In this post, we will see the book *Mathematical Handbook – Higher Mathematics* by M. Vygodsky. We had earlier seen the Mathematical Handbook – Elementary Mathematics by the same author.

# 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.

You can get the book here.

PS: I am thinking of putting this handbook (along with elementary mathematics one) and the ones on physics by Detlaf/Yavorsky as a dedicated website for ready referencing on the internet. This will also create persistent URLs for particular concepts which can be readily referred to. Ideas on what framework/technology to use are welcome. As of now, I have thought of creating the website in plain html with mathml support, but please do suggest any other framework which might be more suitable for the project.

Follow us on The Internet Archive: https://archive.org/details/@mirtitles

Follow Us On Twitter: https://twitter.com/MirTitles

Write to us: mirtitles@gmail.com

Fork us at GitLab: https://gitlab.com/mirtitles/

Add new entries to the detailed book catalog here.

# 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

Odd Numbered Pages have some problem while scanning. starting letters are not clear.

Is there problem in the original book?

LikeLike

the book was very tightly bound, so some initial pages had lot of warping.

edit: I will give another pass at processing the scans to see if the problem can be rectified by manual intervention. Currently, the processing is automated,

LikeLike

Pingback: Mathematical Handbook – Higher Mathematics – Vygodsky – Chet Aero Marine