Fichtenholtz course of differential and integral calculus. Fikhtengolts G.M.

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INTRODUCTION REAL NUMBERS

§ 1. The domain of rational numbers
1. Preliminary remarks
2. Ordering the domain 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. Supporting suggestions
9. Representation of a real number as an infinite decimal fraction
10. Continuity of the domain of real numbers
11. Boundaries of numerical sets

§ 3. Arithmetic operations over real numbers
12. Determination of 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 c rational indicator
19. Power with any real exponent
20. Logarithms
21. Measuring segments

CHAPTER FIRST. THEORY OF LIMITS

§ 1. Variation and its limit
22. Variable value, option
23. Limit options
24. Infinitesimal quantities
25. Examples
26. Some theorems about a variant having a limit
27. Infinitely large quantities

§ 2. Limit theorems that make it easier to find limits
28. Limit passage in equality and inequality
29. Lemmas on infinitesimals
30. Arithmetic operations over variables
31. Vague Expressions
32. Examples for finding limits
33. Stolz's theorem and its applications

§ 3. Monotonous variant
34. Limit of monotonic options
35. Examples
36. Number e
31. Approximate calculation of the number e
38. Lemma about nested intervals

§ 4. Principle of convergence. Partial limits
39. Principle of convergence
40. Partial sequences and partial limits
41. Bolzano-Weierstrass Lemma
42. Largest and smallest limits

CHAPTER TWO. FUNCTIONS OF ONE VARIABLE

§ 1. Concept of function
43. Variable and its scope of change
44. Functional dependence between variables. Examples
45. Definition of the concept of function
46. Analytical method of specifying a function
47. Graph of a function
48. The most important classes of functions
49. Concept of inverse function
50. Inverse trigonometric functions
51. Superposition of functions. Concluding remarks

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

§ 3. Classification of infinitely small and infinitely large quantities
60. Comparison of infinitesimals
61. Infinitesimal scale
62. Equivalent infinitesimals
63. Selecting the main part
64. Tasks
65. Classification of infinitely large

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

§ 5. Properties of continuous functions
80. Theorem on vanishing of a function
81. Application to solving equations
82. Intermediate value theorem
83. Existence of an inverse function
84. Theorem on the boundedness of a function
85. Largest and smallest values of a function
86. The concept of uniform continuity
87. Cantor's theorem
88. Borel's 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 derivative
93. Examples of calculating derivatives
94. Derivative of an inverse function
95. Summary of formulas for derivatives
96. Formula for incrementing a function
97. The simplest rules for calculating derivatives
98. Derivative of a complex function
99. Examples
100. One-sided derivatives
101. Infinite derivatives
102. Further examples special occasions

§ 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 shape of the differential
107. Differentials as a source of approximate formulas
108. Application of differentials when estimating errors

§ 3. Main theorems differential calculus
109. Fermat's theorem
110. Darboux's theorem
111. Rolle's theorem
112. Lagrange formula
113. Limit of derivative
114. Cauchy formula

§ 4. Derivatives and differentials of higher orders
115. Determination of higher order derivatives
116. General formulas for derivatives of any order
117. Leibniz formula
118. Examples
119. Higher order differentials
120. Violation of form invariance for differentials of higher orders
121. Parametric differentiation
122. Finite differences

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

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

CHAPTER FOUR. STUDYING A FUNCTION USING DERIVATIVES

§ 1. Study of the progress of changes in a function
131. Condition for constancy of function
132. Condition for a function to be monotonic
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 sentences 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 type 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 SEVERAL VARIABLES

§ 1. Basic concepts
159. Functional dependence between variables. Examples
160. Functions of two variables and their domains of definition
161. Arithmetic n-dimensional space
162. Examples of regions in n-dimensional space
163. General definition open and closed area
164. Functions of n variables
165. Limit of a function of several variables
166. Reduction to case options
167. Examples
168. Repeated limits

§ 2. Continuous functions
169. Continuity and discontinuities of functions of several variables
170. Operations on continuous functions
171. Functions continuous in a region. Bolzano-Cauchy theorems
172. Bolzano-Weierstrass Lemma
173. Weierstrass's theorems
174. Uniform continuity
175. Borel's Lemma
176. New proofs of the main theorems. Derivatives and differentials of functions of several variables
177. Partial derivatives and partial differentials
178. Full function increment
179. Full differential
180. Geometric interpretation for the case of a function of two variables
181. Derivatives of complex functions
182. Examples
183. Formula for finite increments
184. Derivative in a given direction
185. Invariance of the form of the (first) differential
186. Application of total differential in approximate calculations
187. Homogeneous functions
188. Euler's formula

§ 4. Derivatives into differentials of higher orders
189. Derivatives of higher orders
190. Mixed derivatives theorem
191. Generalization
192. Higher order derivatives of a complex function
193. Differentials of higher orders
194. Differentials of complex functions
195. Taylor's formula

§ 5. Extrema, largest and smallest 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 DETERMINANTS; THEIR APPLICATIONS

§ 1. Formal properties of functional determinants
202. Determination of functional determinants (Jacobians)
203. Multiplication of Jacobians
204. Multiplication of functional 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. Lagrange's method of undetermined multipliers
213. Sufficient conditions for a relative extremum
214. Examples and problems
215. The concept of independence of functions
216. Rank of the Jacobi matrix

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

CHAPTER SEVEN. APPLICATIONS OF DIFFERENTIAL CALCULUS TO GEOMETRY

§ 1. Analytical representation of curves and surfaces
223. Curves on a plane (in rectangular coordinates)
224. Examples
225. Curves mechanical origin
226. Curves on a 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 parametrically defining a curve

