Volume 1. CONTENTS
INTRODUCTION REAL NUMBERS
§ 1. Region rational numbers 11
1. Preliminary remarks 11
2. Ordering the region of rational numbers 12
3. Addition and subtraction of rational numbers 12
4. Multiplication and division of rational numbers 14
5. Axiom of Archimedes 16
§ 2. Introduction of irrational numbers. Region ordering real numbers
6 Definition irrational number 17
7. Ordering the domain of real numbers 19
8. Auxiliary proposals 21
9. Representation of a real number by an infinite decimal 22
10. Continuity of the domain of real numbers 24
11. Limits of Numerical Sets 25

§ 3. Arithmetic operations over real numbers 28
12. Determining the sum of real numbers 28
13. Properties of addition 29
14. Definition of product of real numbers 31
15. Properties of multiplication 3 2
16. Conclusion 34
17. Absolute values 34 § 4. Further properties and applications of real numbers 35
18. Existence of a root. Degree c rational indicator 35
19. Degree with any real exponent 37
20. Logarithms 39
21. Distance measurement 40

CHAPTER FIRST. THEORY OF LIMITS
§ 1. Variants and its limit 43
22. variable, option 43
23. Limit options 46
24. Infinitesimals 47
25. Examples 48
26. Some theorems about a variant with a limit of 52
27. Infinitely large quantities 54

§ 2. Limit theorems that make it easier to find limits 56
28. Limit transition in equality and inequality 56
29. Lemmas on infinitesimals 57
30. Arithmetic operations over variables 58
31. Undefined Expressions 60
32. Examples for Finding Limits 62
33. Stolz's theorem and its applications 67

§ 3. Monotone version 70
34. Limit of monotonic variants 70
35. Examples 72
36. Number e 77
31. Approximate calculation of the number e 79
38. Lemma on nested intervals 82

§ 4. The principle of convergence. Partial limits 83
39. The principle of convergence 83
40. Partial sequences and partial limits 85
41. Bolzano-Weierstrass Lemma 87
42. Maximum and minimum limits 89

CHAPTER TWO. FUNCTIONS OF A SINGLE VARIABLE
§ 1. The concept of a function 93
43. Variable and its range 93
44. Functional dependence between variables. Examples 94
45. Definition of the concept of function 95
46. Analytical method function setting 98
47. Function Graph 100
48. The most important classes of functions 102
49. Concept inverse function 108
50. Reverse trigonometric functions 110
51. Superposition of functions. Concluding remarks 114

§ 2. Limit of a function 115
52. Determining the limit of a function 115
53. Reduction to the case of variants 117
54. Examples 120
55. Extension of the theory of limits 128
56. Examples 130
57. Limit of a monotonic function 133
58. Common feature Bolzano Cosi 134
59. Maximum and minimum limits of a function 135

§ 3. Classification of infinitesimals and infinitesimals large quantities 136
60. Comparison of infinitesimals 136
61. Infinitesimal scale 137
62. Equivalent infinitesimals 139
63. Highlighting the main part 141
64. Tasks 143
65. Classification of infinitely large 145

§ 4. Continuity (and discontinuities) of functions 146
66. Determining the continuity of a function at a point 146
67. Arithmetic operations on continuous functions 148
68. Examples of Continuous Functions 148
69. One-way continuity. Break classification 150
70. Examples discontinuous functions 151
71. Continuity and discontinuities of a monotonic function 154
72. Continuity elementary functions 155
73. Superposition of continuous functions 156
74. Decision of one functional equation 157
75. Functional characteristic exponential, logarithmic and power functions
76. Functional characteristics of trigonometric and hyperbolic cosines
77. Using continuity of functions to calculate limits 162
78. Power-exponent expressions 165
79. Examples 166

§ 5. Properties of continuous functions 168
80. The vanishing theorem 168
81. Application to Solving Equations 170
82. The intermediate value theorem 171
83. Existence of an inverse function 172
84. Boundedness Theorem 174
85. Greatest and smallest value functions 175
86. The concept of uniform continuity 178
87. Cantor's theorem 179

88. Borel Lemma 180
89. New proofs of the main theorems 182
CHAPTER THREE. DERIVATIVES AND DIFFERENTIALS
§ 1. The derivative and its calculation 186
90. The problem of calculating the speed of a moving point 186
91. The problem of drawing a tangent to a curve 187
92. Definition of a derivative 189
93. Examples of calculating derivatives 193
94. Derivative of the inverse function 196
95. Summary of formulas for derivatives 198
96. Formula for function increment 198
97. The simplest rules for calculating derivatives 199
98. Derivative of a complex function 202
99. Examples 203
100. One-sided derivatives 209
101. Infinite derivatives 209
102 Further examples special occasions 211

§ 2. Differential 211
103. Definition of differential 211
104. Connection between differentiability and existence of _ 1. derivative
105. Basic Formulas and differentiation rules 215
106. Invariance of the differential form 216
107. Differentials as a source of approximate formulas 218
108. The use of differentials in the estimation of errors 220

§ 3. Main theorems differential calculus 223
109. Fermat's Theorem 223
110. Darboux theorem 224
111. Rolle's theorem 225
112. Lagrange formula 226
113. Derivative limit 228
114. Cauchy formula 229

§ 4. Derivatives and differentials of higher orders 231
115. Definition of derivatives of higher orders 231
116. General formulas for derivatives of any order 232
117. Leibniz formula 236
118. Examples 238
119. Higher Order Differentials 241
120. Violation of the form invariance for differentials of higher _ ._ orders
121. Parametric differentiation 243
122. Finite differences 244

§ 5. Taylor formula 246
123. Taylor formula for polynomial 246
124. Decay arbitrary function; additional member in Peano form
125. Examples 251
126. Other forms of additional member 254
127 Approximate Formulas 257

§ 6. Interpolation 263
128. The simplest task interpolation. Lagrange formula 263
129. Additional term of the Lagrange formula 264
130. Interpolation with multiple nodes. Hermite formula 265
CHAPTER FOUR. INVESTIGATION OF A FUNCTION WITH THE HELP OF DERIVATIVES
§ 1. Studying the course of a change in function 268
131. Condition of constancy of function 268
132. The condition for the monotonicity of a function 270
133. Proof of inequalities 273
134. Highs and lows; necessary conditions 276
135. Sufficient conditions. First Rule 278
136. Examples 280
137. Second rule 284
138. Use of higher derivatives 286
139. Finding the largest and smallest values ​​288
140. Tasks 290

§ 2. Convex (and concave) functions 294
141. Definition of a convex (concave) function 294
142. The simplest propositions about convex functions 296
143. Conditions for the convexity of a function 298
144. Jensen's inequality and its applications 301
145. Inflection Points 303

§ 3. Construction of graphs of functions 305
146. Statement of the problem 305
147. Scheme for constructing a graph. Examples 306
148. Endless gaps, endless gap. Asymptotes 308
149. Examples 311

§ 4 Disclosure of uncertainties 314
150. Uncertainty of the form 0/0 314
151. Uncertainty of the form oo / oo 320
152. Other types of uncertainties 322

§ 5. Approximate solution of the equation 324
153. Introductory remarks 3 24
154. Rule of proportional parts (method of chords) 325
155. Newton's rule (method of tangents) 328
156. Examples and exercises 331
157. Combined method 335
158. Examples and exercises 336

CHAPTER FIVE. FUNCTIONS OF MULTIPLE VARIABLES
§ 1. Basic concepts 340
159. Functional dependence between variables. Examples 340
160. Functions of two variables and their domains 341
161. Arithmetic n-dimensional space 345
162. Examples of regions in n-dimensional space 348
163. General definition open and closed area 350
164. Functions of n variables 352
165. Limit of a Function of Several Variables 354
166. Reduction to the case of variants 356
167. Examples 358
168. Repeat limits 360
§ 2. Continuous functions 362
169. Continuity and discontinuities of functions of several variables 362
170. Operations on continuous functions 364
171. Functions continuous in a domain. Bolzano-Cauchy theorems 365
172. Lemma of Bolzano-Weierstrass 367
173. Weierstrass' theorems 369
174. Uniform continuity 370
175. Borel Lemma 372
176. New proofs of the main theorems 373
176. Derivatives and Differentials of Functions of Several Variables 373
177. Partial derivatives and partial differentials 375
178. Full increment of a function 378
179. Full differential 381
180. Geometric interpretation for the case of a function of two _ R_ variables
181. Derivatives from complex functions 386
182. Examples 388
183. Finite Increment Formula 390
184. Derivative with respect to given direction 391
185. Invariance of the Form of the (First) Differential 394
186. Application total differential in approximate calculations 396
187. Homogeneous functions 399
188. Euler formula 400

§ 4. Derivatives into higher-order differentials 402
189. Derivatives of Higher Orders 402
190. Mixed derivatives theorem 404
191. Generalization 407
192. Higher Order Derivatives of a Complex Function 408
193. Higher Order Differentials 410
194. Differentials of complex functions 413
195. Taylor Formula 414

§ 5. Extremes, maximum and minimum values ​​417
196. Extrema of a function of several variables. Required. 17 conditions
197. Sufficient conditions (the case of a function of two variables) 419
198. Sufficient conditions (general case) 422
199. Conditions for the absence of an extremum 425
200. The largest and smallest values ​​of functions. Examples 427
201. Tasks 431
CHAPTER SIX. FUNCTIONAL DETERMINERS; THEIR APPS
§ 1. Formal properties of functional determinants 441
202. Definition of functional determinants (Jacobians) 441
203. Multiplication of Jacobians 442
204. Multiplication of function matrices (Jacobi matrices) 444

§ 2. Implicit functions 447
205. The concept of an implicit function of one variable 447
206. Existence of an implicit function 449
207 Differentiability of an implicit function 451
208. Implicit functions of several variables 453
209 Calculating derivatives of implicit functions 460
210. Examples 463

