Mathematical Handbook: Higher Mathematics

✍ Scribed by M.Vygodsky

• Publisher: Mir
• Year: 1980
• Tongue: English
• Leaves: 935
• Edition: 5
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

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
49. The Parabola as the Graph of the Equation y = ax^{2} + bx + c 70
50. The Directrices of the Ellipse and of the Hyperbola 73
51. A General Definition of the Ellipse, Hyperbola and Parabola 75
52. Conic Sections 77
53. The Diameters of a Conic Section 78
54. The Diameters of an Ellipse 79
55. The Diameters of a Hyperbola 80
56. The Diameters of a Parabola 82
57. Second-Order Curves (Quadric Curves) 83
58. General Second-Degree Equation 85
59. Simplifying a Second-Degree Equation. General Remarks 86
60. Preliminary Transformation of a Second-Degree Equation 86
61. Final Transformation of a Second-Degree Equation 88
62. Techniques to Facilitate Simplification of a Second-Degree Equation 95
63. Test for Decomposition of Second-Order Curves 95
    65 Finding Straight Lines that Constitute a Decomposable Second-Order Curve 97
64. Invariants of a Second-Degree Equation 99
65. Three Types of Second-Order Curves 102
66. Central and Noncentral Second-Order Curves (Conics) 104
67. Finding the Centre of a Central Conic 105
68. Simplifying the Equation of a Central Conic 107
69. The Equilateral Hyperbola as the Graph of the Equation y= k/x 109
70. The Equilateral Hyperbola as the Graph of the Equation
    y = (mx + n)/(px + q) 110
71. Polar Coordinates 112
72. Relationship Between Polar and Rectangular Coordinates 114
73. The Spiral of Archimedes 116
74. The Polar Equation of a Straight Line 118