§ 3. Touching curves to each other
238. Envelope of a family of curves
239. Examples
240. Characteristic points
241. The order of tangency of two curves
242. The case of implicitly specifying one of the curves
243. Occupying curve
244. Another approach to osculating curves

§ 4. Length of a plane curve
245. Lemmas
246. Direction on a curve
247. Curve length. 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 involute; evolute search
255. Properties of evolutes and involutes
256. Finding involutes

ADDITION. FUNCTION DISTRIBUTION PROBLEM
257. The case of a function of one variable
258. Statement of the problem for the two-dimensional case
259. Auxiliary sentences
260. Fundamental Theorem of Propagation

Volume 1. CONTENTS
INTRODUCTION REAL NUMBERS
§ 1. Region of rational numbers 11
1. Preliminary remarks 11
2. Ordering the domain 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
6. Definition of the irrational number 17
7. Ordering the domain of real numbers 19
8. Supporting suggestions 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. Determination 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 3 2
16. Conclusion 34
17. Absolute quantities 34 § 4. Further properties and applications of real numbers 35
18. Existence of a root. Power with rational exponent 35
19. Power with any real exponent 37
20. Logarithms 39
21. Measuring segments 40

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

§ 2. Theorems on limits that make it easier to find limits 56
28. Passage to the limit in equality and inequality 56
29. Lemmas on infinitesimals 57
30. Arithmetic operations on variables 58
31. Vague expressions 60
32. Examples for finding limits 62
33. Stolz’s theorem and its applications 67

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

§ 4. Principle of convergence. Partial limits 83
39. Principle of convergence 83
40. Partial sequences and partial limits 85
41. Bolzano-Weierstrass Lemma 87
42. Largest and smallest limits 89

CHAPTER TWO. FUNCTIONS OF ONE VARIABLE
§ 1. Concept of function 93
43. Variable and its scope 93
44. Functional dependence between variables. Examples 94
45. Definition of the concept of function 95
46. Analytical method of specifying a function 98
47. Graph of 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. Determining the limit of a function 115
53. Reduction to case options 117
54. Examples 120
55. Dissemination of the theory of limits 128
56. Examples 130
57. Limit of a monotonic function 133
58. General Bolzano-Cauchy sign 134
59. The largest and smallest limits of a function 135

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

§ 4. Continuity (and discontinuities) of functions 146
66. Determination of continuity of a function at point 146
67. Arithmetic operations on continuous functions 148
68. Examples of continuous functions 148
69. One-way continuity. Classification of ruptures 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 characteristics of exponential, logarithmic and power functions
76. Functional characteristics of trigonometric and hyperbolic cosines
77. Using continuity of functions to calculate limits 162
78. Power-exponential expressions 165
79. Examples 166

§ 5. Properties of continuous functions 168
80. Theorem on the vanishing of a function 168
81. Application to solving equations 170
82. Intermediate value theorem 171
83. Existence of an inverse function 172
84. Theorem on the boundedness of a function 174
85. The largest and smallest values of the function 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. Derivative and its calculation 186
90. Problem of calculating the speed of a moving point 186
91. The problem of drawing a tangent to a curve 187
92. Definition of 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 incrementing a function 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 of special cases 211

§ 2. Differential 211
103. Definition of differential 211
104. Relationship between differentiability and existence of _ 1. derivative
105. Basic formulas and rules of differentiation 215
106. Invariance of the shape of the differential 216
107. Differentials as a source of approximate formulas 218
108. Application of differentials in error estimation 220

§ 3. Basic theorems of differential calculus 223
109. Fermat's Theorem 223
110. Darboux's 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. Determination of higher order derivatives 231
116. General formulas for derivatives of any order 232
117. Leibniz formula 236
118. Examples 238
119. Differentials of higher orders 241
120. Violation of form invariance for differentials of higher _ ._ orders
121. Parametric differentiation 243
122. Finite differences 244

§ 5. Taylor's formula 246
123. Taylor formula for polynomial 246
124. Expansion of an arbitrary function; additional term in Peano form
125. Examples 251
126. Other forms of additional member 254
127. Approximate formulas 257

§ 6. Interpolation 263
128. The simplest interpolation problem. Lagrange Formula 263
129. Additional term of the Lagrange formula 264
130. Interpolation with multiple nodes. Hermite formula 265
CHAPTER FOUR. STUDYING A FUNCTION USING DERIVATIVES
§ 1. Study of the progress of changes in a function 268
131. Condition for constancy of function 268
132. 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. Problems 290

§ 2. Convex (and concave) functions 294
141. Definition of a convex (concave) function 294
142. The simplest sentences about convex functions 296
143. Conditions for 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 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 type oo / oo 320
152. Other types of uncertainties 322

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

CHAPTER FIVE. FUNCTIONS OF SEVERAL VARIABLES
§ 1. Basic concepts 340
159. Functional dependence between variables. Examples 340
160. Functions of two variables and their domains of definition 341
161. Arithmetic n-dimensional space 345
162. Examples of areas in n-dimensional space 348
163. General definition of open and closed area 350
164. Functions of n variables 352
165. Limit of a function of several variables 354
166. Reduction to case options 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 region. Bolzano-Cauchy theorems 365
172. Bolzano-Weierstrass Lemma 367
173. Weierstrass's 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 function 378
179. Full differential 381
180. Geometric interpretation for the case of a function of two _ R_ variables
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 total differential in approximate calculations 396
187. Homogeneous functions 399
188. Euler's formula 400