§ 3. Some applications of the theory of implicit functions 467
211. Relative extremes 467
212. Method indefinite multipliers Lagrange 470
213. Sufficient conditions for a relative extremum 472
214. Examples and tasks 473
215. The concept of independence of functions 477
216. Jacobi matrix rank 479

§ 4. Change of variables 483
217. Functions of one variable 483
218. Examples 485
219. Functions of several variables. Change of independent.„„ variables
220. Method for calculating differentials 489
221. General case change of variables 491
222. Examples 493
CHAPTER SEVEN. APPLICATIONS OF DIFFERENTIAL CALCULUS TO GEOMETRY
§ 1. Analytic representation of curves and surfaces 503
223. Curves in the plane (in rectangular coordinates) 503
224. Examples 505
225. Curves of mechanical origin 508
226. Curves on the plane (in polar coordinates). Examples 511
227. Surfaces and curves in space 516
228. Parametric representation 518
229. Examples 520

§ 2. Tangent and tangent plane 523
230. Tangent to a plane curve in rectangular coordinates 523
231. Examples 525
232. Tangent in polar coordinates 528
233. Examples 529
234. Tangent to a spatial curve. Tangent plane to surface
235. Examples 534
236. Singular points flat curves 535
237. Case parametric task curve 540

§ 3. Tangent between curves 542
238. Envelope of a family of curves 542
239. Examples 545
240. Characteristic points 549
241. The order of touching two curves 551
242. The Case of Implicitly Specifying One of the Curves 553
243. Contiguous Curve 554
244. Another Approach to Contiguous Curves 556

§ 4. The length of a plane curve 557
245. Lemmas 557
246. Curve Direction 558
247. The length of the curve. Arc length additivity 560
248. Sufficient Conditions for Rectifiability. Arc differential 562
249. Arc as a parameter. Positive tangent direction 565

§ 5. Curvature of a Plane Curve 568
250. The concept of curvature 568
251. Circle of Curvature and Radius of Curvature 571
252. Examples 573
253. Coordinates of the center of curvature
254. Definition of evolute and evolvent; search for an evolution
255. Properties of Evolutes and Evolutes
256. Search for evolvents
ADDITION. PROBLEM OF PROPAGATION OF FUNCTIONS
257. The case of a function of one variable
258. Statement of the Problem for the Two-Dimensional Case
259. Auxiliary sentences
260. Main Propagation Theorem
261. Generalization
262. Concluding remarks

Alphabetical index 600

Volume 2. CONTENTS
CHAPTER EIGHT. DERIVATIVE FUNCTION (INDETERMINATE INTEGRAL)
§ 1. Indefinite integral and the simplest methods of its calculation 11
263. Concept antiderivative function(and indefinite integral) 11
264. The Integral and the Area Problem 14
265. Table of basic integrals 17
266. The Simplest Integration Rules 18
267. Examples 19
268. Integration by change of variable 23
269. Examples 27
270. Integration by parts 31
271. Examples 32

§ 2. Integration of rational expressions 36
272. Statement of the problem of integration in final form 36
273. simple fractions and their integration 37
274. Decay proper fractions to simple 38
275. Determination of coefficients. Integration of proper fractions 42
276. Separation of the rational part of the integral 43
277. Examples 47
§ 3. Integration of some expressions containing radicals 50
278. Integration of expressions of the form R .yx + 8
279. Integration of binomial differentials. Examples 51
280. Reduction Formulas 54
281. Integration of expressions of the form K\x, liax2 + bx + c). Substitutions -^ Euler
282. Geometric treatment of Euler substitutions 59
283. Examples 60
284. Other Methods of Calculation 66
285. Examples 72
§ 4. Integration of expressions containing trigonometric and exponential functions 74
286. Integration of differentials i?(sin x, cos x) dx 74
287. Integration of expressions sinv xcosto 76
288. Examples 78
289. Review of other cases 83 § 5. Elliptic integrals 84
290. General remarks and definitions 84
291. Auxiliary transformations 86
292. Reduction to canonical form 88
293. Elliptic integrals of the 1st, 2nd and 3rd kind 90

CHAPTER NINE. DEFINITION INTEGRAL
§ 1. Definition and conditions for the existence of a definite integral 94
294. Another approach to the area problem 94
295. Definition 96
296. Darboux sums 97
297. The Condition for the Existence of the Integral 100
298 Classes of Integrable Functions 101
299. Properties of integrable functions 103
300. Examples and additions 105
301 Lower and Upper Integrals as Limits 106

§ 2. Properties definite integrals 108
302. Integral over an oriented interval 108
303. Properties expressed by equalities 109
304. Properties expressed by inequalities 110
305 Definite Integral as a Function of the Upper Limit 115
306. The Second Mean Value Theorem 117

§ 3. Calculation and transformation of definite integrals 120
307. Calculation with the help of integral sums 120
308. Basic Formula integral calculus 123
309. Examples 125
310. Another derivation of the main formula 128
311. Reduction formulas 130
312. Examples 131
313. The formula for the change of variable in a definite integral 134
314. Examples 135
315. Gauss formula. Landen Transform 141
316. Another derivation of the change of variable formula 143

§ 4. Some applications of definite integrals 145
317. Wallis Formula 145
318. Taylor formula with an additional term 146
319. Transcendence of the number e 146
320. Legendre Polynomials 148
321. Integral inequalities 151

§ 5. Approximate calculation of integrals 153
322. Statement of the problem. Formulas of rectangles and trapezoids 153
323. Parabolic Interpolation 156
324. Splitting the interval of integration 158
325. Additional term of the formula of rectangles 159
326. Additional term of the trapezoid formula 161
327. Additional term of Simpson's formula 162
328. Examples 164
CHAPTER TEN. APPLICATIONS OF THE INTEGRAL CALCULUS TO GEOMETRY, MECHANICS AND PHYSICS
§ 1. The length of a curve 169
329 Calculating Curve Length 169
330. Another approach to the definition of the concept of the length of a curve and its calculation
331. Examples 174
332. natural equation flat curve 180
333. Examples 183
334. The length of the arc of the spatial curve 185

§ 2. Areas and volumes 186
335. Definition of the concept of area. Additivity property 186
336. Area as a limit 188
337. Classes of squaring regions 190
338. Expression of area by integral 192
339. Examples 195
340. Definition of the concept of volume. Its properties 202
341. Classes of bodies having volumes 204
342. Expression of Volume by an Integral 205
343. Examples 208
344. Surface area of ​​rotation 214
345. Examples 217
346. Square cylindrical surface 220
347. Examples 222

§ 3. Calculation of mechanical and physical quantities 225
348. Scheme for applying a definite integral 225
349. Finding the static moments and the center of gravity of a curve 228
350. Examples 229
351. Finding static moments and center of gravity flat figure
352. Examples 232
353. mechanical work 233
354. Examples 235
355. The work of the friction force in a flat foot 237
356. Problems on the summation of infinitesimal elements 239

§ 4. The simplest differential equations 244
357. Basic concepts. First order equations 244
358. Equations of the first degree with respect to the derivative. Separation of variables
359. Tasks 247
360. Compilation notes differential equations 253
361. Tasks 254
CHAPTER ELEVEN. ENDLESS ROWS WITH PERMANENT MEMBERS
§ 1. Introduction 257
362. Basic concepts 257
363. Examples 258
364. Fundamental Theorems 260

§ 2. Convergence of positive series 262
365. Condition for the convergence of a positive series 262
366. Series comparison theorems 264
367. Examples 266
368. Signs of Cauchy and D'Alembert 270
369. Sign of Raabe 272
370. Examples 274
371. Kummer sign 277
372. Gauss sign 279
373. Integral feature Maclaurin-Cauchy 281
374. Sign of Ermakov 285
375. Additions 287

§ 3. Convergence of arbitrary series 293
376. General condition series convergence 293
377 Absolute Convergence 294
378. Examples 296
379. Power series, its interval of convergence 298
380. Expression of the radius of convergence in terms of coefficients 300
381. alternating series 3 02
382. Examples 303
383 Abel Transformation 305
384. Signs of Abel and Dirichlet 307
385. Examples 308

§ 4. Properties of convergent series 313
386. associative property 313
3 87. Commutative property of absolutely convergent series 315
388. The case of non-absolutely convergent series 316
389. Multiplication of rows 320
390. Examples 323
391. General theorem from the theory of limits 325
392. Further theorems on the multiplication of series 327

§ 5. Repeated and double rows 329
393. Repeat rows 329
394. Double rows 333
395. Examples 338
396 Power series with two variables; area of ​​convergence 346
397. Examples 348
398. Multiple rows 350

§ 6. Infinite products 350
399. Basic concepts 350
400. Examples 351
401. Basic theorems. Relationship with rows 353
402. Examples 356

§ 7. Expansions of elementary functions 364
403. Expansion of a function in a power series; taylor row 364
404. Expansion in a series of exponential, basic trigonometric functions, etc.
405. Logarithmic Series 368
406. Sterling formula 369
407. Binomial Series 371
408. Decomposition of sine and cosine into infinite products 374

§ 8. Approximate calculations with the help of series. Series conversion 378
409. General remarks 378
410. Calculating the number to 379
411 Calculating Logarithms 381
412. Calculating Roots 383
413. Euler Series Transformation 3 84
414. Examples 386
415. Kummer Transformation 388
416. Markov Transformation 392

§ 9. Summation of divergent series 394
417. Introduction 394
418 Power Series Method 396
419. Tau Ber's theorem 398
420. Method of Arithmetic Averages 401
421. Relationship between the Poisson-Abel and Cesaro methods 403
422. The Hardy-Landau Theorem 405
423. Application of generalized summation to the multiplication of series 407
424. Other methods of generalized summation of series 408
425. Examples 413
426. General class linear regular summation methods 416
CHAPTER TWELVE. FUNCTIONAL SEQUENCES AND SERIES
§ 1. Uniform convergence 419
427. Introductory remarks 419
428. Uniform and Nonuniform Convergence 421
429. Condition for uniform convergence 425
430. Criteria for uniform convergence of series 427