75. The Polar Equation of a Conic Section 119
    SOLID ANALYTIC GEOMETRY

76. Vectors and Scalars. Fundamentals 120

77. The Vector in Geometry 120
78. Vector Algebra 121
79. Collinear Vectors 121
80. The Null Vector 122
81. Equality of Vectors 122
82. Reduction of Vectors to a Common Origin 123
83. Opposite Vectors 123
84. Addition of Vectors 123
85. The Sum of Several Vectors 125
86. Subtraction of Vectors 126
87. Multiplication and Division of a Vector by a Number 127
88. Mutual Relationship of Collinear Vectors (Division of a Vector
    by a Vector) 128
89. The Projection of a Point on an Axis 129
90. The Projection of a Vector on an Axis 130
91. Principal Theorems on Projections of Vectors 132
92. The Rectangular Coordinate System in Space 133
93. The Coordinates of a Point 134
94. The Coordinates of a Vector 135
95. Expressing a Vector in Terms of Components and in Terms of
    Coordinates 137
96. Operations Involving Vectors Specified by their Coordinates 137 99. Expressing a Vector in Terms of the Radius Vectors of Its Origin and Terminus 137
97. The Length of a Vector. The Distance Between Two Points 138
    101 The Angle Between a Coordinate Axis and aVector 139
98. Criterion of Collinearity (Parallelism) of Vectors 139
99. Division of a Segment in a Given Ratio 140
100. Scalar Product of Two Vectors 141
     104a. The Physical Meaning of a Scalar Product 142
101. Properties of a Scalar Product 142
102. The Scalar Products of Base Vectors 144
103. Expressing a Scalar Product in Terms of the Coordinates of the Factors 145
104. The Perpendicularity Condition of Vectors 146
105. The Angle Between Vectors 146
106. Right-Handed and Left-Handed Systems ofThree Vectors 147
107. The Vector Product of Two Vectors 148
108. The Properties of a Vector Product 150
109. The Vector Products of the Base Vectors 152
110. Expressing a Vector Product in Terms of the Coordinates of
     the Factors 152
111. Coplanar Vectors 154
112. Scalar Triple Product 154
     117 Properties of a Scalar Triple Product 155
113. Third-Order Determinant 156
114. Expressing a Triple Product in Terms of the Coordinates of the
     Factors 169
115. Coplanarity Criterion in Coordinate Form 159
116. Volume of a Parallelepiped 160
117. Vector Triple Product 161
118. The Equation of a Plane 161
119. Special Cases of the Position of a Plane Relative to a Coordi­nate System 162
120. Condition of Parallelism of Planes 163
121. Condition of Perpendicularity of Planes 164
122. Angle Between Two PlaneS 164
123. A Plane Passing Through a Given Point Parallel to a Given Plane 165
124. A Plane Passing Through Three Points 165
125. Intercepts on tne Axes 166
126. Intercept Form of the Equation of a Plane 166
127. A Plane Passing Through Two Points Perpendicular to a Given Plane 167
128. A Plane Passing Through a Given Point Perpendicular to Two Planes 167
129. The Point of Intersection of Three Planes 168
130. The Mutual Positions of a Plane and a Pair of Points 169
131. The Distance from a Point to a Plane 170
132. The Polar Parameters (Coordinates) of a Plane 170
133. The Normal Equation of a Plane 172
134. Reducing the Equation of a Plane to the Normal Form 173
135. Equations of a Straight Line in Space 174
136. Condition Under Which Two First-Degree Equations Represent a Straight Line 176
137. The Intersection of a Straight Line and a Plane 177
138. The Direction Vector 179
139. 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
140. Conditions of Parallelism and Perpendicularity of a Straight Line and a Plane 181
141. A Pencil of Planes 182
142. Projections of a Straight Line on the Coordinate Planes 184
143. Symmetric Form of the Equation of a Straight Line 185
144. Reducing the Equations of a Straight Line to Symmetric Form 187
145. Parametric Equations of a Straight Line 188
146. The Intersection of a Plane with a Straight Line Represented Parametrically 189
147. The Two-Point Form of the Equations of a Straight Line 190
148. The Equation of a Plane Passing Through a Given Point Perpendicular to a Given Straight Line 190
149. The Equations of a Straight Line Passing Through a Given Point Perpendicular to a Given Plane 190
150. The Equation of a Plane Passing Through a Given Point and a Given Straight Line 191
151. The Equation of a Plane Passing Through a Given Point Parallel to Two Given Straight Lines 192
152. The Equation of a Plane Passing Through a Given Straight Line and Parallel to Another Given Straight Line 192
153. The Equation of a Plane Passing Through a Given Straight Line and Perpendicular to a Given Plane 193
154. The Equations of a Perpendicular Dropped from a Given Point onto a Given Straight Line 193
155. The Length of a Perpendicular Dropped from a Given Point onto a Given Straight Line 195
156. The Condition for Two Straight Lines Intersecting or Lying in a Single Plane 196
157. The Equations of a Line Perpendicular to Two Given Straight Lines 197
158. The Shortest Distance Between Two Straight Lines 199
     165a. Right-Handed and Left-Handed Pairs of Straight Lines 201
159. Transformation of Coordinates 202
160. The Equation of a Surface 203 168. Cylindrical Surfaces Whose Generatrices Are Parallel to One of the Coordinate Axes 204
161. The Equations of a Line 205
162. The Projection of a Line on a Coordinate Plane 206
163. Algebraic Surfaces and Their Order 209
164. The Sphere 209
165. The Ellipsoid 210
166. Hyperboloid of One Sheet 213
167. Hyperboloid of Two Sheets 215
168. Quadric Conical Surface 217
169. Elliptic Paraboloid 218
170. Hyperbolic Paraboloid 220
171. Quadric Surfaces Classified 221
172. Straight-Line Generatrices of Quadric Surfaces 224
173. Surfaces of Revolution 225
174. Determinants of Second and Third Order 226
175. Determinants of Higher Order 229
176. Properties of Determinants 231 185. A Practical Technique for Computing Determinants 233
177. Using Determinants to Investigate and Solve Systems of Equations 236
178. Two Equations in Two Unknowns 236
179. Two Equations in Three Unknowns 238