§ 4. Derivatives into differentials of higher orders 402
189. Higher order derivatives 402
190. Theorem on mixed derivatives 404
191. Generalization 407
192. Higher order derivatives of a complex function 408
193. Differentials of higher orders 410
194. Differentials of complex functions 413
195. Taylor formula 414

§ 5. Extrema, largest and smallest values 417
196. Extrema of a function of several variables. Necessary. 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. Problems 431
CHAPTER SIX. FUNCTIONAL DETERMINANTS; THEIR APPLICATIONS
§ 1. Formal properties of functional determinants 441
202. Determination 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. 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 extremes 467
212. Lagrange's method of undetermined multipliers 470
213. Sufficient conditions for a relative extremum 472
214. Examples and problems 473
215. The concept of independence of functions 477
216. Jacobian matrix rank 479

§ 4. Change of variables 483
217. Functions of one variable 483
218. Examples 485
219. Functions of several variables. Replacing independent variables
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. Analytical representation of curves and surfaces 503
223. Curves on a plane (in rectangular coordinates) 503
224. Examples 505
225. Curves of mechanical origin 508
226. Curves on a 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 of plane curves 535
237. The case of parametric specification of curve 540

§ 3. Curves touching each other 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 implicitly specifying one of the curves 553
243. Touching curve 554
244. Another approach to osculating curves 556

§ 4. Length of a plane curve 557
245. Lemmas 557
246. Direction on curve 558
247. Curve length. 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 involute; evolute search
255. Properties of evolutes and involutes
256. Finding involutes
ADDITION. FUNCTION DISTRIBUTION PROBLEM
257. The case of a function of one variable
258. Statement of the problem for the two-dimensional case
259. Auxiliary sentences
260. Fundamental Theorem of Propagation
261. Generalization
262. Concluding remarks

Alphabetical index 600

Volume 2. CONTENTS
CHAPTER EIGHT. ANIMAL FUNCTION (INDEFINITE INTEGRAL)
§ 1. Not definite integral and the simplest methods for calculating it 11
263. The concept of antiderivative function (and indefinite integral) 11
264. Integral and problem of determining area 14
265. Table of basic integrals 17
266. The simplest rules of integration 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 integration problem in final form 36
273. Simple fractions and their integration 37
274. Decomposition proper fractions to simple 38
275. Determination of coefficients. Integrating Proper Fractions 42
276. Isolating the rational part of the integral 43
277. Examples 47
§ 3. Integration of some expressions containing radicals 50
278. Integrating expressions of the form R .ух + 8
279. Integration of binomial differentials. Examples 51
280. Reduction formulas 54
281. Integration of expressions of the form K\x,l1ax2 + bx + c). Substitutions -^ Euler
282. Geometric interpretation of Euler substitutions 59
283. Examples 60
284. Other calculation techniques 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. Integrating 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. DEFINITE 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. 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 like limits 106

§ 2. Properties of 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. Second Mean Value Theorem 117

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

§ 4. Some applications of definite integrals 145
317. Wallis formula 145
318. Taylor formula with 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. Division of the integration interval 158
325. Additional term of the rectangle formula 159
326. Additional term of the trapezoidal formula 161
327. Additional term of Simpson's formula 162
328. Examples 164
CHAPTER TEN. APPLICATIONS OF INTEGRAL CALCULUS TO GEOMETRY, MECHANICS AND PHYSICS
§ 1. Length of the curve 169
329. Calculating the length of a curve 169
330. Another approach to defining the concept of curve length and calculating it
331. Examples 174
332. Natural equation flat curve 180
333. Examples 183
334. Arc length 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 squarable areas 190
338. Expressing 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. Expressing volume by integral 205
343. Examples 208
344. Surface area of rotation 214
345. Examples 217
346. Area cylindrical surface 220
347. Examples 222

§ 3. Calculation of mechanical and physical quantities 225
348. Scheme for applying a definite integral 225
349. Finding static moments and the center of gravity of a curve 228
350. Examples 229
351. Finding the static moments and center of gravity of a plane figure
352. Examples 232
353. Mechanical work 233
354. Examples 235
355. Work of friction force in a flat heel 237
356. Problems involving 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. Separating Variables
359. Problems 247
360. Notes on drafting differential equations 253
361. Problems 254
CHAPTER ELEVEN. ENDLESS RANKS WITH PERMANENT MEMBERS
§ 1. Introduction 257
362. Basic concepts 257
363. Examples 258
364. Basic theorems 260

§ 2. Convergence of positive series 262
365. Convergence condition positive series 262
366. Theorems for comparison of series 264
367. Examples 266
368. Signs of Cauchy and D'Alembert 270
369. Raabe's sign 272
370. Examples 274
371. Kummer's sign 277
372. Gaussian test 279
373. Maclaurin-Cauchy integral test 281
374. Ermakov’s sign 285
375. Additions 287

§ 3. Convergence of arbitrary series 293
376. General condition convergence of series 293
377. Absolute convergence 294
378. Examples 296
379. Power series, its interval of convergence 298
380. Expressing the radius of convergence through coefficients 300
381. Alternating series 3 02
382. Examples 303
383. Abel transformation 305
384. Abel and Dirichlet tests 307
385. Examples 308

§ 4. Properties of convergent series 313
386. Matching property 313
3 87. Commutative property of absolutely convergent series 315
388. The case of non-absolutely convergent series 316
389. Multiplying rows 320
390. Examples 323
391. General theorem from the theory of limits 325
392. Further theorems on 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; convergence region 346
397. Examples 348
398. Multiple rows 350

