Mathematical analysis test paper for the second semester

20 1 1 year in the Department of Mathematics of Harbin Institute of Technology, is that ok?

20 1 1 Entrance Examination for Postgraduates in Mathematics Department of Harbin Institute of Technology

[6 12] Mathematical Analysis Examination Outline

Name of Examination Subject: Mathematical Analysis Examination Subject Code: [6 12]

First, the examination requirements:

1) requires candidates to master the basic concepts, theories and methods of mathematical analysis.

2) Candidates are required to have strict mathematical demonstration ability, counterexample ability and basic calculation ability.

3) Candidates are required to know the actual source and historical background of the basic concepts, theories and methods in mathematical analysis, know their geometric and physical meanings, and have the ability to apply mathematical analysis to solve practical problems.

Second, the examination content:

1), limit and continuity

A master the concepts of sequence limit and function limit, including the upper and lower limits of sequence and the left and right limits of function.

Master the properties of limit and four operations, especially the principle of double-sided clamping and two special limits.

C. Grasp the basic theorems of real number system: interval set theorem, supremum existence theorem, monotone bounded principle, Bolzano-Weierstrass theorem, Heine-Borel finite covering theorem, Cauchy convergence criterion; And understand the relationship.

D. master the concept of functional continuity and related discontinuous point types. Be able to use the properties of four continuous operations and compound operations of functions and the corresponding properties of infinitesimal quantities; And understand the relationship between them.

E. master the properties of continuous functions on closed intervals: boundedness theorem, maximum theorem, intermediate value theorem and control theorem.

2) Differential calculus of unary function

A. Understand the concepts of derivative and differential and their relationship, understand the geometric meaning and physical meaning of derivative, and understand the relationship between derivability and continuity of function.

B be familiar with the algorithm of function derivative and differential, including the algorithm of high-order intermediate derivative, and find the derivative of piecewise function.

C master Rolle's mean value theorem, Lagrange's mean value theorem, Cauchy's mean value theorem and Taylor's formula.

D monotonicity, extreme value, maximum value and convexity of functions can be studied by derivatives.

E. master the method of finding the limit of infinitive with lobida's law.

3) Integral of unary function

A. understand the concept of indefinite integral. Master the basic formulas of indefinite integral, substitution integral method and partial integral, and you can find the integral of rational function, triangular rational function and simple meta-function.

B master the concept of definite integral, including Darboux sum, upper and lower integral, integrable condition and integrable function class.

C. Grasp the nature of definite integral, the basic theorem of calculus, method of substitution and integration by parts of definite integral.

D. Grasp the representation and calculation of some geometric physical quantities by definite integral (area of plane figure, arc length of plane straight line, volume and lateral area of rotating body, solid volume with known parallel cross-sectional area, variable force work, mass and centroid of object).

E. understand the concept of generalized integral. Proficiently master the comparison discriminant method, Abel discriminant method and Dirichlet discriminant method to judge the convergence of generalized integral; Include that second mean value theorem of integration.

4), infinite series

A. Understand the concept of convergence and divergence of polynomial series and master the basic properties of polynomial series.

B master the necessary conditions of convergence and divergence of positive series, such as comparison discrimination, Cauchy discrimination, D 'Alembert discrimination and integral discrimination.

C master the concepts of absolute convergence and conditional convergence of arbitrary series and their relations. Familiar with Leibniz discriminant method of staggered series. Master the properties of absolute convergence series.

D master the concept of uniform convergence of function term series and Weierstrass discriminant method to discriminate uniform convergence. Abel discriminant and Dirichlet discriminant. Master the properties of uniformly convergent series.

E master the concepts of power series and its convergence radius, including Cauchy-Hadamard theorem and Abel's first theorem.

F. master the properties of power series. Functions can be expanded into power series. Understand the approximation theorem of Wilstrass.

G. Understand the concept and properties of Fourier series and the discrimination method of convergence and divergence.

5) Calculus of multivariate functions

A. Understand the concepts of limit and continuity, partial derivative and total differential of multivariate function, and find the partial derivative and total differential of multivariate function.

B. master the existence theorem of implicit function.

C. Know the extremum and unconditional extremum of multivariate functions and understand the geometric application of partial derivatives.

Master the concepts and calculations of multiple integral, curve integral and surface integral.

E. master gauss formula, green formula and stoke formula and their applications.

6), including parameter variable integration

A. Understand the concept and properties of constant integral with parametric variables.

B. master the concept of uniform convergence of generalized integrals with parameters and its discrimination method. Master the properties of uniformly convergent generalized integrals with parametric variables.

Third, the examination paper structure:

1) Examination time: 180 minutes, full mark: 150 points.

2) Problem structure

Answer: Arguments and Counterexamples (105- 135)

B: Basic calculation (15-45 points)

Fourth, the bibliography:

1. Mathematical Analysis (Volume I and Volume II), editor-in-chief of Mathematics Department of Fudan University, Higher Education Press, 2007, 2nd edition.

2. Mathematical analysis problem set, edited by Peking University Mathematics Department, Higher Education Press.

20 1 1 Entrance Examination for Postgraduates in Mathematics Department of Harbin Institute of Technology

[83 1] Advanced Algebra Examination Outline

Exam Subject Name: Advanced Algebra Exam Subject Code: [83 1]

First, the examination requirements

multinomial

1. Understand the concepts of number field, polynomial, divisibility, greatest common factor, coprime, irreducibility, k factor and multifactor. Understand the concepts of polynomial ring, WeChat service, primitive polynomial, dictionary sorting method, symmetric polynomial, elementary symmetric polynomial, homogeneous polynomial and polynomial function.

