МИНИCTEPCTBO ОБРАЗОВАНИЯ И НАУКИ РОССИЙСКОЙ ФЕДЕРАЦИИ ФЕДЕРАЛЬНОЕ ГОСУДАРСТВЕННОЕ АВТОНОМНОЕ ОБРАЗОВАТЕЛЬНОЕ УЧРЕЖДЕНИЕ ВЫСШЕГО ПРОФЕССИОНАЛЬНОГО ОБРАЗОВАНИЯ «СЕВЕРО-КАВКАЗСКИЙ ФЕДЕРАЛЬНЫЙ УНИВЕРСИТЕТ» MATHEMATICS (МАТЕМАТИКА) УЧЕБНОЕ ПОСОБИЕ Направление подготовки 21.03.01 – Нефтегазовое дело Бакалавриат Ставрополь 2015 1 УДК 51 (075.8) ББК 22.1 я73 М 34 Печатается по решению редакционно-издательского совета Северо-Кавказского федерального университета М 34 MATHEMATICS (Математика): учебное пособие / авт.-сост.: Н. <...> Торопцев © ФГАОУ ВПО «Северо-Кавказский федеральный университет», 2014 2 СHAPTER 1 Section 1. <...> Linear algebra and analytical geometry Topic 1. <...> This lecture is devoted to the discussion of the following concepts: matrix, matrix equality, matrix operations, transposes and symmetric matrices. <...> A matrix is a rectangular array of numbers. <...> For example, here is a matrix 3 12 5 17 иш жц зч - Definition VSM: Vector Space of mЧn Matrices. <...> The vector space M(mЧn) is the set of all mЧn matrices with entries from the set of complex numbers. <...> The above matrix is a 3 Ч 4 matrix because there are two rows and three columns. <...> The first row is (3 -1 2) , the second row is (5 -1 7) The first column is 3 5 жц. зч иш When specifying the size of a matrix, you always list the number of rows before the number of columns. <...> For example, 7 is in position 2, 3 because it is in the second row and the third column. <...> Whenever a theorem has a conclusion saying two matrices are equal (think about your objects), we will consider appealing to this definition as a way of formulating the toplevel structure of the proof. <...> Again, we will overload a symbol (`+') and a convention (juxtaposition for scalar multiplication). <...> Given the mЧn matrices A and B, define the sum of A and B as an mЧn matrix, written A+B, according to [A+B] ij = [A] ij +[B] ij , 1≤i≤m,1≤j≤n So matrix addition takes two matrices of the same size and combines them (in a natural way!) to create a new matrix of the same size. <...> As with vectors, in this context we call a number a scalar <...>
MATHEMATICS_(Математика)_Учебное_пособие._Направление_подготовки_21.03.01_–_Нефтегазовое_дело._Бакалавриат.pdf
УДК 51 (075.8)
ББК 22.1 я73
М 34
Печатается по решению
редакционно-издательского совета
Северо-Кавказского федерального
университета
М 34
MATHEMATICS (Математика): учебное пособие / авт.-сост.:
Н. В. Ширяева, А.С. Мараховский. – Ставрополь: Изд-во СКФУ,
2015. – 236 с.
Пособие составлено в соответствии с требованиями ФГОС ВПО и
предназначено для обучения иностранных студентов основам математики.
Пособие включает три раздела: курс лекций, практикум и методические
рекомендации по организации самостоятельной работы.
Рекомендовано для иностранных студентов, изучающих математику на
английском языке.
УДК 51 (075.8)
ББК 22.1 я73
Авторы-составители:
канд. психол. наук, доцент Н. В. Ширяева,
канд. физ-мат. наук, д-р. экон. наук, доцент А. С. Мараховский
Рецензенты:
канд. физ-мат. наук, профессор А. С. Адамчук,
д-р. экон. наук, профессор Е. Л. Торопцев
© ФГАОУ ВПО «Северо-Кавказский
федеральный университет», 2014
2
Стр.2
СHAPTER 1
Section 1. Linear algebra
and analytical geometry
Topic 1. Matrix. Matrix Operations
Introduction. This lecture is devoted to the discussion of the following
concepts: matrix, matrix equality, matrix operations, transposes and
symmetric matrices.
We begin with a definition of a totally general set of matrices, and see
where that takes us.
A matrix is a rectangular array of numbers. Several of them are re-
ferred
to as matrices. For example, here is a matrix 3 12
5 17
иш
жц
зч
-
Definition VSM: Vector Space of m×n Matrices. The vector space
M(m×n) is the set of all m×n matrices with entries from the set of complex
numbers.
Just as we made, and used, a careful definition of equality for column
vectors, so too, we have precise definitions for matrices.
The size or dimension of a matrix is defined as m Ч n where m is the
number of rows and n is the number of columns. The above matrix is a 3
Ч 4 matrix because there are two rows and three columns.
The first row is (3 -1 2) , the second row is (5 -1 7) The first column
is 3
5
æö.
зч
иш
When specifying the size of a matrix, you always list the number of rows
before the number of columns. Also, you can remember the columns are like
columns in a Greek temple. They stand up right while the rows just lay there
like rows made by a tractor in a plowed field. Elements of the matrix are identified
according to position in the matrix. For example, 7 is in position 2, 3
because it is in the second row and the third column.
Using this notation on the above matrix, a 23 = 7.
3
.
Стр.3
СОДЕРЖАНИЕ
СHAPTER 2
SECTION 1. Linear algebra and analytical geometry
Topic 1. Matrix. Matrix Operations………………………………..
Topic 2. Matrix Multiplication…………………………………….