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

§ 7. Expansions of elementary functions 364
403. Expansion of a function into a power series; Taylor series 364
404. Series expansion 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 using series. Converting series 378
409. General remarks 378
410. Calculating the number to 379
411. Calculation of logarithms 381
412. Calculation of roots 383
413. Transformation of series according to Euler 3 84
414. Examples 386
415. Kummer transformation 388
416. Markov transformation 392

§ 9. Summation of divergent series 394
417. Introduction 394
418. Method of power series 396
419.Tauber's theorem 398
420. Method of arithmetic averages 401
421. Relationship between the Poisson-Abel and Cesaro methods 403
422. Hardy-Landau theorem 405
423. Application of generalized summation to 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 non-uniform convergence 421
429. Condition for uniform convergence 425
430. Signs of uniform convergence of series 427

§ 2. Functional properties of the sum of series 430
431. Continuity of the sum of series 430
432. Remark on quasi-uniform convergence 432
433. Term-by-term passage to the limit 434
434. Term by term integration of series 436
435. Term-by-term differentiation of series 438
436. Sequence Point of View 441
437. Continuity of the sum of a power series 444
438. Integration and differentiation of power series 447

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

§ 4. Additional information about power series 481
445. Actions on power series 481
446. Substitution of series into series 485
447. Examples 487
448. Division of power series 492
449. Bernoulli numbers and expansions in which they occur 494
450. Solving equations in series 498
451. Inversion of a power series 502
452. Lagrange series 505

§ 5. Elementary functions complex variable 508
453. Complex numbers 508
454. Complex option 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 member 542
467. Examples of calculations using the Euler-Maclaurin formula 544
468. Another type of Euler-Maclaurin formula 547
469. Formula and Sterling series 550

CHAPTER THIRTEEN. IMPROPER INTEGRALS
§ 1. Improper integrals with demons finite 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. Abel and Dirichlet tests 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. Note regarding singular points 581
481. Application of the basic formula of integral calculus. Examples
482. Conditions and signs for the existence of the integral 584
483. Examples 587
484. Principal values of 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 Moves calculations of improper integrals 611
492. Some remarkable integrals 611
493. Calculation of improper integrals using integral sums. The case of integrals with finite limits
494. The case of integrals with infinite limit 617
495. Frullani integrals 621
496. Integrals of rational functions between infinite 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. A note on the approximate calculation of proper integrals
501. Approximate calculation of improper integrals with infinite limit
502. Using asymptotic expansions 650
CHAPTER FOURTEEN. INTEGRALS DEPENDING ON PARAMETER
§ 1. Elementary theory 654
503. Statement of the problem 654
504. Uniform tendency towards the limiting function 654
505. Permutation of two limit passages 657
506. Passage to the limit under the integral sign 659
507. Differentiation under the integral sign 661
508. Integration under the integral sign 663
509. The case when the limits of the integral also 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. Determination of uniform convergence of integrals 682
514. Condition for uniform convergence. Connection with rows 684
515. Sufficient criteria for uniform convergence 684
516. Another case of uniform convergence 687
517. Examples 689

§ 3. Use of uniform convergence of integrals 694
518. Passage to the limit under the integral sign 694
519. Examples 697
520. Continuity and differentiability of the integral with respect to parameter 710
521. Integration of the integral over parameter 714
522. Application to the calculation of some integrals 717
523. Examples of differentiation under the integral sign 723
524. Examples of integration under the integral sign 733

§ 4. Additions 743
525. Arzela Lemma 743
526. Passage to the limit under the integral sign 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 of the function Г by its properties 760
533. Other functional characteristic functions G 762
534. Examples 764
535. Logarithmic derivative of the function Г 770
536. Multiplication theorem for function Г 772
537. Some series expansions and products 774
538. Examples and additions 775
539. Calculation of some definite integrals 782
540. Sterling Formula 789
541. Calculation of the Euler constant 792
542. Compiling a table of decimal logarithms of the function Г 793
Alphabetical index 795
Alphabetical index

A fundamental textbook on mathematical analysis, which has gone through many editions and been translated into several foreign languages, is distinguished, on the one hand, by the systematicity and rigor of presentation, and on the other hand, in simple language, detailed explanations and numerous examples illustrating the theory.
"Course..." is intended for university students, pedagogical and technical universities and has been used for a long time in various educational institutions as one of the main teaching aids. It allows the student not only to master theoretical material, but also to gain the most important practical skills. “Course...” is highly valued by mathematicians as a unique collection of various facts of analysis, some of which cannot be found in other books in Russian.

Chapter 7. Applications of differential calculus to geometry

Volume 2

The second volume of the “Course...” is devoted to the theory of the integral of a function of one real variable and the theory of series and is intended, first of all, for students of the first two years of non-humanitarian universities. The presentation is exceptionally detailed, complete and provided with numerous examples, including such classic sections of analysis as indefinite integral and methods for its calculation, the definite Riemann integral, improper integral, numerical and functional series, integrals depending on a parameter, etc. Some of them that are poorly represented or not presented at all are presented in detail. elementary textbooks topics: infinite products, the Euler-Maclaurin summation formula and its applications, asymptotic expansions, summation theory and approximate calculations using divergent series, etc. Being one of the best systematic textbooks on integral calculus and, at the same time, a unique collection specific facts related to series and integrals, this book will certainly be useful for both students and teachers higher mathematics, as well as specialists in various fields who use mathematics in their work, including mathematicians, physicists and engineers.
The first edition was published in 1948.