§ 2. Functional properties row sums 430
431. Continuity of the sum of a series 430
432. Remark on quasi-uniform convergence 432
433. Term by Term to the Limit 434
434. Termwise Integration of Series 436
435. Term Differentiation of Series 438
436. Sequence Viewpoint 441
437. Continuity of the sum of a power series 444
438. Integration and differentiation of power series 447

§ 3 Appendices 450
439. Examples on the continuity of the sum of a series and on the passage to the limit term by term
440. Examples for term-by-term integration of series 457
441. Examples for term-by-term differentiation of series 468
442. Method successive approximations in the theory of implicit functions 474
443. Analytical definition trigonometric functions 477
444. An example of a continuous function without a derivative 479

§ four. additional information about power series 481
445. Actions on power series 481
446. Substituting a row into a row 485
447. Examples 487
448. Division of power series 492
449. Bernoulli numbers and expansions in which they occur 494
450. Solving equations by series 498
451. Power series inversion 502
452. Lagrange Series 505

§ 5. Elementary functions of a complex variable 508
453. Complex numbers 508
454. Complex variant and its limit 511
455. Functions of a Complex Variable 513
456. Power series 515
457. Exponential function 518
458. logarithmic function 520
459. Trigonometric functions and their inverses 522
460 Power Function 526
461. Examples 527

§ 6. Enveloping and asymptotic series. Euler-Maclaurin formula 531
462. Examples 531
463. Definitions 533
464. Basic properties of asymptotic expansions 536
465. Derivation of the Euler-Maclaurin Formula 540
466. Study of an additional term 542
467. Examples of Calculations Using the Euler-Maclaurin Formula 544
468. Another form of the Euler-Maclaurin formula 547
469. Sterling's formula and series 550

CHAPTER THIRTEEN. Improper integrals
§ 1. Improper integrals with infinite limits 552
470. Definition of Integrals with Infinite Limits 552
471. Application of the basic formula of integral calculus 554
472. Examples 555
473. Analogy with series. The simplest theorems 558
474. Convergence of the Integral in the Case positive function 559
475 Convergence of the Integral in the General Case 561
476. Signs of Abel and Dirichlet 563
477. Reducing an Improper Integral to an Infinite Series 566
478. Examples 569

§ 2. Improper integrals of unbounded functions 577
479. Definition of integrals of unbounded functions 577
480. Remark on Singular Points 581
481. Application of the basic formula of integral calculus. Examples
482. Conditions and signs of the existence of an integral 584
483. Examples 587
484. Principal Meanings improper integrals 590
485. Remark on generalized values ​​of divergent integrals 595

§ 3. Properties and transformation of improper integrals 597
486. The simplest properties 597
487. Mean Value Theorems 600
488 Integration by Parts in the Case of Improper Integrals 602
489. Examples 602
490 Change of Variables in Improper Integrals 604
491. Examples 605

§ 4. Special methods for calculating improper integrals 611
492 Some Remarkable Integrals 611
493. Calculation of improper integrals with the help of integral sums. The case of integrals with finite limits
494. The case of integrals with endless limit 617
495 Frullani Integrals 621
496. Integrals of rational functions between endless limits
497. Mixed examples and exercises 629

§ 5. Approximate calculation of improper integrals 641
498. Integrals with finite limits; highlighting features 641
499. Examples 642
500. Remark on Approximate Calculation of Eigenintegrals
501. Approximate calculation of improper integrals with an infinite limit
502 Using asymptotic expansions 650
CHAPTER FOURTEEN. INTEGRALS DEPENDING ON A PARAMETER
§ one. elementary theory 654
503. Statement of the problem 654
504. Uniform striving for limit function 654
505. Permutation of two passages to the limit 657
506. Passing to the limit under the sign of the integral 659
507. Differentiation under the Integral Sign 661
508 Integration Under the Integral Sign 663
509. The case when and the limits of the integral depend on the parameter 665
510. Introduction of a multiplier depending only on x 668
511. Examples 669
512. Gaussian proof of the fundamental theorem of algebra 680
§ 2. Uniform convergence of integrals 682
513. Definition of uniform convergence of integrals 682
514. Condition for uniform convergence. Relationship with Rows 684
515. Sufficient signs uniform convergence 684
516. Another case of uniform convergence 687
517. Examples 689

§ 3. Use of the uniform convergence of integrals 694
518. Passing to the limit under the sign of the integral 694
519. Examples 697
520 Continuity and differentiability of an integral with respect to a parameter 710
521 Integration of the integral with respect to a parameter 714
522. Application to the calculation of certain integrals 717
523. Examples for Differentiation under the Integral Sign 723
524. Examples for Integration Under the Integral Sign 733

§ 4. Additions 743
525. Artzel's Lemma 743
526. Passing to the limit under the sign of the integral 745
527. Differentiation under the integral sign 748
528 Integration Under the Integral Sign 749

§ 5. Euler integrals 750
529. Euler integral of the first kind 750
530. Euler integral of the second kind 753
531. The Simplest Properties of the Function Г 754
532. Unambiguous definition function Γ by its properties 760
533. Another functional characteristic of the function Г 762
534. Examples 764
535. Logarithmic derivative of the function Г 770
536. Multiplication Theorem for the Function Г 772
537. Some expansions into series and products 774
538. Examples and additions 775
539. Calculation of certain definite integrals 782
540. Sterling formula 789
541 Calculating the Euler constant 792
542. Compiling a table decimal logarithms G functions 793
Alphabetical index 795
Alphabetical index

G.M. Fikhtengolts
COURSE OF DIFFERENTIAL AND INTEGRAL CALCULUS
VOLUME 1
Content
INTRODUCTION
REAL NUMBERS
§ 1. The region of rational numbers 11 1. Preliminary remarks 11 2. Ordering of the region of rational numbers 12 3. Addition and subtraction of rational numbers 12 4. Multiplication and division of rational numbers 14 5. Axiom of Archimedes 16
§ 2. Introduction of irrational numbers. Ordering the domain of real numbers
17 6. Definition of an irrational number 17 7. Ordering of the domain of real numbers 19 8. Auxiliary sentences 21 9. Representation of a real number by an infinite decimal fraction 22 10. Continuity of the domain of real numbers 24 11. Boundaries of numerical sets 25
§ 3. Arithmetic operations on real numbers 28 12. Definition of the sum of real numbers 28 13. Properties of addition 29 14. Definition of the product of real numbers 31 15. Properties of multiplication 32 16. Conclusion 34 17. Absolute values ​​34
§ 4. Further properties and applications of real numbers 35 18. Existence of a root. Power with a rational exponent 35 19. Power with any real exponent 37 20. Logarithms 39 21. Measurement of segments 40
CHAPTER FIRST. THEORY OF LIMITS
§ 1. Options and its limit 43 22. Variable value, options 43 23. Options limit 46

24. Infinitely small quantities 47 25. Examples 48 26. Some theorems about a variant with a limit 52 27. Infinitely large quantities 54
§ 2. Limit theorems that make it easier to find limits 56 28. Passing to the limit in equality and inequality 56 29. Infinitely small lemmas 57 30. Arithmetic operations on variables 58 31. Indefinite expressions 60 32. Examples for finding limits 62 33. Stolz's theorem and its applications 67
§ 3. Monotone Variant 70 34. Limit of Monotone Variant 70 35. Examples 72 36. Number e 77 37. Approximate calculation of the number e 79 38. Lemma on nested intervals 82
§ 4. The principle of convergence. Partial limits 83 39. Convergence principle 83 40. Partial sequences and partial limits 85 41. Bolzano-Weierstrass lemma 87 42. Maximum and minimum limits 89
CHAPTER TWO. FUNCTIONS OF A SINGLE VARIABLE
§ 1. The concept of a function 93 43. A variable and its range 93 44. Functional dependence between variables. Examples 94 45. Definition of the concept of a function 95 46. Analytical way of defining a function 98 47. Graph of a function 100 48. The most important classes of functions 102 49. The concept of an inverse function 108 50. Inverse trigonometric functions 110 51. Superposition of functions. Concluding remarks 114
§ 2. Limit of a function 115 52. Definition of the limit of a function 115

53. Reduction to the Variant Case 117 54. Examples 120 55. Extension of the Theory of Limits 128 56. Examples 130 57. Limit of a Monotonic Function 133 58. Common Bolzano-Cauchy Test 134 59. Maximum and Minimum Limits of a Function 135
§ 3. Classification of infinitesimals and infinitesimals 136 60. Comparison of infinitesimals 136 61. Scale of infinitesimals 137 62. Equivalent infinitesimals 139 63. Isolation of the main part 141 64. Problems 143 65. Classification of infinitesimals 145
§ 4. Continuity (and discontinuities) of functions 146 66. Determination of the continuity of a function at a point 146 67. Arithmetic operations on continuous functions 148 68. Examples of continuous functions 148 69. One-sided continuity. Classification of discontinuities 150 70. Examples of discontinuous functions 151 71. Continuity and discontinuities of a monotonic function 154 72. Continuity of elementary functions 155 73. Superposition of continuous functions 156 74. Solution of one functional equation 157 75. Functional characteristic of exponential, logarithmic and power functions
158 76. Functional characteristics of trigonometric and hyperbolic cosines
160 77. Using continuity of functions to calculate limits 162 78. Power-exponent expressions 165 79. Examples 166
§ 5. Properties of continuous functions 168 80. The vanishing theorem 168 81. Application to the solution of equations 170 82. The intermediate value theorem 171