180. 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

181. Introductory Remarks 247

182. Rational Numbers 248
183. Real Numbers 248
184. The Number Line 249
185. Variable and Constant Quantities 250
186. Function 250
187. Ways of Representing Functions 252
188. The Domain of Definition of a Function 254
189. Intervals 257
190. Classification of Functions 258
191. Basic Elementary Functions 259
192. Functional Notation 259
193. The Limit of a Sequence 261
194. The Limit of a Function 262
195. The Limit of a Function Defined 264
196. The Limit of a Constant 265
197. Infinitesimals 265
198. Infinities 266
199. The Relationship Between Infinities and Infinitesimals 267
200. Bounded Quantities 267
201. An Extension of the Limit Concept 267
202. Basic Properties of Infinitesimals 269
203. Basic Limit Theorems 270
204. The Number e 271
205. The Limit of sin x / x as x → 0 273
206. Equivalent Infinitesimals 273
207. Comparison of Infinitesimals 274
     217a. The Increment of a Variable Quantity 276
208. The Continuity of a Function at a Point 277
209. The Properties of Functions Continuous at a Point 278
     219a. One-Sided (Unilateral) Limits. The Jump of a Function 278
210. The Continuity of a Function on a Closed Interval 279

211. The Properties of Functions Continuous on a Closed Interval 280
     DIFFERENTIAL CALCULUS

212. Introductory Remarks 282

213. Velocity 282
214. The Derivative Defined 284
215. Tangent Line 285
216. The Derivatives of Some Elementary Functions 287
217. Properties of a Derivative 288
218. The Differential 289
219. The Mechanical Interpretation of a Differential 290
220. The Geometrical Interpretation of a Differential 291
221. Differentiable Functions 291
222. The Differentials of Some Elementary Functions 294
223. Properties of a Differential 294
224. The Invariance of the Expression f'(x) dx 294
225. Expressing a Derivative in Terms of Differentials 295
226. The Function of a Function (Composite Function) 296
227. The Differential of a Composite Function 296
228. The Derivative of a Composite Function 297
229. Differentiation of a Product 298
230. Differentiation of a Quotient (Fraction) 299
231. Inverse Function 300
232. Natural Logarithms 302
233. Differentiation of a Logarithmic Function 303
234. Logarithmic Differentiation 304
235. Differentiating an Exponential Function 306
236. Differentiating Trigonometrie Functions 307
237. Differentiating Inverse Trigonometrie Functions 308
     247a. Some Instructive Examples 309
238. The Differential in Approximate Calculations 311
239. Using the Differential to Estimate Errors in Formulas 318
240. Differentiation of Implicit Functions 315
241. Parametric Representation of a Curve 316
242. Parametric Representation of a Function 318
243. The Cycloid 320
244. The Equation of a Tangent Line to a Plane Curve 321
     254a. Tangent Lines to Quadric Curves 323
245. The Equation of a Normal 323
246. Higher-Order Derivatives 324
247. Mechanical Meaning of the Second Derivative 325
248. Higher-Order Differentials 326
249. Expressing Higher Derivatives in Terms of Differentials 329
250. Higher Derivatives of Functions Represented Parametrically 330 261. Higher Derivatives of Implicit Functions 331
251. Leibniz Rule 332
252. Rolle’s Theorem 334
253. Lagrange’s Mean-Value Theorem 335
254. Formula of Finite Increments 337
255. Generalized Mean-Value Theorem (Cauchy) 339
256. Evaluating the Indeterminate Form 0/0 341
257. Evaluating the Indeterminate Form ∞/∞ 344
258. Other indeterminate Expressions 345
259. Taylor’s Formula (Historical Background) 347
260. Taylor’s Formula 351
261. Taylor’s Formula for Computing the Values of a Function 353
262. Increase and Decrease of a Function 360
263. 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
264. Maxima and Minima 364
265. Necessary Condition for a Maximum and a Minimum 365
266. The First Sufficient Condition for a Maximum and a Minimum 366 278. Rule for Finding Maxima and Minima 366
267. The Second Sufficient Condition for a Maximum and a Minimum 372 280. Finding Greatest and Least Values of a Function 372
268. The Convexity of Plane Curves. Point of Inflection 379
269. Direction of Concavity 380
270. Rule for Finding Points of Inflection 381
271. Asymptotes 383
272. Finding Asymptotes Parallel to the CoordinateAxes 383
273. Finding Asymptotes Not Parallel to the Axis ofOrdinates 386
274. Construction of Graphs (Examples) 388
275. Solution of Equations. General Remarks 392
276. Solution of Equations. Method of Chords 394
277. Solution of Equations. Method of Tangents 396