Chapter 8. Antiderivative function (indefinite integral)
Chapter 10. Applications of integral calculus to geometry, mechanics and physics

Volume 3

The third and final volume contains detailed statement such sections of differential and integral calculus as the theory of multiple, curvilinear and surface integrals, elements of vector analysis, the theory of functions of limited variation and the Stieltjes integral, Fourier series and integrals. Using simple geometric language makes the text much easier to understand; at the same time, many complex theoretical issues are presented more fully than in any other educational publication. Special attention focused on applications general theory: a large number of specific formulas and facts, examples and problems of both a purely mathematical and applied nature turns the “Course...” into a unique textbook useful for students of non-humanitarian universities for whom it is directly intended, as well as mathematicians, physicists, engineers and others specialists who use mathematics in their work.
The first edition was published in 1949.

Fikhtengolts G. M.Course of differential and integral calculus. In 3 vols. T. I / Prev. and approx. A.A. Florinsky. - 8th ed. - M.: FIZMATLIT, 2003. - 680 p. - ISBN 5-9221-0156-0.

The fundamental textbook on mathematical analysis, which has gone through many editions and translated into a number of foreign languages, is distinguished, on the one hand, by its systematic and rigorous presentation, and on the other, by its simple language, detailed explanations and numerous examples illustrating the theory.

The course is intended for students of universities, pedagogical and technical universities and has been used for a long time in various educational institutions as one of the main teaching aids. It allows the student not only to master theoretical material, but also to gain the most important practical skills. The course is highly valued by mathematicians as a unique collection of various facts of analysis, some of which cannot be found in other books in Russian.

The first edition was published in 1948.

EDITOR'S FOREWORD

Course of differential and integral calculus Grigory Mikhailovich Fikhtengolts is an outstanding work of scientific and pedagogical literature, which has gone through many editions and translated into a number of foreign languages. The course has no equal in terms of the volume of factual material covered and the number of various applications general theorems in geometry, algebra, mechanics, physics and technology. Many famous modern mathematicians note that it was the Course of G. M. Fikhtengolts that instilled in them student years taste and love for mathematical analysis gave the first clear understanding of this subject.

Over the 50 years that have passed since the release of the first edition of the Course, its text has practically not become outdated and currently can still be used and is being used by university students as well as various technical and pedagogical universities as one of the main textbooks on mathematical analysis and higher mathematics courses. Moreover, despite the appearance of new good textbooks, the audience of readers of the Course by G. M. Fikhtengolts during its existence has only expanded and now includes students from a number of physics and mathematics lyceums, students of advanced courses mathematical qualification engineers.

High level The demand for the Course is explained by its unique features. Basic theoretical material included in the Course is a classic part of modern mathematical analysis, which was finally formed by the beginning of the 20th century (does not contain measure theory and general set theory). This part of analysis is taught in the first two years of universities and is included (in whole or in large part) in the programs of all technical and pedagogical universities. Volume I of the Course includes differential calculus of one and several real variables and its main applications, Volume II is devoted to the theory of the Riemann integral and the theory of series, Volume III- multiple, curvilinear and surface integrals, Stieltjes integral, series and Fourier transform.

A huge number of examples and applications, usually very interesting, some of which cannot be found in other literature in Russian, constitute one of the main features of the Course, already mentioned above.

Another significant feature is the accessibility, detail and thoroughness of the presentation of the material. The significant volume of the Course does not become an obstacle to its absorption. On the contrary, it allows the author to pay sufficient attention to the motivations for new definitions and problem statements, detailed and thorough proofs of the main theorems, and many other aspects that make it easier for the reader to understand the subject. In general, the problem of combining clarity and rigor of presentation (the absence of the latter simply leads to distortion math facts) is well known to any teacher. Huge pedagogical skill Grigory Mikhailovich allows him throughout the Course to give many examples of solving this problem; along with other circumstances, this turns the Course into an indispensable model for a beginning lecturer and an object of research for specialists in the methods of teaching higher mathematics.

Another feature of the Course is the very slight use of any elements of set theory (including notation). At the same time, the full rigor of the presentation is maintained; in general, just like 50 years ago, this approach makes it easier for a significant part of the readership to initially master the subject.

In the new edition of the Course by G. M. Fikhtengolts, which we bring to the attention of the reader, typos found in a number of previous editions have been eliminated. In addition, the publication is equipped brief comments, relating to those places in the text (very few), when working with which the reader may experience certain inconveniences; notes are made, in particular, in cases where the term or figure of speech used by the author differs in some way from the most common at present. Responsibility for the content of the notes lies entirely with the editor of the publication.

The editor is deeply grateful to Professor B. M. Makarov, who read the texts of all the notes and made a number of valuable opinions. I would also like to thank all the employees of the Department of Mathematical Analysis of the Faculty of Mathematics and Mechanics of St. Petersburg state university, who discussed with the author of these lines various issues related to the texts of previous editions and the idea of a new edition of the Course.

The editors thank in advance all readers who, with their comments, would like to contribute to further improvement of the quality of the publication.

A. A. Florinsky

Fikhtengolts G.M. (2003) Course of differential and integral calculus. T.1.