Topic 3. Determinant of a Matrix …………………………………
Topic 4. Matrix Inverses and Systems of Linear Equations……….
Topic 5. Systems of Equations, Algebraic Procedures (GaussJordan
procedure)……………………………………….
Topic 6. Systems of Equations…………………………………….
Topic 7. The Cramer’s rule………………………………………..
Topic 8. А analytical geometry…………………………………….
SECTION 2. Basic calculus
Topic 9. Limits……………………………………………………..
Topic 10. Derivative and differential of a function………………..
Topic 11. Using derivatives in curve tracing………………………
SECTION 3. Differential calculation of multi-variable functions
Topic
12. Multi-variable function. Derivatives and differentials of
multi-variable functions…………………………………
SECTION 4. Integral calculation of single-variable functions
Topic 13. Indefinite integral……………………………………….
Topic 14. Methods of integration………………………………….
Topic 15. Definite integral. Calculation of a definite integral…….
Topic 16. Applications of a definite integral. Improper integrals…
SECTION 5. Integral calculation of multi-variable functions
Topic 17. Multiple integral………………………………………...
Topic 18. Contour and surface integrals…………………………..
232
41
43
45
48
50
54
55
3
8
11
15
18
23
26
28
32
34
37
Стр.232
SECTION 6. Differential equations
Topic 19. Basic differential equations……………………………..
Topic 20. Liner differential equations……………………………..
SECTION 7. Elements of field theory
Topic 21. Field theory……………………………………………...
SECTION 8. Series
Topic 22. Number series…………………………………………...
Topic 23. Exponential and functional series……………………….
SECTION 9. Elements of the complex variable function theory
Topic 24. Complex variable function theory………………………
SECTION 10. Basics of probability analysis
Topic 25. Probability theory……………………………………….
Topic 26. Elements of mathematical statistics……………………..
GLOSSARY……………………………………………………….
СHAPTER 2
SECTION 1. Linear algebra and analytical geometry
Topic 1. Matrix. Matrix Operations……………………………….
Topic 2. Matrix Multiplication…………………………………….
Topic 3. Determinant of a Matrix………………………………….
Topic 4. Matrix Inverses and Systems of Linear Equations……….
Topic 5. Systems of Equations, Algebraic Procedures. (GaussJordan
procedure)………………………………………..
Topic 6. Systems of Equations……………………………………
Topic 7. The Cramer’s rule……………………………………….
Topic 8. А analytical geometry……………………………………
SECTION 2. Basics of calculus
Topic 9. The limit of a function at a point and infinity……………
Topic 10. Derivative and differential of a function……………….
Topic 11. Using derivatives in curve tracing………………………
233
58
60
63
66
69
73
78
80
83
86
87
88
93
96
98
100
102
115
120
123
Стр.233
SECTION 3. Differential calculation of multi-variable functions
Topic
12. Multi-variable function. Derivatives and differentials of
multi-variable functions………………………………..
SECTION 4. Integral calculation of single-variable functions
Topic 13. Indefinite integral……………………………………….
Topic 14. Methods of integration………………………………….
Topic 15. Definite integral. Calculation of a definite integral…….
Topic 16. Applications of a definite integral. Improper integrals…
SECTION 5. Integral calculation of multi-variable functions
Topic 17. Multiple integrals……………………………………….
Topic 18. Contour and surface integrals…………………………...
SECTION 6. Differential equations
Topic 19. Basic differential equations……………………………..
Topic 20. Liner differential equations……………………………..
SECTION 7. Elements of field theory
Topic 21. Field theory……………………………………………..
SECTION 8. Series
Topic 22. Number series…………………………………………..
Topic 23. Exponential and functional series……………………….
SECTION 9. Elements of the complex variable function theory
Topic 24. Complex variable function theory………………………
SECTION 10. Basics of probability analysis
Topic 25. Probability theory……………………………………….
Topic 26. Elements of mathematical statistics ……………………
CHAPTER 3
SECTION 1. Linear algebra and analytical geometry
Topic 1. Matrix. Matrix Operations………………………………..
Topic 2. Matrix Multiplication…………………………………….
Topic 3. Determinant of a Matrix………………………………….
234
179
180
182
132
133
137
139
140
145
148
153
155
157
164
168
170
174
175
Стр.234
Topic 4. Matrix Inverses and Systems of Linear Equations……….
Topic 5. Systems of Equations, Algebraic Procedures (GaussJordan
procedure)……………………………………….
Topic 6. Systems of Equations…………………………………….
Topic 7. The Cramer’s rule………………………………………..
Topic 8. Аnalytical geometry……………………………………...
SECTION 2. Basics of calculus
Topic 9. Limits…………………………………………………….
Topic 10. Derivative and differential of a function ………………
Topic 11. Using derivatives in curve tracing………………………
SECTION 3. Differential calculation of multi-variable functions
Topic
12. Multi-variable function. Derivatives and differentials of
multi-variable functions………………………………...
SECTION 4. Integral calculation of single-variable functions
Topic 13 Indefinite integral………………………………………..
Topic 14. Methods of integration………………………………….
Topic 15. Definite integral. Calculation of a definite integral…….
Topic 16. Applications of a definite integral. Improper integrals…
SECTION 5. Integral calculation of multi-variable functions
Topic 17. Multiple integral………………………………………...
Topic 18. Contour and surface integrals ………………………….
SECTION 6. Differential equations
Topic 19. Basic differential equations……………………………..
Topic 20. Liner differential equations……………………………..
211
212
217
218
221
224
227
228
230
183
185
186
187
189
191
194
207
235
Стр.235