83. The existence of an inverse function 172 84. The boundedness theorem for a function 174 85. The largest and smallest values ​​of a function 175 86. The concept of uniform continuity 178 87. Cantor's theorem 179 88. Borel's lemma 180 89. New proofs of the main theorems 182
CHAPTER THREE. DERIVATIVES AND DIFFERENTIALS
§ 1. The derivative and its calculation 186 90. The problem of calculating the speed of a moving point 186 91. The problem of drawing a tangent to a curve 187 92. Defining a derivative 189 93. Examples of calculating derivatives 193 94. Derivative of an inverse function 196 95. Summary of formulas for derivatives 198 96 Formula for incrementing a function 198 97 Simple rules for calculating derivatives 199 98 Derivative of a complex function 202 99 Examples 203 100 One-sided derivatives 209 101 Infinite derivatives 209 102 Further examples of special cases 211
§ 2. The differential 211 103. The definition of the differential 211 104. The connection between differentiability and the existence of a derivative
213 105. Basic formulas and rules of differentiation 215 106. Invariance of the form of a differential 216 107. Differentials as a source of approximate formulas 218 108. Application of differentials in estimating errors 220
§ 3. Fundamental theorems of differential calculus 223 109. Fermat's theorem 223 110. Darboux's theorem 224 111. Rolle's theorem 225 112. Lagrange's formula 226

113 Derivative limit 228 114 Cauchy formula 229
§ 4. Derivatives and differentials of higher orders 231 115. Definition of derivatives of higher orders 231 116. General formulas for derivatives of any order 232 117. Leibniz formula 236 118. Examples 238 119. Differentials of higher orders 241 120. Form invariance violation for differentials of higher orders
242 121. Parametric differentiation 243 122. Finite differences 244
§ 5. Taylor's formula 246 123. Taylor's formula for a polynomial 246 124. Decomposition of an arbitrary function; additional term in the form
Peano
248 125. Examples 251 126. Other forms of the additional term 254 127. Approximate formulas 257
§ 6. Interpolation 263 128. The simplest problem of interpolation. Lagrange formula 263 129. Additional term of Lagrange formula 264 130. Interpolation with multiple nodes. Hermite formula 265
CHAPTER FOUR. EXPLORING A FUNCTION WITH THE HELP
DERIVATIVES
§ 1. The study of the course of change of a function 268 131. The condition for the constancy of a function 268 132. The condition for the monotonicity of a function 270 133. The proof of inequalities 273 134. Maxima and minima; necessary conditions 276 135. Sufficient conditions. First rule 278 136. Examples 280 137. Second rule 284 138. Using higher derivatives 286 139. Finding the largest and smallest values ​​288

140. Tasks 290
§ 2. Convex (and concave) functions 294 141. Definition of a convex (concave) function 294 142. The simplest propositions about convex functions 296 143. Conditions for the convexity of a function 298 144. Jensen's inequality and its applications 301 145. Inflection points 303
§ 3. Construction of graphs of functions 305 146. Statement of the problem 305 147. Scheme for constructing a graph. Examples 306 148. Infinite gaps, infinite gap. Asymptotes 308 149. Examples 311
§ 4. Disclosure of uncertainties 314 150. Uncertainty of the form 0/0 314 151. Uncertainty of the form
∞
∞ /
320 152. Other types of uncertainties 322
§ 5. Approximate solution of an equation 324 153. Introductory remarks 324 154. Rule of proportional parts (method of chords) 325 155. Newton's rule (method of tangents) 328 156. Examples and exercises 331 157. Combined method 335 158. Examples and exercises 336
CHAPTER FIVE. FUNCTIONS OF MULTIPLE VARIABLES
§ 1. Basic concepts 340 159. Functional dependence between variables. Examples 340 160. Functions of two variables and their domains 341 161. Arithmetic n-dimensional space 345 162. Examples of regions in n-dimensional space 348 163. General definition of an open and closed region 350 164. Functions of n variables 352 165. Limit of a function of several variables 354 166. Reduction to the case of variants 356 167. Examples 358 168. Repeated limits 360

§ 2. Continuous functions 362 169. Continuity and discontinuities of functions of several variables 362 170. Operations on continuous functions 364 171. Functions continuous in a domain. Bolzano-Cauchy theorems 365 172. Bolzano-Weierstrass lemma 367 173. Weierstrass theorems 369 174. Uniform continuity 370 175. Borel lemma 372 176. New proofs of the main theorems 373 176. Derivatives and differentials of functions of several variables 373 177. Partial derivatives and partial differentials 375 178 Total increment of a function 378 179 Total differential 381 180 Geometric interpretation for the case of a function of two variables
383 181. Derivatives of complex functions 386 182. Examples 388 183. Finite increment formula 390 184. Derivative in a given direction 391 185. Invariance of the form of the (first) differential 394 186. Application of the total differential in approximate calculations 396 187. Homogeneous functions 399 188. Euler formula 400
§ 4. Derivatives to higher-order differentials 402 189. Higher-order derivatives 402 190. Mixed derivatives theorem 404 191. Generalization 407 192. Higher-order derivatives of a complex function 408 193. Higher-order differentials 410 194. Differentials of complex functions 413 195. Formula Taylor 414
§ 5. Extrema, maximum and minimum values ​​417 196. Extrema of a function of several variables. The necessary conditions
417 197. Sufficient conditions (the case of a function of two variables) 419

198. Sufficient conditions (general case) 422 199. Conditions for the absence of an extremum 425 200. The largest and smallest values ​​of functions. Examples 427 201. Tasks 431
CHAPTER SIX. FUNCTIONAL DETERMINERS; THEM
APPS
§ 1. Formal properties of functional determinants 441 202. Definition of functional determinants (Jacobians) 441 203. Multiplication of Jacobians 442 204. Multiplication of functional matrices (Jacobi matrices) 444
§ 2. Implicit functions 447 205. The concept of an implicit function of one variable 447 206. The existence of an implicit function 449 207. Differentiability of an implicit function 451 208. Implicit functions of several variables 453 209. Calculation of derivatives of implicit functions 460 210. Examples 463
§ 3. Some applications of the theory of implicit functions 467 211. Relative extrema 467 212. The method of indefinite Lagrange multipliers 470 213. Sufficient conditions for a relative extremum 472 214. Examples and problems 473 215. The concept of independence of functions 477 216. The rank of the Jacobian matrix 479
§ 4. Change of variables 483 217. Functions of one variable 483 218. Examples 485 219. Functions of several variables. Change of independent variables
488 220. Method for calculating differentials 489 221. General case of change of variables 491 222. Examples 493
CHAPTER SEVEN. APPLICATIONS OF DIFFERENTIAL
CALCULUS TO GEOMETRY
§ 1. Analytic representation of curves and surfaces 503

223. Curves on the plane (in rectangular coordinates) 503 224. Examples 505 225. Curves of mechanical origin 508 226. Curves on the plane (in polar coordinates). Examples 511 227. Surfaces and curves in space 516 228. Parametric representation 518 229. Examples 520
§ 2. Tangent and tangent plane 523 230. Tangent to a plane curve in rectangular coordinates 523 231. Examples 525 232. Tangent in polar coordinates 528 233. Examples 529 234. Tangent to a spatial curve. Tangent plane to surface
530 235. Examples 534 236. Singular points of plane curves 535 237. The case of parametrically defining a curve 540
§ 3. Tangency of curves 542 238. Envelope of a family of curves 542 239. Examples 545 240. Characteristic points 549 241. The order of tangency of two curves 551 242. The case of implicit specification of one of the curves 553 243. Touching curve 554 244. Another approach to touching curves 556
§ 4. Length of a plane curve 557 245. Lemmas 557 246. Direction on a curve 558 247. Length of a curve. Arc length additivity 560 248. Sufficient conditions for rectifiability. Arc differential 562 249. Arc as a parameter. Positive tangent direction 565
§ 5. Curvature of a plane curve 568 250. Concept of curvature 568 251. Circle of curvature and radius of curvature 571 252. Examples 573

253. Coordinates of the center of curvature 577 254. Definition of evolute and involute; search for an involute 578 255. Properties of evolutes and evolvents 581 256. Search for involutes 585
ADDITION. PROBLEM OF PROPAGATION OF FUNCTIONS
257 Case of a function of one variable 587 258 Statement of the problem for the two-dimensional case 588 259 Auxiliary propositions 590 260 Main propagation theorem 594 261 Generalization 595 262 Concluding remarks 597
Alphabetical index 600
Alphabetical index
Absolute value 14, 31, 34
Absolute extreme 469
Algebraic function 448
Analytical way of defining a function 97, 98
Analytic expression functions
98
- representation of curves 503, 517
- - surfaces 517
Anomaly (eccentric) of the planet
174
Function argument 95, 341
The arithmetic value of the root
(radical) 36, 103
- space 345
Arcsine, arccosine, etc. 110
Archimedes 64
Archimedes axiom 16, 34
Archimedean spiral 512, 529
Asymptote 309
Asymptotic point 513, 514
Astroid 506, 511, 526, 546, 573, 583
barometric formula 95
Bernoulli, John 206, 314
- Jacob 38
- lemniscate 515, 530, 575, 577
Infinite decimal 22
- derivative 209
Infinitely large value 54,
117
- - - classification 145
- - - order 145
- small value 47, 117
- - - higher order [designation
O(
α)] 136, 137
- - - classification 136
- - - Lemmas 57
- - - order 137
- - - equivalence 139
Infinity
,
−∞
+∞
26, 55
Infinite Gap 94, 308
- gap 309
Boyle-Mariotte law 94
Bolzano 84
Bolzano method 88
Bolzano-Weierstrass Lemma 87,
367
Bolzano-Cauchy theorems 1st and 2nd
168, 171, 182, 366
- - condition 84, 134
Borel Lemma 181, 372
Option 44, 344
- increasing (non-decreasing) 70
- having a limit of 52
- as function icon 96