G.M. Fikhtengolts
COURSE OF DIFFERENTIAL AND INTEGRAL CALCULUS
VOLUME 1
Content
INTRODUCTION
REAL NUMBERS
§ 1. The domain of rational numbers 11 1. Preliminary remarks 11 2. Ordering the domain 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 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. Determination of the sum of real numbers 28 13. Properties of addition 29 14. Determination of the product of real numbers 31 15. Properties of multiplication 32 16. Conclusion 34 17. Absolute quantities 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. Measuring segments 40
CHAPTER FIRST. THEORY OF LIMITS
§ 1. Variant and its limit 43 22. Variable value, variant 43 23. Limit variant 46

24. Infinitesimal quantities 47 25. Examples 48 26. Some theorems about the variant having 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. Lemmas about infinitesimals 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. Monotonic variant 70 34. Limit of monotonic variant 70 35. Examples 72 36. Number e 77 37. Approximate calculation of the number e 79 38. Lemma on nested intervals 82
§ 4. Principle of convergence. Partial limits 83 39. Convergence principle 83 40. Partial sequences and partial limits 85 41. Bolzano-Weierstrass lemma 87 42. Largest and smallest limits 89
CHAPTER TWO. FUNCTIONS OF ONE VARIABLE
§ 1. The concept of a function 93 43. A variable and the area of its change 93 44. Functional dependence between variables. Examples 94 45. Definition of the concept of a function 95 46. Analytical method 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. Determination of the limit of a function 115

53. Reduction to case variants 117 54. Examples 120 55. Extension of the theory of limits 128 56. Examples 130 57. Limit of a monotonic function 133 58. General Bolzano-Cauchy test 134 59. The largest and smallest limits of a function 135
§ 3. Classification of infinitesimal and infinitely large quantities 136 60. Comparison of infinitesimals 136 61. Infinitesimal scale 137 62. Equivalent infinitesimals 139 63. Identification of the main part 141 64. Problems 143 65. Classification of infinitely large 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 characteristics 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-exponential expressions 165 79. Examples 166
§ 5. Properties of continuous functions 168 80. Theorem on the vanishing of a function 168 81. Application to solving equations 170 82. Theorem on the intermediate value 171

83. Existence of an inverse function 172 84. Theorem on the boundedness of 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. 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 the 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 the increment of a function 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 of special cases 211
§ 2. Differential 211 103. Definition of differential 211 104. Relationship 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. Basic 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. Limit of derivative 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. Violation of form invariance 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. Expansion of an arbitrary function; additional member in the form
Peano
248 125. Examples 251 126. Other forms of an additional term 254 127. Approximate formulas 257
§ 6. Interpolation 263 128. The simplest interpolation problem. Lagrange formula 263 129. Additional term of Lagrange formula 264 130. Interpolation with multiple nodes. Hermite formula 265
CHAPTER FOUR. STUDYING A FUNCTION USING
DERIVATIVES
§ 1. Study of the progress of changes in a function 268 131. Condition for constancy of a function 268 132. Condition for monotonicity of a function 270 133. Proof of inequalities 273 134. Maxima and minimum; 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. Problems 290
§ 2. Convex (and concave) functions 294 141. Definition of a convex (concave) function 294 142. The simplest sentences 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 the 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 SEVERAL VARIABLES
§ 1. Basic concepts 340 159. Functional dependence between variables. Examples 340 160. Functions of two variables and their domains of definition 341 161. Arithmetic n-dimensional space 345 162. Examples of domains in n-dimensional space 348 163. General definition of an open and closed domain 350 164. Functions of n variables 352 165. Limit of a function of several variables 354 166. Reduction to case options 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's lemma 372 176. New proofs of the main theorems 373 176. Derivatives and differentials of functions of several variables 373 177. Particulars 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 from complex functions 386 182. Examples 388 183. The formula of final increments 390 184. The derivative of the given direction 391 185. The invariability of the form (first) differential 394 186. The application of a complete differential in close computing 396 187. Uniform functions 399 188. Euler's formula 400
§ 4. Derivatives of higher order differentials 402 189. Higher order derivatives 402 190. Theorem on mixed derivatives 404 191. Generalization 407 192. Higher order derivatives of a complex function 408 193. Higher order differentials 410 194. Differentials of complex functions 413 1 95. Formula Taylor 414
§ 5. Extrema, largest and smallest 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. Problems 431
CHAPTER SIX. FUNCTIONAL DETERMINANTS; THEIR
APPLICATIONS
§ 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. Lagrange’s method of indefinite multipliers 470 213. Sufficient conditions for a relative extremum 472 214. Examples and problems 473 215. The concept of independence of functions 477 216. Rank of the Jacobian matrix 479
§ 4. Replacement of variables 483 217. Functions of one variable 483 218. Examples 485 219. Functions of several variables. Replacing independent variables
488 220. Method for calculating differentials 489 221. General case of changing variables 491 222. Examples 493
CHAPTER SEVEN. DIFFERENTIAL APPLICATIONS
CALCULUS TO GEOMETRY
§ 1. Analytical representation of curves and surfaces 503

223. Curves on a plane (in rectangular coordinates) 503 224. Examples 505 225. Curves of mechanical origin 508 226. Curves on a 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 parametric definition of a curve 540
§ 3. Touching curves to each other 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 implicitly specifying one of the curves 553 243. Occupying 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. Additivity of arc length 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 577 254. Definition of evolute and involute; finding an evolute 578 255. Properties of evolutes and involutes 581 256. Finding involutes 585
ADDITION. FUNCTION DISTRIBUTION PROBLEM
257. The case of a function of one variable 587 258. Statement of the problem for the two-dimensional case 588 259. Auxiliary propositions 590 260. The 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 method for specifying functions 97, 98
Analytical expression functions
98
- presentation of curves 503, 517
- - surfaces 517
Anomaly (eccentric) planet
174
Function argument 95, 341
Arithmetic value of 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
- Yakov 38
- lemniscate 515, 530, 575, 577
Infinite decimal 22
- derivative 209
Endlessly 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 span 94, 308
- gap 309
Boyle-Marriott Law 94
Bolzano 84
Bolzano method 88
Bolzano-Weierstrasse 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 of icon 96