2. Master the properties of divisibility, division theorem with remainder, greatest common factor theorem, discrimination and properties of coprime polynomials, discrimination and properties of irreducible polynomials, unique factorization theorem of polynomials, remainder theorem, factorization theorem, basic theorem of algebra, Vieta theorem, Gauss lemma, Eisenstein discrimination theorem and basic theorem of symmetric polynomials.

3. Master the necessary and sufficient conditions of non-repetitive factorial, discriminant conditions, Lagrangian interpolation formula, polynomial factorization theory in complex number field, real number field and rational number field, and rational root range of rational polynomial.

4. Take turns to master division and comprehensive division. Master the method of transforming symmetric polynomials into polynomials of elementary symmetric polynomials.

(2) Determinant

1. Understand the concept of determinant, and understand the concepts of subform, remainder and algebraic remainder of determinant.

2. Master the properties of determinant, expand theorem by ranks, Cramer's law, Laplace's theorem and determinant multiplication formula.

3. I will use the properties and expansion theorem of determinant to calculate determinant and master the basic method of calculating determinant.

(3) Linear equation

1. Understand the concepts of vector linear correlation, vector group equivalence, maximal independent group, vector group rank, matrix rank, basic solution system, solution space, etc.

2. Master the discriminant theorem of linear equations and the structure of solutions of linear equations.

3. Master the method of solving linear equations with row elementary transformation.

(4) Matrix

1. Understand the concept of matrix, and understand the concepts and properties of identity matrix, diagonal matrix, triangular matrix, symmetric matrix and antisymmetric matrix.

2. Master the linear operation, multiplication, transposition and operation rules of matrices.

3. Understand the concept of inverse matrix, master the properties of inverse matrix and the necessary and sufficient conditions for matrix reversibility. Understand the concept and properties of adjoint matrix.

4. Master the elementary transformation of matrix, master the properties of elementary matrix, understand the concept of matrix equivalence, and use the elementary transformation method to find the rank and inverse matrix of matrix.

5. Understand the block matrix and master the operation and elementary transformation of the block matrix.

(5) quadratic form

1. The concept of quadratic form and matrix representation, understand the concept of quadratic form rank, master the concepts of canonical form and canonical form of quadratic form and the law of inertia.

2. Master the method of transforming quadratic form into standard form by contract transformation and orthogonal transformation.

3. Master the positive definite, semi-positive definite, negative definite and semi-negative definite of quadratic form and corresponding matrix and their discrimination methods.

(6) Linear space

1. Understand the concepts of linear space, subspace, generating subspace, basis, dimension, coordinate, transfer matrix, sum and direct sum of subspaces. Understand the concept of isomorphism in linear space.

2. Master the basic continuation theorem and dimension formula, and master the necessary and sufficient conditions of direct sum.

3. Can find basis, dimension, coordinate and transformation matrix.

(7) Linear transformation

1. Understand the concepts of linear transformation, eigenvalue, eigenvector, characteristic polynomial, characteristic subspace, invariant subspace, matrix of linear transformation, similar transformation, similar matrix, range and kernel of linear transformation, Jardan canonical form, minimal polynomial, etc.

2. Master the properties of linear transformation, similarity matrix, eigenvalues and eigenvectors, kernel space and value domain, and invariant subspace. Master Hamilton-Cayley theorem and the conditions and methods of decomposing linear space V into A- invariant subspace, and understand the minimum polynomial theory.

3. Master the matrix representation method of linear transformation, the method of finding the eigenvalues and eigenvectors of linear transformation, and the conditions and methods of matrix similarity diagonalization. Mastering the thinking method of "mutual change" between linear transformation and matrix will solve the related problems of various special subspaces.

8) Matrix

1. Understand the concepts of matrix, invertible matrix, determinant factor, invariant factor and elementary factor of matrix, and understand the standard form of matrix.

2. Grasp the necessary and sufficient conditions of matrix reversibility, matrix equivalence and digital matrix similarity, and understand the theoretical derivation of Jordan canonical form.

3. Find the canonical form and invariant factor of the matrix. Jordan canonical form of number matrix.

(9) Euclidean space

1. Master the concepts of inner product, Euclidean space, vector length, included angle, distance, metric matrix, standard orthogonal basis, orthogonal complement, orthogonal transformation, orthogonal matrix, symmetric transformation and isomorphism.

2. Master Schmidt orthogonalization method. Master the properties of standard orthogonal basis, orthogonal transformation, orthogonal matrix, symmetric transformation and canonical form.

3. Master the properties of eigenvalues and eigenvectors of real symmetric matrices. Orthogonal similarity transformation will be used to diagonalize the similarity of real symmetric matrices.

Second, the examination content

Note: The "chapter" and "section" in this paper refer to the "chapter" and "section" in Higher Algebra (Geometry and Algebra Department of Peking University Mathematics Department, Higher Education Press, 3rd edition, 2003).

1) Polynomial (Part 1:11)

2) Determinant (Section 1-8 in Chapter II)

3) Linear equation (Section 1-6 in Chapter III)

4) Matrix (Chapter 4, section 1-7)

5) Quadratic type (Section 1-4 in Chapter 5)

6) Linear space (Chapter 6, section 65,438+0-8)

7) Linear transformation (Section 65,438+0-9 in Chapter 7)

8) Matrix (Section 1-6 in Chapter 8)

9) Euclidean space (Chapter 9, section 65,438+0-6)

Third, the examination paper structure

1) Examination time: 180 minutes, full mark: 150 points.

2) Problem structure