Monotone 70
- limited 53
- decreasing (non-increasing) 70
Weierstrass-Bolzano Lemma 87,
367
- theorems 1 and 2 175, 176, 183,
369, 370, 373
Vertical asymptote 309
Upper bound number set 26
- - - - fine 26
Real numbers 19
- - subtraction 31
- - division 34
- - decimal approximation 22
- - area 24 continuity
- - density (enhanced) area 21
- - equality 19
- - addition 28
- - multiplication 31
- - area ordering 19
Viviani curve 521, 535
Helix 521, 534
- surface 523, 535
Nested spans, Lemma 83
internal point sets 350
Concave (convex up) functions or curves 295
- - - - concavity conditions 298
Return point 539, 541
Increasing option 70
- feature 133
Rotation surface 522
Convex (convex down) functions or curves 294
- - - - convexity conditions 298
- strictly functions or curves 298
Higher order infinitesimal
[designation o(
α)] 136, 137
- - differentials 241
- - - functions of several variables
410
- - derivatives 231, 232
245
- - - private 402
harmonic oscillation 208
Gauss 74, 439
Holder-Cauchy inequality 275,
302
Geographical coordinates 522
Geometric interpretation of the differential 214
- - full differential 386
- - derivative 190
Hyperbole 506, 575, 580
- isosceles 102, 103
Hyperbolic Spiral 529
Hyperbolic sine, cosine, etc. 107
- functions, continuity 149
- - reverse 108-109
- - derivatives 205
Hypocycloid 509
The main branch (principal value) of the arcsine, arccosine, etc.
110, 114
- part ( main member) infinitely small 141
Smooth curve 594
Horizontal asymptote 309
Function Gradient 394
Area border 351
- number set (upper, lower) 25-28
- - - fine 26
Function Graph 100
- - building 305
- - spatial 343
Huygens formula 260
Darboux theorem 224
Movement Equation 187
Double Curve Point 538
Double function limit 360
Two variable function 341
Dedekind 17
Dedekind Main Theorem 25

Real numbers, cm.
Real numbers
Cartesian sheet 507, 538
Decimal approximation of a real number 22
Decimal Logarithms 79
Dot set diameter 371
Dirichlet function 99, 102, 153
Discriminant curve 545, 550
Differential 211, 215
- order, 1st, 2nd, n 241
- geometric interpretation 214
- arches 562, 567
- shape invariance 216
- full 382
- - order, 1st, 2nd, n 410
- - geometric interpretation 386
- - shape invariance 394
- - method of calculation (when changing variables) 489
- application to approximate calculations 218, 220, 396
- private 378, 411
Differentiation 215
- parametric 243
- rules 215, 395
Differentiable function 212, 382
Implicit function differentiability 451
Length of segments 40
- flat curve 560
- - - additivity 560
- spatial curve 567
Additional formula term
Taylor 249, 257, 415
- - - Lagrange 263
- - - Ermita 266
Fractional rational function 103
- - - continuity 148
- - - multiple variables 353
e(number) 78, 148
- irrationality 82
- approximate calculation 81
Unit 14, 32
Dependent functions 478
Change of variables 483
Enclosed area 351
- sphere 351
Closed set 351
Closed box 351
Closed gap 93
- simplex 351
Point point 539
damped oscillation 208, 282
Signs rule (when multiplying) 16,
32
Jensen 295
Jensen inequality 301
Distance measurement 40
Isolated Curve Point 536, 539
Invariance of the differential form 216, 394
Interpolation 263
Interpolating nodes 263
- - multiples of 266
Interpolation formula
Lagrange 263
- - Ermita 266
Irrational numbers 19
Cantor theorem 179, 184, 370, 374
Cardioid 510, 515, 530
Touch curves 542
- - order 551
Tangent 188, 210, 386, 523, 530,
533, 555
- one-sided 209
- cut 524
- - polar 528
- plane 384, 532
- positive direction 567
Tangent transformation 485,
487, 493, 500
Tangent method (approximate solution of equations) 328
Cassini oval 515
quadratic form 423

Maximum and minimum values ​​476
- - undefined 425
- - defined 423
- - semi-defined 427
Kepler Equation 174
Clapeyron formula 340, 377
Smooth curve class 594
Classification of infinitely large
145
- - small 136
Function classes 102
Harmonic oscillation 208
- damped 208, 282
- functions 177, 370
Combined method
(approximate solution of equations) 335
Compressor 433
Finite differences 244
Finite increments formula 227,
390
Cone go, order, 2, 535
Coordinate lines (surfaces)
520
Coordinates n- measuring point 345
Real number root, existence 35
- equations (functions), existence 170
- - approximate calculation 170,
324
Cosine 103
- functional characteristic
160
- hyperbolic 107
160
Cosecant 103
Cotangent 103
- hyperbolic 107
Cauchy 67, 69, 84, 192
Cauchy-Bolzano theorems 1 and 2
168, 171, 182, 366
- - condition 84, 134
- additional member form 257
- formula 229
Multiple curve point 505, 519, 538,
540
Curvature 568
- circle 571
- radius 571
- average 568
- center 571
Curves, see corresponding title
- in space 517, 518
- in n-dimensional space 347
- on plane 503, 508, 511
- transitional 576
Kronecker 99
Cube n-dimensional 348
Piecewise smooth curve 595
Lagrange 192, 257, 470
Lagrange interpolation formula 263
- - - additional member 265
- theorem, formula 226, 227
- additional member form 257,
415
Lebesgue 181
Legendre polynomials 240
Legendre transformation 487, 499,
500
Leibniz 192, 215, 241
Leibniz formula 238, 241
Lemniscate Bernoulli 515, 530, 575,
577
Logarithm, existence 39
- decimal 50, 79
- natural (or neperov) 78
- - change to decimal 79
Logarithmic spiral 514, 529,
574, 581
- feature 103
- - continuity 155, 174
- - derivative 195, 197

Functional characteristic
159
broken line (in n-dimensional space)
347
L'Hopital rule 314, 320
Maclaurin formula 247, 251
Maximum, see extreme
Matrix functional (Jacobi)
444, 478
- - grade 468, 471, 479
Multiplication Matrices 444
Mere 44
Minimum, see extreme
Minkowski inequality 276
Multivalued function 96, 109, 341,
447, 453
The set of points is closed 351
- - limited 352
- numeric, limited from above, from below 26
Multipliers indefinite, method
470
Modulus of conversion from natural logarithms to decimal logarithms 79
Monotone option 70
- feature 133
- - continuity, breaks 154
Function monotonicity condition 270
n variable function 352
n-multiple curve point 540
n-multiple limit 360
n-dimensional sphere 349, 351
n-dimensional space 345
n-dimensional box 348, 351
n-dimensional simplex 349, 351
Highest value functions 176,
286
Highest limit options 89
- - functions 136
The smallest value of the function is 176,
289
- - - multiple variables 427
Least limit options 89
- - functions 136
Least squares method 438
Oblique asymptote 310
Function overlay 114
Curve Direction 558
natural logarithm 78
Independence of functions 478
Independent variables 94, 341,
352
Uncertainties disclosure 62, 314
- type 0/0 60, 314
- -
∞
∞ / 61, 320
- -
∞
⋅
0 61, 322
- -
∞
−
∞
62, 323
- -
0 0
,
0
,
1
∞
∞
166, 323
Indefinite multipliers, method
470
Napier, Napier logarithms 78
Continuity of the domain of real numbers 24
- straight 42
- functions in area 365
- - in the interval 148
- - at point 146, 362
- - one-sided 150
- - uniform 178, 370
Continuous functions, operations on them 148, 364
- - properties 168-185, 365-374
- - superposition 114, 364
Inequalities, proof 122,
273, 302
Cauchy's inequality 275, 346
- Cauchy-Gelder 275, 302
- Jensen 301
- Minkowski 276
Improper numbers (points) 26, 55,
355
Implicit Functions 447, 453
- - calculation of derivatives 460
- - existence and properties 449,
451, 453

Lower limit of the number set 26
- - - - fine 26
Curve normal 523
- - - cut 524
- - - - polar 528
Surface normal 532, 534
Newton method (approximate solution of equations) 328
Relative extreme 467
Section, measurement 40
- tangent, normal 524
- - - polar 528
Error estimation 220, 396
Region in n-dimensional space
350
- variable changes
(variables) 95, 341
- closed 351
- function definitions 95, 341
- open 350
- liaison 352
Inverse function 108
- - continuity 172
- - derivative 196
- - existence 172
Inverse trigonometric functions 110
- - - continuity 156, 174
- - - derivatives 197
Ordinary point (curve or surface) 504, 505, 520
Oval Cassini 515
Curve Family Envelope 543
Limited option 53
Bounded point set
352
- - numeric 26
Boundedness of a continuous function, Theorems 175, 183,
369, 373
Single value function 96, 341
Homogeneous function 399
One-way continuity and discontinuities of function 150
One-sided tangent 209
- derivative 209
- - higher order 232
Neighborhood of point 115
- -n-dimensional 348, 349
Determinant, derivative 388
- functional (Jacobi) 441
Singular point (curve or surface) 504, 505, 517, 518,
519, 531, 533, 535, 537
- - insulated 536
- - double 538
- - multiple of 505, 519, 538, 540
Ostrohradsky 442
open area 350
- sphere 349, 350
Open span 93
- box 348, 350
- simplex 349, 350
Relative error 140, 218,
397
Parabola 64, 103, 525, 546, 575, 579
Paraboloid of revolution 344
Parallelepiped n-dimensional 348
Parameter 217, 504
Parametric differentiation 243
- curve representation 217, 504, 512
- - - in space 518
- - surfaces 519
Peano form of the additional term
249
Inflection point 303
Variable 43, 93
- independent 94, 341, 352
Variable replacement 483
Commutative property of addition, multiplication 12, 14,
29, 32
Permutation differentiation
405, 407
- limit transitions 361, 406