Monotonous 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 limit number set 26
- - - - exact 26
Real numbers 19
- - subtraction 31
- - division 34
- - decimal approximation 22
- - continuity of area 24
- - density (enhanced) area 21
- - equality 19
- - addition 28
- - multiplication 31
- - ordering area 19
Viviani curve 521, 535
Helix 521, 534
- surface 523, 535
Nested intervals, Lemma 83
Inner point sets 350
Concave (convex upward) functions or curves 295
- - - - concavity conditions 298
Return point 539, 541
Ascending option 70
- function 133
Rotation surface 522
Convex (convex down) functions or curves 294
- - - - convexity conditions 298
- strictly functions or curves 298
Higher order infinitesimals
[designation o(
α)] 136, 137
- - differentials 241
- - - functions of several variables
410
- - derivatives 231, 232
245
- - - private 402
Harmonic oscillation 208
Gauss 74, 439
Hölder-Cauchy inequality 275,
302
Geographical coordinates 522
Geometric interpretation of differential 214
- - full differential 386
- - derivative 190
Hyperbole 506, 575, 580
- equilateral 102, 103
Hyperbolic spiral 529
Hyperbolic sine, cosine, etc. 107
- functions, continuity 149
- - reverse 108-109
- - derivatives 205
Hypocycloid 509
Main branch (main value) of arcsine, arccosine, etc.
110, 114
- Part ( main member) infinitesimal 141
Smooth curve 594
Horizontal asymptote 309
Gradient function 394
Region border 351
- numerical set (upper, lower) 25-28
- - - exact 26
Graph of function 100
- - construction 305
- - spatial 343
Huygens formula 260
Darboux's theorem 224
Movement equation 187
Double curve point 538
Double limit 360 function
Two variable function 341
Dedekind 17
Dedekind's main theorem 25

Real numbers, cm.
Real numbers
Cartesian sheet 507, 538
Decimal approximation of the real number 22
Decimal logarithms 79
Point set diameter 371
Dirichlet function 99, 102, 153
Discriminant curve 545, 550
Differential 211, 215
- order, 1st, 2nd, n th 241
- geometric interpretation 214
- arcs 562, 567
- form invariance 216
- full 382
- - order, 1st, 2nd, n th 410
- - geometric interpretation 386
- - form invariance 394
- - calculation method (when replacing variables) 489
- application to approximate calculations 218, 220, 396
- private 378, 411
Differentiation 215
- parametric 243
- rules 215, 395
Differentiable function 212, 382
Differentiability of the Implicit Function 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
- - - several variables 353
e(number) 78, 148
- irrationality 82
- approximate calculation 81
Unit 14, 32
Dependent functions 478
Replacing variables 483
Enclosed area 351
- sphere 351
Closed set 351
Closed parallelepiped 351
Closed gap 93
- simplex 351
Sharpening point 539
Damped oscillation 208, 282
Sign rule (for multiplication) 16,
32
Jensen 295
Jensen inequality 301
Measuring segments 40
Isolated curve point 536, 539
Invariance of differential shape 216, 394
Interpolation 263
Interpolation nodes 263
- - multiples of 266
Interpolation formula
Lagrange 263
- - Ermita 266
Irrational numbers 19
Cantor's theorem 179, 184, 370, 374
Cardioid 510, 515, 530
Touching curves 542
- - order 551
Tangent 188, 210, 386, 523, 530,
533, 555
- one-sided 209
- segment 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 shape 423

Highest and lowest values 476
- - undefined 425
- - defined 423
- - semi-definite 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
Final increments formula 227,
390
Cone of go, order, 2, 535
Coordinate lines (surfaces)
520
Coordinates n-measurement point 345
Root of a real number, existence of 35
- equations (functions), existence 170
- - approximate calculation 170,
324
Cosine 103
- functional characteristics
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
- form of additional member 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
- V n-dimensional space 347
- on the 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
- form of additional member 257,
415
Lebesgue 181
Legendre polynomials 240
Legendre transformation 487, 499,
500
Leibniz 192, 215, 241
Leibniz formula 238, 241
Bernoulli's Lemniscate 515, 530, 575,
577
Logarithm, existence 39
- decimal 50, 79
- natural (or neper) 78
- - change to decimal 79
Logarithmic spiral 514, 529,
574, 581
- function 103
- - continuity 155, 174
- - derivative 195, 197

Functional characteristics
159
Broken line (in n-dimensional space)
347
L'Hopital rule 314, 320
Maclaurin formula 247, 251
Maximum, see Extreme
Functional matrix (Jacobi)
444, 478
- - rank 468, 471, 479
Multiplication matrices 444
Mere 44
Minimum, see Extreme
Minkowski inequality 276
Multi-valued function 96, 109, 341,
447, 453
Closed set of points 351
- - limited 352
- numeric, limited above, below 26
Undetermined multipliers, method
470
Module for converting natural logarithms to decimal ones 79
Monotonous option 70
- function 133
- - continuity, discontinuities 154
Monotonicity of the function 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 parallelepiped 348, 351
n-dimensional simplex 349, 351
The highest value of the function is 176,
286
Highest limit options 89
- - functions 136
The smallest value of the function is 176,
289
- - - several variables 427
Lowest limit options 89
- - functions 136
Least squares method 438
Oblique asymptote 310
Function overlay 114
Direction on curve 558
Natural logarithm 78
Independence of functions 478
Independent variables 94, 341,
352
Uncertainty disclosure 62, 314
- type 0/0 60, 314
- -
∞
∞ / 61, 320
- -
∞
⋅
0 61, 322
- -
∞
−
∞
62, 323
- -
0 0
,
0
,
1
∞
∞
166, 323
Uncertain multipliers, method
470
Neper, Neper logarithms 78
Continuity of the domain of real numbers 24
- straight 42
- functions in the 365 area
- - 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-Helder 275, 302
- Jensen 301
- Minkowski 276
Improper numbers (dots) 26, 55,
355
Implicit functions 447, 453
- - calculation of derivatives 460
- - existence and properties 449,
451, 453