278. Combined Chord and Tangent Method 398
     INTEGRAL CALCULUS

279. Introductory Remarks 401

280. Antiderivative 403
281. Indefinite Integral 404
282. Geometrical Interpretation of Integration 406
283. Computing the Integration Constant from Initial Data 409
284. Properties of the Indefinite Integral 410
285. Table of Integrais 411
286. Direct integration 413
287. Integration by Substitution (Change of Variable) 414
288. Integration by Parts 418
289. Integration of Some Trigonometric Expressions 421
290. Trigonometrie Substitutions 426
291. Rational Functions 426
     304a. Taking out the Integral Part 426
292. Techniques for Integrating Rational Fractions 427
293. Integration of Partial Rational Fractions 428
294. Integration of Rational Functions (General Method) 431
295. Factoring a Polynomial 438
296. On the Integrability of Elementary Functions 439
297. Some Integrais Dependent on Radicals 439
298. The Integral of a Binomial Differential 441
299. Integrais of the Form ∫ R (x, √(ax^{2} + bx + c) dx 443
300. Integrais of the Form ∫ R (sin x, cos x) dx 445
301. The Definite Integral 446
302. Properties of the Definite Integral 450
303. Geometrical Interpretation of the Definite Integral 452
304. Mechanical Interpretation of the Definite Integral 453
305. Evaluating a Definite Integral 455
     318a. The Bunyakovsky Inequality 456
306. The Mean-Value Theorem of Integral Calculus 456
307. The Definite Integral as a Function of the Upper Limit 458
308. The Differential of an Integral 460
309. The Integral of a Differential. The Newton-Leibniz Formula 462 323. Computing a Definite Integral by Means of the Indefinite
     Integral 464
310. Definite Integration by Parts 465
311. The Method of Substitution in a Definite Integral 466
312. On Improper Integrais 471
313. Integrais with Infinite Limits 472
314. The Integral of a Function with a Discontinuity 476
315. Approximate Integration 480
316. Rectangle Formulas 483
317. Trapezoid Rule 485
318. Simpson’s Rule (for Parabolic Trapezoids) 486
319. Areas of Figures Referred to Rectangular Coordinates 488
320. Scheme for Employing the Definite Integral 490
321. Areas of Figures Referred to Polar Coordinates 492
322. The Volume of a Solid Computed by the Shell Method 494
323. The Volume of a Solid of Revolution 496
324. The Arc Length of a Plane Curve 497
325. Differential of Arc Length 499
326. The Arc Length and Its Differential inPolarCoordinates 499

327. The Area of a Surface of Revolution 501
     PLANE AND SPACE CURVES (FUNDAMENTALS)

328. Curvature 503

329. The Centre, Radius and Circle of Curvature of a Plane Curve 504
330. Formulas for the Curvature, Radius and Centre of Curvature of a Plane Curve 505
331. The Evolute of a Plane Curve 508
332. The Properties of the Evolute of a Plane Curve 510
333. Involute of a Plane Curve 511
334. Parametric Representation of a Space Curve 512
335. Helix 514
336. The Arc Length of a Space Curve 515
337. A Tangent to a Space Curve 516
338. Normal Planes 518
339. The Vector Function of a Scalar Argument 519
340. The Limit of a Vector Function 520
341. The Derivative Vector Function 521
342. The Differential of a Vector Function 523
343. The Properties of the Derivative and Differential of a Vector Function 524
344. Osculating Plane 525
345. Principal Normal. The Moving Trihedron 527
346. Mutual Positions of a Curve and a Plane 529
347. The Base Vectors of the Moving Trihedron 529
348. The Centre, Axis and Radius of Curvature of a Space Curve 530
349. Formulas for the Curvature, and the Radius and Centre of Cur­vature of a Space Curve 531
350. On the Sign of the Curvature 534