Spirals 576
Periodic decimal 24
Surface 343, 517, 519
- rotations 522
Repeated limit of a function of several variables 360
Subtangent 207, 524
- polar 528
Subnormal 524
- polar 528
Subsequence 85
Border Point 351
Error absolute, relative 139, 140, 218,
221, 397
exponential function 103
- - continuity 149, 155
- - derivative 194
- - functional characteristic
158
Full function increment 378
Full differential 381, 396
- - higher order 410, 413
- - geometric interpretation 386
- - shape invariance 394
- - applications to approximate calculations 396
Semicubic parabola 506, 540,
548, 579
Semi-open span 93
Polar subtangent, subnormal 528
Polar Curve Equation 511
Polar coordinates 493, 495, 512
Polar segment of the tangent, normal 528
Order of infinite magnitude 145
- - small size 137
- differential 241
- touch curves 551
- derivative 231
Sequence 44
Function constancy condition 268
Rule, see related title
Limit options 46, 48
- - endless 55
- - uniqueness 54
- - monotonous 71
- - largest, smallest 89
- - partial 86
- relationship 59
- works 59
- derivative 228
- differences 59
- amounts 59
- functions 115, 117
- - monotonous 139
- - largest, smallest 135
- - multiple variables 354, 357
- - - - repeated 360
- - partial 135
Passage to the limit in equality, in inequality 56
Legendre transformation 487, 499,
500
- point (planes, spaces)
485, 493
Approximate solution of the equation
324
Approximate calculations, differential application
218, 220, 396
Approximate formulas 140, 143,
218, 257-263
Variable Increment 147
- functions, formula 199
- multiple variables complete, formula 379
- - - - private 375
Increment final formula 227,
390
Product variant, limit 59, 61
- functions, limit 129, 130
- - continuity 148, 364
216, 236, 241, 395

Product of numbers 14, 31
Derivative see also, name, functions, 189
- endless 209
- higher order 231
- - - connection with finite differences
245
- geometric interpretation 190
- non-existence 211
- one-sided 209
- in a given direction 391
- calculation rules 199
- gap 211
- private 375
- - higher order 402
Gap 82
- closed, semi-open, open, finite, endless 93, 94
Intermediate value, theorem
171
Proportional parts, rule
325
simple point(curve or surface) 505, 520
Spatial graph of a function
343
Space n-dimensional
(arithmetic) 345
Direct to n-dimensional space 347
Uniform continuity of a function 178, 370
Radical, arithmetic value
36, 103
Radius of curvature 571
difference option, etc., see sum
- numbers 13, 31
Derivative break 211
- 146 functions
- - monotonous 154
- - ordinary, kind, go, and, go, 1, 2,
151
- - multiple variables 362
Matrix rank 468, 471, 479
Disclosure of Uncertainties 62,
314
distribution property multiplications 15, 34
Feature distribution 587
Distance between points in n- dimensional space 345
Rational function 102
- - continuity 148
- - multiple variables 353
- - - - continuity 358, 563
Rational numbers, subtraction 13
Rational numbers division 15
- - density 12
- - addition 12
- - multiplication 14
- - ordering 12
Riemann 154
Rolle's Theorem 225
Rosha and Schlemilha additional member form 257
Links Equation 467
Connected area 352
Condensations point 115, 116, 117, 351
Sekans 103
Curve family 542
Section in the numerical area 17, 24
Signum (function) 29
Current 192
Sylvester 423
Simplex n-dimensional 349, 351
Sinus 103
- hyperbolic 107
- limit of relation to the arc 122
Sinusoid 106, 304
Point movement speed 186
- in this moment 187, 190
- medium 186
Complex function 115, 353
- - continuity 156, 365
- - derivatives and differentials
202, 216, 242, 386, 395, 413, 414
Mixed derivatives, theorem
404

Contiguous Curve 554
- straight 555
Contact circle 555, 571
Associative property of addition, multiplication 13, 14, 29, 32
Comparison of infinitesimals 136
Arithmetic-harmonic mean
74
- - - geometric 74
- arithmetic 275, 430
- harmonic 74, 303
- geometric 74, 275, 303, 430
- value, theorem 227
- - generalized theorem 230
Average curvature 568
- speed 186, 190
Stationary point 277, 418
Power function 103
- - continuity 156
- - derivative 194
- - functional characteristic
158
exponential function
(two variables) 353
Power exponential function limit 358, 359
- - - - continuity 363
- - - - differentiation 376
Power exponential expression, limit 165
- - - - derivative 206, 388
Degree with real exponent 37
Sum option, limit 59, 62
- functions, limit 129, 130
- functions, continuity 148, 364
- - derivative and differential 200,
216, 233, 395
- numbers 12, 28
Superposition of functions 114, 353, 364
Sphere 344
-n-dimensional 349, 350
Spherical coordinates 495
Convergence principle 84, 134
Tabular way function assignments
97
Tangent 103
- hyperbolic 107
Geometric body 345
Heat capacity 191
Point, see related name
Function points 352
Fine limit (upper, lower) 26
Trigonometric functions 103
- - continuity 149
- - derivatives 195
Triple point 540
Triple Limit 360
Taylor formula 246, 249, 257, 415
Descending option 70
- feature 133
corner point 209
Interpolation nodes 263
- - multiples of 266
Whitney 590
Snail 514, 529
Curve equation 100, 230, 503, 511,
518
- surfaces 343, 517, 519
- approximate solution 170, 324
- existence of roots 170
Acceleration 191, 231
Farm theorem 223
Form quadratic 423
Formula see also, corresponding, name, 97,
98
Functional dependency 94, 340
- matrix 444, 478
Functional Equation 157, 158,
160
Functional identifier 441
Function see also, name, functions, 95
- study 268
- several variables 341, 352
- from function (or from functions) 115,
353

Characteristic point on a curve
539
Hestins 590
Function change progress 268
Chord method of approximate solution of equations 325
An entire rational function 102
- - - continuity 149
- - - several variables 353
- - - - - continuity 358, 363
- part of the number [ E(R)] 48
Center of curvature 571, 577
Chain line 207, 505, 573
Cycloid 508, 526, 574, 581
Projecting cylinder 518
Partial Sequence 85
Partial limit options 86
- - functions 135
Partial derivative 375
- - higher order 402
Private option, limit 59, 60
- function value 96
- increment 375
- functions, limit 129, 130
- - continuity 148, 364
- - derivative and differential 201,
216, 395
- numbers 15
Private differential 378, 411
Chebyshev formula 262
Numbers, see Rational,
irrational,
Real numbers
Numerical axis 42
- sequence 44
Schwarz 407
Schlemilha and Rocha additional member form 257
Stolz Theorem 67
Involute 578, 582-583, 585
- circle 511, 527, 574
Evolute 579, 582, 583, 585
Euler 78
Euler formula 401
Equivalent infinitesimals (sign) 139
Extreme (maximum, minimum) 277
- search rules 277, 278, 284,
287
- own, not own 277
- functions of several variables
417
- - - - absolute 469
- - - - relative 467
Electrical network 436, 474
Elementary Functions 102
- - continuity 155
- - derivatives 193, 197, 233
Ellipse 448, 506, 525, 547, 575, 579
Ellipsoid 535
Hermite interpolation formula
266
- - - additional member 267
Epicycloid 509, 527
Jacobi 376
- matrix 444, 478
- determinant (jacobian) 441

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CHAPTER EIGHT. DERIVATIVE FUNCTION (INDETERMINATE INTEGRAL)

§ 1. Indefinite integral and the simplest methods of its calculation
263. Concept of antiderivative function (and indefinite integral)
264. The Integral and the Area Problem
265. Table of basic integrals
266. The Simplest Integration Rules
267. Examples
268. Integration by Change of Variable
269. Examples
270. Integration by parts
271. Examples

§ 2. Integration rational expressions
272. Statement of the Integration Problem in the Final Form
273. Simple fractions and their integration
274. Decomposition of proper fractions into simple ones
275. Determination of coefficients. Integration of proper fractions
276. Separation of the rational part of the integral
277. Examples

§ 3. Integration of some expressions containing radicals
278. Integration of expressions
279. Integration of binomial differentials. Examples
280. Reduction formulas
281. Integration of expressions. Euler substitutions
282. Geometric treatment of Euler substitutions
283. Examples
284. Other Methods of Calculation
285. Examples

§ 4. Integration of expressions containing trigonometric and exponential functions
286. Integration of Differentials R(sin x, cos x)
287. Integration of expressions
288. Examples
289. Review of other cases

§ 5. Elliptic integrals
290. General remarks and definitions
291. Auxiliary transformations
292. Reduction to canonical form
293. Elliptic integrals of the 1st, 2nd and 3rd kind

CHAPTER NINE. DEFINITION INTEGRAL

§ 1. Definition and conditions for the existence of a definite integral
294. Another approach to the area problem
295. Definition
296. Darboux sums
297. The Condition for the Existence of an Integral
298. Classes of Integrable Functions
299. Properties of Integrable Functions
300. Examples and additions
301. Lower and Upper Integrals as Limits

§ 2. Properties of definite integrals
302. Integral over an oriented interval
303. Properties expressed by equalities
304. Properties Expressed by Inequalities PO
305. Definite Integral as a Function of the Upper Limit
306. Second Mean Value Theorem

§ 3. Calculation and transformation of definite integrals
307. Calculation with the help of integral sums
308. Basic Formula of Integral Calculus
309. Examples
310. Another derivation of the main formula
311. Reduction formulas
312. Examples
313. The formula for the change of variable in a definite integral
314. Examples
315. Gauss formula. Landen transform
316. Another derivation of the change of variable formula

§ 4. Some applications of definite integrals
317. Wallis Formula
318. Taylor formula with an additional term
319. Transcendence of the number e
320. Legendre Polynomials
321. Integral inequalities

§ 5. Approximate calculation of integrals
322. Statement of the problem. Formulas for rectangles and trapezoids
323 Parabolic Interpolation
324. Splitting the Interval of Integration
325. Additional term of the formula of rectangles
326. Additional term of the trapezoid formula
327. Additional term of Simpson's formula
328. Examples

