504
344. Formulas for the Curvature, Radius and Centre of Curvature of a Plane Curve 505
345. The Evolute of a Plane Curve 508
346. The Properties of the Evolute of a Plane Curve 510
347. Involute of a Plane Curve 511
348. Parametric Representation of a Space Curve 512
349. Helix 514
350. The Arc Length of a Space Curve 515
351. A Tangent to a Space Curve 516
352. Normal Planes 518
353. The Vector Function of a Scalar Argument 519
354. The Limit of a Vector Function 520
355. The Derivative Vector Function 521
356. The Differential of a Vector Function 523
357. The Properties of the Derivative and Differential of a Vector Function 524
358. Osculating Plane 525
359. Principal Normal. The Moving Trihedron 527
360. Mutual Positions of a Curve and a Plane 529
361. The Base Vectors of the Moving Trihedron 529
362. The Centre, Axis and Radius of Curvature of a Space Curve 530
363. Formulas for the Curvature, and the Radius and Centre of Curvature of a Space Curve 531
364. On the Sign of the Curvature 534
365. Torsion 535
SERIES
366. Introductory Remarks 637
367. The Definition of a Series 537
368. Convergent and Divergent Series 538
369. A Necessary Condition for Convergence of a Series 540
370. The Remainder of a Series 542
371. Elementary Operations on Series 543
372. Positive Series 545
373. Comparing Positive Series 545
374. D’Alembert’s Test for a Positive Series 548
375. The Integral Test for Convergence 549
376. Alternating Series. Leibniz’ Test 552
377. Absolute and Conditional Convergence 553
378. D’Alembert’s Test for an Arbitrary Series 555
379. Rearranging the Terms of a Series 555
380. Grouping the Terms of a Series 556
381. Multiplication of Series 558
382. Division of Series 561
383. Functional Series 562
384. The Domain of Convergence of a Functional Series 563
385. On Uniform and Nonuniform Convergence 565
386. Uniform and Nonuniform Convergence Defined 568
387. A Geometrical Interpretation of Uniform and Nonuniform Convergence 568
388. A Test for Uniform Convergence. Regular Series 569
389. Continuity of the Sum of a Series 570
390. Integration of Series 571
391. Differentiation of Series 575
392. Power Series 576
393. The Interval and Radius of Convergence of a Power Series 577
394. Finding the Radius of Convergence 578
395. The Domain of Convergence of a Series Arranged in Powers of x – x_{0} 580
396. Abel’s Theorem 581
397. Operations on Power Series 582
398. Differentiation and Integration of a Power Series 584
399. Taylor’s Series 586
400. Expansion of a Function in a Power Series 587
401. Power-Series Expansions of Elementary Functions 589
402. The Use of Series in Computing Integrais 594
403. Hyperbolic Functions 595
404. Inverse Hyperbolic Functions 598
405. On the Origin of the Names of the Hyperbolic Functions 600
406. Complex Numbers 601
407. A Complex Function of a Real Argument 602
408. The Derivative of a Complex Function 604
409. Raising a Positive Number to a Complex Power 605
410. Euler’s Formula 607
411. Trigonometrie Series 608
412. Trigonometrie Series (Historical Background) 608
413. The Orthogonality of the System of Functions cos nx, sin nx 609 414. Euler-Fourier Formulas 611
415. Fourier Series 615
416. The Fourier Series of a Continuous Function 615
417. The Fourier Series of Even and Odd Functions 618
418. The Fourier Series of a Discontinuous Function 622
DIFFERENTIATION AND INTEGRATION OF FUNCTIONS OF SEVERAL VARIABLES
419. A Function of Two Arguments 626
420. A Function of Three and More Arguments 627
421. Modes of Representing Functions of Several Arguments 628
422. The Limit of a Function of Several Arguments 630
423. On the Order of Smallness of a Function of Several Arguments 632 424. Continuity of a Function of Several Arguments 633
425. Partial Derivatives 634
426. A Geometrical Interpretation of Partial Derivatives for the Case of Two Arguments 635
427. Total and Partial Increments 636
428. Partial Differential 636
429. Expressing a Partial Derivative in Terms of a Differential 637
430. Total Differential 638
431. Geometrical Interpretation of the Total Differential (for the Case of Two Arguments) 640
432. Invariance of the differential Expression f’x dx +f’y dy +f’z dz
