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Author: M. Vygodsky

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Added Date: 2022-06-01

Language: eng

Subjects: mathematics; partial differential equations; calculus, analytic geometry, curves, series, analysis, reference books, mir publishers

Publishers: Mir Publishers

Collections: mir-titles, additional collections

Pages Count: 300

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Year: 1987

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Description

 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 Moscow

Translated from the Russian by George Yankovsky

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. 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 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 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 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 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 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 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 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 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 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 Indefini te Integrais 846
Index 854


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