CHAPTER TEN. APPLICATIONS OF THE INTEGRAL CALCULUS TO GEOMETRY, MECHANICS AND PHYSICS

§ 1. Curve length
329 Calculating the Length of a Curve
330. Another approach to the definition of the concept of the length of a curve and its calculation
331. Examples
332. Natural Equation of a Plane Curve
333. Examples
334. Arc Length of a Space Curve

§ 2. Areas and volumes
335. Definition of the concept of area. Additivity property
336. Area as a Limit
337. Classes of squaring regions
338. Expression of area by integral
339. Examples
340. Definition of the concept of volume. Its properties
341. Classes of bodies having volumes
342. Expression of Volume by an Integral
343. Examples
344. Surface area of ​​rotation
345. Examples
346. Area of ​​a cylindrical surface
347. Examples

§ 3. Calculation of mechanical and physical quantities
348. Scheme of Application of a Definite Integral
349. Finding the Static Moments and the Center of Gravity of a Curve
350. Examples
351. Finding the static moments and the center of gravity of a plane figure
352. Examples
353. Mechanical work
354. Examples
355. The work of the friction force in a flat heel
356. Problems for the summation of infinitesimal elements

§ 4. The simplest differential equations
357. Basic concepts. First order equations
358. Equations of the first degree with respect to the derivative. Separation of variables
359. Tasks
360. Remarks on the Compilation of Differential Equations
361. Tasks

CHAPTER ELEVEN. ENDLESS ROWS WITH PERMANENT MEMBERS

§ 1. Introduction
362. Basic concepts
363. Examples
364. Fundamental Theorems

§ 2. Convergence of positive series
365. Condition for the Convergence of a Positive Series
366. Series Comparison Theorems
367. Examples
368. Signs of Cauchy and D'Alembert
369. Sign of Raabe
370. Examples
371. Sign of Kummer
372. Gauss sign
373. Integral sign of Maclaurin-Cauchy
374. Sign of Ermakov
375. Additions

§ 3. Convergence of arbitrary series
376. General Condition for the Convergence of a Series
377. Absolute Convergence
378. Examples
379. Power Series, Its Interval of Convergence
380. Expression of the radius of convergence in terms of coefficients
381. Alternating Series
382. Examples
383. Abel Transform
384. Signs of Abel and Dirichlet
385. Examples

§ 4. Properties of convergent series
386. Associative Property
387. Commutative property of absolutely convergent series
388. The Case of Nonabsolutely Convergent Series
389. Multiplication of rows
390. Examples
391. General theorem from the theory of limits
392. Further theorems on the multiplication of series

§ 5. Repeated and double rows
393. Repeated rows
394. Double rows
395. Examples
396 Power series with two variables; region of convergence
397. Examples
398. Multiple rows

§ 6. Infinite products
399. Basic concepts
400. Examples
401. Basic theorems. Relationship with rows
402. Examples

§ 7. Expansions of elementary functions
403. Expansion of a function in a power series; Taylor series
404. Expansion in a series of exponential, basic trigonometric functions, etc.
405. Logarithmic Series
406. Stirling formula
407. Binomial Series
408. Decomposition of sine and cosine into infinite products

§ 8. Approximate calculations with the help of series. Series conversion
409. General remarks
410. Calculating the number of tt
411. Calculating Logarithms
412. Calculating Roots
413. Euler Series Transformation
414. Examples
415. Kummer's Transformation
416. Markov Transform

§ 9. Summation of divergent series
417. Introduction
418. Power Series Method
419. Tauber's theorem
420. Method of Arithmetic Averages
421. Relationship between Poisson-Abel and Cesaro methods
422. Hardy-Landau Theorem
423. Application of generalized summation to multiplication of series
424. Other methods of generalized summation of series
425. Examples
426. General class of linear regular summation methods

CHAPTER TWELVE. FUNCTIONAL SEQUENCES AND SERIES

§ 1. Uniform convergence
427. Introductory remarks
428. Uniform and non-uniform convergence
429. Condition for uniform convergence
430. Criteria for Uniform Convergence of Series

§ 2. Functional properties of the sum of a series
431. Continuity of the sum of a series
432. A remark on quasi-uniform convergence
433. Transition to the limit term by term
434. Termwise Integration of Series
435. Term Differentiation of Series
436. Sequence Point of View
437. Continuity of the sum of a power series
438. Integration and differentiation of power series

§ 3. Applications
439. Examples on the continuity of the sum of a series and on the passage to the limit term by term
440. Examples for term-by-term integration of series
441. Examples for term-by-term differentiation of series
442. Method of Successive Approximations in the Theory of Implicit Functions
443. Analytical Definition of Trigonometric Functions
444. An example of a continuous function without a derivative

§ 4. Additional information about power series
445. Actions on power series
446. Substituting a row into a row
447. Examples
448. Division of power series
449. Bernoulli numbers and expansions in which they occur
450. Solving Equations in Series
451. Power series inversion
452. Lagrange series

§ 5. Elementary functions of a complex variable
453. Complex Numbers
454. Complex variant and its limit
455. Functions of a Complex Variable
456. Power Series
457. Exponential function
458. Logarithmic function
459. Trigonometric Functions and Their Inverses
460. Power Function
461. Examples

§ 6. Enveloping and asymptotic series. Euler-Maclaurin formula
462. Examples
463. Definitions
464. Basic Properties of Asymptotic Expansions
465. Derivation of the Euler-Maclaurin Formula
466. Study of an additional member
467. Examples of Calculations Using the Euler-Maclaurin Formula
468. Another form of the Euler-Maclaurin formula
469. Sterling's Formula and Series

CHAPTER THIRTEEN. Improper integrals

§ 1. Improper integrals with infinite limits
470. Definition of integrals with infinite limits
471. Application of the basic formula of integral calculus
472. Examples
473. Analogy with series. The simplest theorems
474. Convergence of the Integral in the Case of a Positive Function
475. Convergence of the Integral in the General Case
476. Signs of Abel and Dirichlet
477. Reducing an Improper Integral to an Infinite Series
478. Examples

§ 2. Improper integrals of unbounded functions
479. Definition of Integrals of Unbounded Functions
480. A note on singular points
481. Application of the basic formula of integral calculus. Examples
482. Conditions and signs of the existence of an integral
483. Examples
484. Principal Values ​​of Improper Integrals
485. A Remark on Generalized Values ​​of Divergent Integrals

§ 3. Properties and transformation of improper integrals
486. The Simplest Properties
487. Mean Value Theorems
488 Integration by Parts in the Case of Improper Integrals
489. Examples
490. Change of Variables in Improper Integrals
491. Examples

§ 4. Special methods for calculating improper integrals
492. Some Remarkable Integrals
493. Calculation of improper integrals with the help of integral sums. The case of integrals with finite limits
494. The Case of Integrals with an Infinite Limit
495 Frullani Integrals
496. Integrals of Rational Functions between Infinite Limits
497. Mixed examples and exercises

§ 5. Approximate calculation of improper integrals
498. Integrals with finite limits; highlighting features
499. Examples
500. Remark on Approximate Calculation of Eigenintegrals
501. Approximate calculation of improper integrals with an infinite limit
502. Use of asymptotic expansions

CHAPTER FOURTEEN. INTEGRALS DEPENDING ON A PARAMETER

§ 1. Elementary theory
503. Statement of the problem
504. Uniform Aspiration to the Limit Function
505. Permutation of two passages to the limit
506. Passing to the limit under the integral sign
507. Differentiation under the Integral Sign
508. Integration under the integral sign
509. The Case When And The Limits Of The Integral Depend On The Parameter
510. Introduction of a multiplier depending only on x
511. Examples
512. Gaussian proof of the fundamental theorem of algebra

§ 2. Uniform convergence of integrals
513. Definition of uniform convergence of integrals
514. Condition for uniform convergence. Relationship with rows
515. Sufficient Tests for Uniform Convergence
516. Another case of uniform convergence
517. Examples

§ 3. Use of uniform convergence of integrals
518. Passing to the limit under the integral sign
519. Examples
520. Continuity and differentiability of an integral with respect to a parameter
521. Integration over a parameter
522. Application to the calculation of certain integrals
523. Examples for Differentiation under the Integral Sign
524. Examples for integration under the integral sign

§ 4. Additions
525. Arzel's Lemma
526. Passing to the limit under the integral sign
527. Differentiation under the Integral Sign
528. Integration under the integral sign

§ 5. Euler integrals
529. Euler integral of the first kind
530. Euler integral of the second kind
531. The Simplest Properties of the Function Γ
532. Unique definition of the function Γ by its properties
533. Another functional characteristic of the function Г
534. Examples
535. The logarithmic derivative of the function Г
536. The multiplication theorem for the function Г
537. Some expansions into series and products
538. Examples and additions
539. Calculation of certain definite integrals
540. Stirling formula 9
541 Calculating the Euler Constant
542. Drawing up a table of decimal logarithms of the function G

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INTRODUCTION REAL NUMBERS

§ 1. The region of rational numbers
1. Preliminary remarks
2. Ordering the region of rational numbers
3. Addition and subtraction of rational numbers
4. Multiplication and division of rational numbers
5. Axiom of Archimedes

§ 2. Introduction of irrational numbers. Ordering the domain of real numbers
6. Definition of an irrational number
7. Ordering the domain of real numbers
8. Auxiliary sentences
9. Representation of a real number by an infinite decimal fraction
10. Continuity of the domain of real numbers
11. Boundaries of Numerical Sets

§ 3. Arithmetic operations on real numbers
12. Determining the sum of real numbers
13. Properties of addition
14. Definition of the product of real numbers
15. Properties of multiplication
16. Conclusion
17. Absolute values

§ 4. Further properties and applications of real numbers
18. Existence of a root. Degree with rational exponent
19. Degree with any real exponent
20. Logarithms
21. Measuring segments

CHAPTER FIRST. THEORY OF LIMITS

§ 1. Variants and its limit
22. Variable, options
23. Limit options
24. Infinitesimals
25. Examples
26. Some theorems about a variant with a limit
27. Infinitely large quantities