351. Torsion 535
     SERIES

352. Introductory Remarks 637

353. The Definition of a Series 537
354. Convergent and Divergent Series 538
355. A Necessary Condition for Convergence of a Series 540
356. The Remainder of a Series 542
357. Elementary Operations on Series 543
358. Positive Series 545
359. Comparing Positive Series 545
360. D’Alembert’s Test for a Positive Series 548
361. The Integral Test for Convergence 549
362. Alternating Series. Leibniz’ Test 552
363. Absolute and Conditional Convergence 553
364. D’Alembert’s Test for an Arbitrary Series 555
365. Rearranging the Terms of a Series 555
366. Grouping the Terms of a Series 556
367. Multiplication of Series 558
368. Division of Series 561
369. Functional Series 562
370. The Domain of Convergence of a Functional Series 563
371. On Uniform and Nonuniform Convergence 565
372. Uniform and Nonuniform Convergence Defined 568
373. A Geometrical Interpretation of Uniform and Nonuniform Con­vergence 568
374. A Test for Uniform Convergence. Regular Series 569
375. Continuity of the Sum of a Series 570
376. Integration of Series 571
377. Differentiation of Series 575
378. Power Series 576
379. The Interval and Radius of Convergence of a Power Series 577
380. Finding the Radius of Convergence 578
381. The Domain of Convergence of a Series Arranged in Powers of x – x_{0} 580
382. Abel’s Theorem 581
383. Operations on Power Series 582
384. Differentiation and Integration of a Power Series 584
385. Taylor’s Series 586
386. Expansion of a Function in a Power Series 587
387. Power-Series Expansions of Elementary Functions 589
388. The Use of Series in Computing Integrais 594
389. Hyperbolic Functions 595
390. Inverse Hyperbolic Functions 598
391. On the Origin of the Names of the Hyperbolic Functions 600
392. Complex Numbers 601
393. A Complex Function of a Real Argument 602
394. The Derivative of a Complex Function 604
395. Raising a Positive Number to a Complex Power 605
396. Euler’s Formula 607
397. Trigonometrie Series 608
398. Trigonometrie Series (Historical Background) 608
399. The Orthogonality of the System of Functions cos nx, sin nx 609 414. Euler-Fourier Formulas 611
400. Fourier Series 615
401. The Fourier Series of a Continuous Function 615
402. The Fourier Series of Even and Odd Functions 618

403. The Fourier Series of a Discontinuous Function 622
     DIFFERENTIATION AND INTEGRATION OF FUNCTIONS OF SEVERAL VARIABLES

