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 6r how to expand a function in a trigonometrie 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). Ail rules are abundantly illustra- ted with examples, which form an integral part of the hand­ book (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 tne 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 read­ ing;; however, this does not mean it is not important.Considerable attention has been paid to the historical background of mathematical entities, their origin and develop­ ment. 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 formai exposition. Also, biographical information from the lives of mathematicians has been included where deemed advisable.Fifth Reprinting 1987 Mir Publishers MoscowTranslated from the Russian by George YankovskyContentsPLANE ANALYTIC GEOMETRY1. 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 2510. The Distance Between Two Points 25 11. Dividing a Line-Segment in a Given Ratio 261la. Midpoint of a Line-Segment12. Second-Order Determinant13. The Area of a Triangle14. 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 3622. The Condition for Three Points Lying on OneStraight Line 3823. 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 4530. 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) 5135. 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 6747. 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 10771. The Equilateral Hyperbola as the Graph of the Equation y= k/x 10972. The Equilateral Hyperbola as the Graph of the Equation y = (mx + n)/(px + q) 11073. 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 119SOLID ANALYTIC GEOMETRY78. Vectors and Scalars. Fundamentals 120 79. The Vector in Geometry 120 80. Vector Algebra 121 81. Collinear Vectors 121 82. TheNu11Vector 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 Vectorby 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 13395. The Coordinates of a Point 13496. The Coordinates of a Vector 13597. Expressing a Vector in Terms of Components and in Terms ofCoordinates 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 ofthe 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 theFactors 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 165130. 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 179144. 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 CoordinatePlanes 184 150. Symmetric Form of theEquation of a StraightLine 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 193161. The Equations of a Perpendicular Dropped from a Given Point onto a Given Straight Line 193162. The Length of a Perpendicular Dropped from a Given Point onto a Given Straight Line 195163. The Condition for Two Straight Lines Intersecting or Lying in a Single Plane 196164. The Equations of a Line Perpendicular to Two Given Straight Lines 197 165. The Shortest Distance Between Two StraightLines 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 ComputingDeterminants 233 186. Using Determinants to Investigate and Solve Systems of Equations 236 187. Two Equations in Two Unknowns 236188. 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 246FUNDAMENTALS OF MATHEMATICAL ANALYSIS191. Introductory Remarks 247192. Rational Numbers 248193. Real Numbers 248194. The Number Line 249195. Variable and Constant Quantities 250196. Function 250197. Ways of Representing Functions 252198. The Domain of Definition of a Function 254 199. Intervals 257 200. Classification of Functions 258201. Basic Elementary Functions 259 202. Functional Notation 259203. The Limit of a Sequence 261204. The Limit of a Function 262205. The Limit of a Function Defined 264206. The Limit of a Constant 265207. Infinitesimals 265208. Infinities 266209. The Relationship Between Infinities and Infinitesimals 267210. Bounded Quantities 267 211. An Extension of the Limit Concept 267212. Basic Properties of Infinitesimals 269213. Basic Limit Theorems 270214. The Number e 271215. The Limit of sin x / x as x → 0 273216. 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 280DIFFERENTIAL CALCULUS222. 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 291232. 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 309248. The Differential in Approximate Calculations 311249. Using the Differential to Estimate Errors in Formulas 318 250. Differentiation of Implicit Functions 315 251. Parametric Representation of a Curve 316252. Parametric Representation of a Function 318253. The Cycloid 320254. 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) 339267. Evaluating the Indeterminate Form 0/0 341268. Evaluating the Indeterminate Form ∞/∞ 344269. Other indeterminate Expressions 345270. Taylor’s Formula (Historical Background) 347271. Taylor’s Formula 351272. 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 383285. 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 398INTEGRAL CALCULUS292. 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 Trigonometrie 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 441312. Integrais of the Form ∫ R (x, √(ax^{2} + bx + c) dx 443 313. Integrais of the Form ∫ R (sin x, cos x) dx 445314. 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 IndefiniteIntegral 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 488334. 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 501Plane and Space Curves (FUNDAMENTALS)342. Curvature 503 343. The Centre, Radius and Circle of Curvature of a Plane Curve 504344. Formulas for the Curvature, Radius and Centre of Curvature of a Plane Curve 505345. 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 523357. The Properties of the Derivative and Differential of a Vector Function 524 358. Osculating Plane 525 359. Principal Normal. The Moving Trihedron 527360. Mutual Positions of a Curve and a Plane 529361. The Base Vectors of the Moving Trihedron 529362. The Centre, Axis and Radius of Curvature of a Space Curve 530363. 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 535SERIES366. 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 556381. 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 568387. 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 578395. The Domain of Convergence of a Series Arranged in Powers of x - x_{0} 580396. 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 594403. 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) 608413. 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 622Differentiation and Integration of Functions of Several Variables419. 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 634426. A Geometrical Interpretation of Partial Derivatives for the Case of Two Arguments 635 427. Total and Partial Increments 636 428. Partial Differential 636429. Expressing a Partial Derivative in Terms of a Differential 637 430. Total Differential 638431. Geometrical Interpretation of the Total Differential (for the Case of Two Arguments) 640432. Invariance of the differential Expression f'x dx +f'y dy +f'z dzof the Total Di­fferential 640433. The Technique of Differentiation 641434. Differentiable Functions 642435. The Tangent Plane and the Normal to a Surface 643436. The Equation of the Tangent Plane 644437. The Equation of the Normal 646438. Differentiation of a Composite Function 646439. Changing from Rectangular to Polar Coordinates 647440. Formulas for Derivatives of a Composite Function 648441. 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 654445. The Technique of Repeated Differentiation 656 446. Symbolism of Differentials 657 447. Taylor’s Formula for a Function of Several Arguments 658448. 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 663452. Geometrical Interpretation of a Double Integral 665453. Properties of a Double Integral 666454. Estimating a Double Integral 666455. 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 675459. The Area of a Piece of Surface 677460. Triple Integral 681461. Computing a Triple Integral (Simplest Case) 681462. 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 693471. Line Integrals 695472. Mechanical Meaning of a Line Integral 697473. Computing a Line Integral 698474. Green’s Formula 700475. Condition Under Which Line Integral Is Independent of Path 701 476. An Alternative Form of the Condition Given in Sec. 475 703 DIFFERENTIAL EQUATIONS477. Fundamentals 706 478. First-Order Equation 708 479. Geometrical Interpretation of a First-Order Equation 708 480. Isoclines 711481. Particular and General Solutions of a First-Order Equation 712 482. Equations with Variables Siparated 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 754SOME REMARKABLE CURVES503. 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 833TABLESI. Natural Logarithms 839 II. Table for Changing from Natural Logarithms to Common Lo­garithms 843 III. Table for Changing from Common Logarithms to Natural Loga­rithmsIV. The Exponential Function e^{x} 844V. Table of Indefini te Integrais 846Index 854