§ 2. Limit theorems that make it easier to find limits
28. Passage to the Limit in Equality and Inequality
29. Lemmas on infinitesimals
30. Arithmetic operations on variables
31. Indefinite expressions
32. Examples for finding limits
33. Stolz's theorem and its applications

§ 3. Monotonic variant
34. Limit of monotonic variants
35. Examples
36. Number e
31. Approximate calculation of the number e
38. Lemma on nested intervals

§ 4. The principle of convergence. Partial Limits
39. The principle of convergence
40. Partial sequences and partial limits
41. Bolzano-Weierstrass Lemma
42. Greatest and smallest limits

CHAPTER TWO. FUNCTIONS OF A SINGLE VARIABLE

§ 1. The concept of a function
43. Variable and area of ​​its change
44. Functional dependence between variables. Examples
45. Definition of the concept of a function
46. ​​Analytical way of defining a function
47. Function Graph
48. The most important classes of functions
49. The concept of an inverse function
50. Inverse Trigonometric Functions
51. Superposition of functions. Final remarks

§ 2. Limit of a function
52. Definition of the limit of a function
53. Reduction to the case of variants
54. Examples
55. Extension of the theory of limits
56. Examples
57. Limit of a monotonic function
58. Common sign of Bolzano-Cauchy
59. The largest and smallest limits of a function

§ 3. Classification of infinitesimal and infinitely large quantities
60. Comparison of infinitesimals
61. Scale of infinitesimals
62. Equivalent infinitesimals
63. Highlighting the main part
64. Tasks
65. Classification of infinitely large

§ 4. Continuity (and discontinuities) of functions
66. Determining the continuity of a function at a point
67. Arithmetic operations on continuous functions
68. Examples of Continuous Functions
69. One-way continuity. Classification of breaks
70. Examples of discontinuous functions
71. Continuity and discontinuities of a monotonic function
72. Continuity of elementary functions
73. Superposition of continuous functions
74. Solution of one functional equation
75. Functional characteristics of exponential, logarithmic and power functions
76. Functional characteristics of trigonometric and hyperbolic cosines
77. Using the Continuity of Functions to Calculate Limits
78. Power and exponential expressions

§ 5. Properties of continuous functions
80. The vanishing theorem
81. Application to solution of equations
82. Intermediate Value Theorem
83. Existence of an inverse function
84. Boundedness Theorem
85. The largest and smallest values ​​of a function
86. The concept of uniform continuity
87. Cantor's theorem
88. Borel Lemma
89. New proofs of the main theorems

CHAPTER THREE. DERIVATIVES AND DIFFERENTIALS

§ 1. Derivative and its calculation
90. The problem of calculating the speed of a moving point
91. The problem of drawing a tangent to a curve
92. Definition of a derivative
93. Examples of calculating derivatives
94. Derivative of the inverse function
95. Summary of formulas for derivatives
96. Formula for function increment
97. The simplest rules for calculating derivatives
98. Derivative of a compound function
99. Examples
100. One-sided derivatives
101. Infinite derivatives
102. Further examples of special cases

§ 2. Differential
103. Definition of differential
104. Relationship between differentiability and the existence of a derivative
105. Basic formulas and rules of differentiation
106. Invariance of the differential form
107. Differentials as a source of approximate formulas
108. The use of differentials in the estimation of errors

§ 3. Basic theorems of differential calculus
109. Fermat's Theorem
110. Darboux theorem
111. Rolle's theorem
112. Lagrange formula
113. Derivative limit
114. Cauchy formula

§ 4. Derivatives and differentials of higher orders
115. Definition of derivatives of higher orders
116. General formulas for derivatives of any order
117. Leibniz formula
118. Examples
119. Higher Order Differentials
120. Form Invariance Violation for Higher-Order Differentials
121. Parametric differentiation
122. Finite Differences

§ 5. Taylor formula
123. Taylor formula for a polynomial
124. Decomposition of an arbitrary function; additional member in Peano form
125. Examples
126. Other forms of additional term
127. Approximate Formulas

§ 6. Interpolation
128. The simplest problem of interpolation. Lagrange formula
129. Additional term of the Lagrange formula
130. Interpolation with multiple nodes. Hermite formula

CHAPTER FOUR. INVESTIGATION OF A FUNCTION WITH THE HELP OF DERIVATIVES

§ 1. Study of the course of change of a function
131. Condition of constancy of a function
132. Condition of monotonicity of a function
133. Proof of Inequalities
134. Highs and lows; the necessary conditions
135. Sufficient conditions. First rule
136. Examples
137. Second Rule
138. Use of Higher Derivatives
139. Finding the largest and smallest values
140. Tasks

§ 2. Convex (and concave) functions
141. Definition of a convex (concave) function
142. The simplest propositions about convex functions
143. Conditions for the Convexity of a Function
144. Jensen's inequality and its applications
145. Inflection Points

§ 3. Construction of graphs of functions
146. Statement of the problem
147. Scheme for constructing a graph. Examples
148. Endless gaps, endless gap. Asymptotes
149. Examples

§ 4. Disclosure of uncertainties
150. Uncertainty of the form 0/0
151. Uncertainty of the form oo/oo
152. Other types of uncertainties

§ 5. Approximate solution of the equation
153. Introductory remarks
154. Rule of proportional parts (method of chords)
155. Newton's rule (tangent method)
156. Examples and exercises
157. Combined method
158. Examples and exercises

CHAPTER FIVE. FUNCTIONS OF MULTIPLE VARIABLES

§ 1. Basic concepts
159. Functional dependence between variables. Examples
160. Functions of Two Variables and Their Domains
161. Arithmetic n-dimensional space
162. Examples of regions in n-dimensional space
163. General definition of open and closed area
164. Functions of n Variables
165. Limit of a Function of Several Variables
166. Reduction to the case of variants
167. Examples
168. Repeat limits

§ 2. Continuous functions
169. Continuity and Discontinuities of Functions of Several Variables
170. Operations on Continuous Functions
171. Functions continuous in a domain. Bolzano-Cauchy theorems
172. Bolzano-Weierstrass Lemma
173. Weierstrass' theorems
174. Uniform continuity
175. Borel Lemma
176. New proofs of the main theorems. Derivatives and differentials of functions of several variables
177. Partial Derivatives and Partial Differentials
178. Complete increment of a function
179. Full differential
180. Geometric Interpretation for the Case of a Function of Two Variables
181. Derivatives of complex functions
182. Examples
183. Finite Increment Formula
184. Derivative with respect to a given direction
185. Invariance of the Form of the (First) Differential
186. Application of the Total Differential in Approximate Calculations
187. Homogeneous functions
188. Euler formula

§ 4. Derivatives into Higher-Order Differentials
189. Derivatives of Higher Orders
190. Mixed Derivatives Theorem
191. Generalization
192. Higher Order Derivatives of a Complex Function
193. Higher Order Differentials
194. Differentials of Complex Functions
195. Taylor formula

§ 5. Extremes, maximum and minimum values
196. Extrema of a function of several variables. The necessary conditions
197. Sufficient conditions (the case of a function of two variables)
198. Sufficient conditions (general case)
199. Conditions for the absence of an extremum
200. The largest and smallest values ​​of functions. Examples
201. Tasks

CHAPTER SIX. FUNCTIONAL DETERMINERS; THEIR APPS

§ 1. Formal properties of functional determinants
202. Definition of functional determinants (Jacobians)
203. Multiplication of Jacobians
204. Multiplication of function matrices (Jacobi matrices)

§ 2. Implicit functions
205. The concept of an implicit function of one variable
206. Existence of an implicit function
207. Differentiability of an Implicit Function
208. Implicit functions of several variables
209. Calculation of derivatives of implicit functions
210. Examples

§ 3. Some applications of the theory of implicit functions
211. Relative Extremes
212. Method of indefinite Lagrange multipliers
213. Sufficient conditions for a relative extremum
214. Examples and tasks
215. The concept of independence of functions
216. The rank of the Jacobian matrix

§ 4. Change of variables
217. Functions of one variable
218. Examples
219. Functions of several variables. Change of independent variables
220. Method for calculating differentials
221. General case of change of variables
222. Examples

CHAPTER SEVEN. APPLICATIONS OF DIFFERENTIAL CALCULUS TO GEOMETRY

§ 1. Analytic representation of curves and surfaces
223. Curves on the plane (in rectangular coordinates)
224. Examples
225. Curves of mechanical origin
226. Curves on the plane (in polar coordinates). Examples
227. Surfaces and curves in space
228. Parametric representation
229. Examples

§ 2. Tangent and tangent plane
230. Tangent to a plane curve in rectangular coordinates
231. Examples
232. Tangent in polar coordinates
233. Examples
234. Tangent to a spatial curve. Tangent plane to surface
235. Examples
236. Singular points of plane curves
237. The Case of Parametric Curve Specification

§ 3. Tangent between curves
238. Envelope of a family of curves
239. Examples
240. Characteristic points
241. The order of touching two curves
242. The Case of Implicitly Specifying One of the Curves
243. Contiguous Curve
244. Another approach to contiguous curves

§ 4. Length of a plane curve
245. Lemmas
246. Curve Direction
247. The length of the curve. Arc length additivity
248. Sufficient Conditions for Rectifiability. Arc differential
249. Arc as a parameter. Positive tangent direction

§ 5. Curvature of a Plane Curve
250. The concept of curvature
251. Circle of Curvature and Radius of Curvature
252. Examples
253. Coordinates of the center of curvature
254. Definition of evolute and evolvent; search for an evolution
255. Properties of Evolutes and Evolutes
256. Search for evolvents

ADDITION. PROBLEM OF PROPAGATION OF FUNCTIONS
257. The case of a function of one variable
258. Statement of the Problem for the Two-Dimensional Case
259. Auxiliary sentences
260. Main Propagation Theorem