404. A Function of Two Arguments 626

405. A Function of Three and More Arguments 627
406. Modes of Representing Functions of Several Arguments 628
407. The Limit of a Function of Several Arguments 630
408. On the Order of Smallness of a Function of Several Arguments 632 424. Continuity of a Function of Several Arguments 633
409. Partial Derivatives 634
410. A Geometrical Interpretation of Partial Derivatives for the Case of Two Arguments 635
411. Total and Partial Increments 636
412. Partial Differential 636
413. Expressing a Partial Derivative in Terms of a Differential 637
414. Total Differential 638
415. Geometrical Interpretation of the Total Differential (for the Case of Two Arguments) 640
416. Invariance of the differential Expression f’x dx +f’y dy +f’z dz
     of the Total Di­fferential 640
417. The Technique of Differentiation 641
418. Differentiable Functions 642
419. The Tangent Plane and the Normal to a Surface 643
420. The Equation of the Tangent Plane 644
421. The Equation of the Normal 646
422. Differentiation of a Composite Function 646
423. Changing from Rectangular to Polar Coordinates 647
424. Formulas for Derivatives of a Composite Function 648
425. Total Derivative 649
426. Differentiation of an Implicit Function of Several Variables 650 443. Higher-Order Partial Derivatives 653
427. Total Differentials of Higher Orders 654
428. The Technique of Repeated Differentiation 656
429. Symbolism of Differentials 657
430. Taylor’s Formula for a Function of Several Arguments 658
431. The Extremum (Maximum or Minimum) of a Function of Seve­ral Arguments 660
432. Rule for Finding an Extremum 660
433. Sufficient Conditions for an Extremum (for the Case of Two Arguments) 662
434. Double Integral 663
435. Geometrical Interpretation of a Double Integral 665
436. Properties of a Double Integral 666
437. Estimating a Double Integral 666
438. Computing a Double Integral (Simplest Case) 667
439. Computing a Double Integral (General Case) 670
440. Point Function 674
441. Expressing a Double Integral in Polar Coordinates 675
442. The Area of a Piece of Surface 677
443. Triple Integral 681
444. Computing a Triple Integral (Simplest Case) 681
445. Computing a Triple Integral (General Case) 682
446. Cylindrical Coordinates 685
447. Expressing a Triple Integral in Cylindrical Coordinates 685
448. Spherical Coordinates 686
449. Expressing a Triple Integral in Spherical Coordinates 687
450. Scheme for Applying Double and Triple Integrais 688
451. Moment of Inertia 689
452. Expressing Certain Physical and Geometrical Quantities in Terms of Double Integrais 691
453. Expressing Certain Physical and Geometrical Quantities in Terms of Triple Integrals 693
454. Line Integrals 695
455. Mechanical Meaning of a Line Integral 697
456. Computing a Line Integral 698
457. Green’s Formula 700
458. Condition Under Which Line Integral Is Independent of Path 701

459. An Alternative Form of the Condition Given in Sec. 475 703
     DIFFERENTIAL EQUATIONS

460. Fundamentals 706

461. First-Order Equation 708
462. Geometrical Interpretation of a First-Order Equation 708
463. Isoclines 711
464. Particular and General Solutions of a First-Order Equation 712
465. Equations with Variables Separated 713
466. Separation of Variables. General Solution 714
467. Total Differential Equation 716 484a. Integrating Factor 717
468. Homogeneous Equation 718
469. First-Order Linear Equation 720
470. Clairaut’s Equation 722
471. Envelope 724
472. On the Integrability of Differential Equations 726
473. Approximate Integration of First-Order Equations by Euler’s Method 726
474. Integration of Differential Equations by Means of Series 728
475. Forming Differential Equations 730
476. Second-Order Equations 734
477. Equations of the nth Order 736
478. Reducing the Order of an Equation 736
479. Second-Order Linear Differential Equations 738
480. Second-Order Linear Equations with Constant Coefficients 742
481. Second-Order Homogeneous Linear Equations with Constant Coefficients 742
     498a. Connection Between Cases 1 and 3 in Sec. 498 744
482. Second-Order Nonhomogeneous Linear Equations with Constant Coefficients 744
483. Linear Equations of Any Order 750
484. Method of Variation of Constants (Parameters) 752
485. Systems of Differential Equations. Linear Systems 754

SOME REMARKABLE CURVES

1. Strophoid 756
2. Cissoid of Diodes 758
3. Leaf of Descartes 760
4. Versiera 763
5. Conchoid of Nicomedes 766
6. Limaçon. Cardioid 770
7. Cassinian Curves 774
8. Lemniscate of Bernoulli 779
9. Spiral of Archimedes 782
10. Involute of a Circle 785
11. Logarithmic Spiral 789
12. Cycloids 795
13. Epicycloids and Hypocycloids 810
14. Tractrix 826
15. 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

✦ Subjects

Higher mathematics, Calculus, Analytical Geometry, Differential Equations