Bottom line number set 26
- - - - exact 26
Normal to curve 523
- - - segment 524
- - - - polar 528
Normal to surface 532, 534
Newton's method (approximate solution of equations) 328
Relative extreme 467
Line segment, dimension 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
Ovals of Cassini 515
Curve family envelope 543
Limited variant 53
Limited set spot
352
- - numeric 26
Boundedness of a continuous function, Theorems 175, 183,
369, 373
Single value function 96, 341
Homogeneous function 399
One-sided continuities and discontinuities of function 150
One-way 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
- - isolated 536
- - double 538
- - multiple of 505, 519, 538, 540
Ostrogradsky 442
Open area 350
- sphere 349, 350
Open span 93
- parallelepiped 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 extra penis
249
Inflection point 303
Variable 43, 93
- independent 94, 341, 352
Variable replacement 483
Commutative property of addition, multiplication 12, 14,
29, 32
Rearrangement of differentiation
405, 407
- limit passages 361, 406

Transition curves 576
Periodic decimal fraction 24
Surface 343, 517, 519
- rotation 522
Repeated limit of a function of several variables 360
Podcastnaya 207, 524
- polar 528
Subnormal 524
- polar 528
Subsequence 85
Border point 351
Absolute, relative error 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
- - form invariance 394
- - applications to approximate calculations 396
Semi-cubic parabola 506, 540,
548, 579
Half-open gap 93
Polar subtangent, subnormal 528
Polar Curve Equation 511
Polar coordinates 493, 495, 512
Polar tangent segment, normal 528
The order is infinite large size 145
- - small size 137
- differential 241
- touching curves 551
- derivative 231
Sequence 44
Function constancy condition 268
Rule, see corresponding title
Limit options 46, 48
- - endless 55
- - uniqueness 54
- - monotonous 71
- - largest, smallest 89
- - partial 86
- relationships 59
- works 59
- derivative 228
- differences 59
- amounts 59
- functions 115, 117
- - monotonous 139
- - largest, smallest 135
- - several variables 354, 357
- - - - repeated 360
- - partial 135
Passage to the limit in equality, in inequality 56
Legendre's transformation 487, 499,
500
- point (plane, space)
485, 493
Approximate solution of the equation
324
Approximate calculations, application of differential
218, 220, 396
Approximate formulas 140, 143,
218, 257-263
Variable increment 147
- functions, formula 199
- several variables complete, formula 379
- - - - private 375
Increments finite 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, half-open, open, finite, infinite 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 function 178, 370
Radical, arithmetic value
36, 103
Radius of curvature 571
Difference option, etc., see Amount
- numbers 13, 31
Derivative discontinuity 211
- functions 146
- - monotonous 154
- - ordinary, kind, go, and, go, 1, 2,
151
- - several variables 362
Matrix rank 468, 471, 479
Unraveling Uncertainties 62,
314
Distributive property of multiplication 15, 34
Function Propagation 587
Distance between points in n- dimensional space 345
Rational function 102
- - continuity 148
- - several 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
Rocha and Schlemilha form of additional member 257
Relationships Equation 467
Communication area 352
Condensation point 115, 116, 117, 351
Sekans 103
Curve family 542
Section in the numerical domain 17, 24
Signum (function) 29
Current strength 192
Sylvester 423
Simplex n-dimensional 349, 351
Sine 103
- hyperbolic 107
- arc relation limit 122
Sine wave 106, 304
Point movement speed 186
- V this moment 187, 190
- average 186
Complex function 115, 353
- - continuity 156, 365
- - derivatives and differentials
202, 216, 242, 386, 395, 413, 414
Mixed derivatives, theorem
404

Occupant curve 554
- straight 555
Touching circle 555, 571
Combinative 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
Power-exponential function
(two variables) 353
Power-exponential function limit 358, 359
- - - - continuity 363
- - - - differentiation 376
Power exponential expression, limit 165
- - - - derivative 206, 388
Power with real exponent 37
Amount 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 method function assignments
97
Tangent 103
- hyperbolic 107
Geometric body 345
Heat capacity 191
Period, see corresponding title
Function points 352
Exact border (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
- function 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
Fermat's theorem 223
Quadratic shape 423
Formula see, also, corresponding, name, 97,
98
Functional dependence 94, 340
- matrix 444, 478
Functional equation 157, 158,
160
Functional determinant 441
Function see also, name, functions, 95
- study 268
- several variables 341, 352
- from function (or from functions) 115,
353

Characteristic point on the curve
539
Hestins 590
Function change progress 268
Chord method for approximate solution of equations 325
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
Particular option, limit 59, 60
- function value 96
- increment 375
- functions, limit 129, 130
- - continuity 148, 364
- - derivative and differential 201,
216, 395
- numbers 15
Partial differential 378, 411
Chebyshev formula 262
Numbers, see Rational,
Irrational,
Real numbers
Number axis 42
- sequence 44
Schwartz 407
Schlemilha and Rosha form of additional member 257
Stolz's Theorem 67
Involute 578, 582-583, 585
- circles 511, 527, 574
Evolute 579, 582, 583, 585
Euler 78
Euler formula 401
Equivalent infinitesimal quantities (sign) 139
Extreme (maximum, minimum) 277
- search rules 277, 278, 284,
287
- own, improper 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