of the Total Differential 640
433. The Technique of Differentiation 641
434. Differentiable Functions 642
435. The Tangent Plane and the Normal to a Surface 643
436. The Equation of the Tangent Plane 644
437. The Equation of the Normal 646
438. Differentiation of a Composite Function 646
439. Changing from Rectangular to Polar Coordinates 647
440. Formulas for Derivatives of a Composite Function 648
441. Total Derivative 649
442. Differentiation of an Implicit Function of Several Variables 650 443. Higher-Order Partial Derivatives 653
444. Total Differentials of Higher Orders 654
445. The Technique of Repeated Differentiation 656
446. Symbolism of Differentials 657
447. Taylor’s Formula for a Function of Several Arguments 658
448. The Extremum (Maximum or Minimum) of a Function of Several Arguments 660
449. Rule for Finding an Extremum 660
450. Sufficient Conditions for an Extremum (for the Case of Two Arguments) 662
451. Double Integral 663
452. Geometrical Interpretation of a Double Integral 665
453. Properties of a Double Integral 666
454. Estimating a Double Integral 666
455. Computing a Double Integral (Simplest Case) 667
456. Computing a Double Integral (General Case) 670
457. Point Function 674
458. Expressing a Double Integral in Polar Coordinates 675
459. The Area of a Piece of Surface 677
460. Triple Integral 681
461. Computing a Triple Integral (Simplest Case) 681
462. Computing a Triple Integral (General Case) 682
463. Cylindrical Coordinates 685
464. Expressing a Triple Integral in Cylindrical Coordinates 685
465. Spherical Coordinates 686
466. Expressing a Triple Integral in Spherical Coordinates 687
467. Scheme for Applying Double and Triple Integrais 688
468. Moment of Inertia 689
469. Expressing Certain Physical and Geometrical Quantities in Terms of Double Integrais 691
470. Expressing Certain Physical and Geometrical Quantities in Terms of Triple Integrals 693
471. Line Integrals 695
472. Mechanical Meaning of a Line Integral 697
473. Computing a Line Integral 698
474. Green’s Formula 700
475. Condition Under Which Line Integral Is Independent of Path 701
476. An Alternative Form of the Condition Given in Sec. 475 703
DIFFERENTIAL EQUATIONS
477. Fundamentals 706
478. First-Order Equation 708
479. Geometrical Interpretation of a First-Order Equation 708
480. Isoclines 711
481. Particular and General Solutions of a First-Order Equation 712
482. Equations with Variables Separated 713
483. Separation of Variables. General Solution 714
484. Total Differential Equation 716 484a. Integrating Factor 717
485. Homogeneous Equation 718
486. First-Order Linear Equation 720
487. Clairaut’s Equation 722
488. Envelope 724
489. On the Integrability of Differential Equations 726
490. Approximate Integration of First-Order Equations by Euler’s Method 726
491. Integration of Differential Equations by Means of Series 728
492. Forming Differential Equations 730
493. Second-Order Equations 734
494. Equations of the nth Order 736
495. Reducing the Order of an Equation 736
496. Second-Order Linear Differential Equations 738
497. Second-Order Linear Equations with Constant Coefficients 742
498. Second-Order Homogeneous Linear Equations with Constant Coefficients 742
498a. Connection Between Cases 1 and 3 in Sec. 498 744
499. Second-Order Nonhomogeneous Linear Equations with Constant Coefficients 744
500. Linear Equations of Any Order 750
501. Method of Variation of Constants (Parameters) 752
502. Systems of Differential Equations. Linear Systems 754
SOME REMARKABLE CURVES
503. Strophoid 756
504. Cissoid of Diodes 758
505. Leaf of Descartes 760
506. Versiera 763
507. Conchoid of Nicomedes 766
508. Limaçon. Cardioid 770
509. Cassinian Curves 774
510. Lemniscate of Bernoulli 779
511. Spiral of Archimedes 782
512. Involute of a Circle 785
513. Logarithmic Spiral 789
514. Cycloids 795
515. Epicycloids and Hypocycloids 810
516. Tractrix 826
517. Catenary 833
TABLES
I. Natural Logarithms 839
II. Table for Changing from Natural Logarithms to Common Logarithms 843
III. Table for Changing from Common Logarithms to Natural Logarithms
IV. The Exponential Function e^{x} 844
V. Table of Indefinite Integrals 846
Index 854
