I. Function, Limit and Continuity

Examination content

The concept and representation of function, boundedness, monotonicity, periodicity and parity of function, the properties of basic elementary functions of inverse function, piecewise function and implicit function, and the establishment of functional relationship of graphic elementary function.

Definitions and properties of sequence limit and function limit, left limit and right limit of function, concepts and relationships of infinitesimal and infinitesimal, properties of infinitesimal and four operational limits of infinitesimal, two important limits: monotone bounded criterion and pinch criterion.

Concept of Function Continuity Types of Discontinuous Points of Functions Continuity of Elementary Functions Properties of Continuous Functions on Closed Interval

Examination requirements

Understand the concept of function, master the representation of function, and establish the functional relationship of application problems.

Understand the boundedness, monotonicity, periodicity and parity of functions.

Understand the concepts of compound function and piecewise function, inverse function and implicit function.

Master the nature and graphics of basic elementary functions and understand the concept of elementary functions.

5. Understand the concept of limit, the concepts of left limit and right limit of function and the relationship between the existence of function limit and left limit and right limit.

6. Master the nature of limit and four algorithms.

7. Master two criteria for the existence of limit, and use them to find the limit, and master the method of using two important limits to find the limit.

8. Understand the concepts of infinitesimal and infinitesimal, master the comparison method of infinitesimal, and find the limit with equivalent infinitesimal.

9. Understanding the concept of function continuity (including left continuity and right continuity) will distinguish the types of function discontinuity points.

10. Understand the properties of continuous function and continuity of elementary function, understand the properties of continuous function on closed interval (boundedness, maximum theorem, mean value theorem), and apply these properties.

Second, the differential calculus of unary function

Examination content

Concepts of derivative and differential, geometric and physical meaning of derivative, relationship between derivability and continuity of function, tangent and normal of plane curve, four operations of derivative and differential, derivative of basic elementary function, differential method of compound function, inverse function, implicit function and function determined by parameter equation, higher derivative, invariance of first-order differential form, Lobida's law of differential mean value theorem, and discrimination of function monotonicity. Extreme value of function, concavity and convexity of function graph, inflection point and asymptote, description of function graph, maximum and minimum value of function, arc differential, concept of curvature, curvature circle and radius.

Examination requirements

1. Understand the concepts of derivative and differential, understand the relationship between derivative and differential, understand the geometric meaning of derivative, find the tangent equation and normal equation of plane curve, understand the physical meaning of derivative, describe some physical quantities with derivative, and understand the relationship between function derivability and continuity.

2. Master the four algorithms of derivative and the derivative rule of compound function, and master the derivative formula of basic elementary function. Knowing the four algorithms of differential and the invariance of first-order differential form, we can find the differential of function.

3. If you understand the concept of higher derivative, you will find the higher derivative of simple function.

4. We can find the derivative of piecewise function, implicit function, function determined by parametric equation and inverse function.

5. Understand and apply Rolle theorem, Lagrange mean value theorem, Taylor theorem, and Cauchy mean value theorem.

6. Master the method of finding the limit of infinitive with L'H?pital's law.

7. Understand the concept of extreme value of function, master the method of judging monotonicity of function and finding extreme value of function with derivative, and master the method of finding maximum and minimum value of function and its application.

8. Understand the concepts of curvature, circle of curvature and radius of curvature, and calculate curvature and radius of curvature.

3. Integral calculus of unary function

Examination content

Concepts of original function and indefinite integral, basic properties of indefinite integral, basic integral formula, concept and basic properties of definite integral, mean value theorem of definite integral, upper bound function of integral and its derivative, Newton-Leibniz formula, substitution integration method of indefinite integral and definite integral, rational and simple irrational functions of partial integral, rational function and trigonometric function, abnormal (generalized) integral and definite integral.

Examination requirements

1. Understand the concepts of original function and indefinite integral and definite integral.

2. Master the basic formula of indefinite integral, the properties of indefinite integral and definite integral and the mean value theorem of definite integral, and master the integration methods of method of substitution and integration by parts.

3. Know the integral of rational function, rational trigonometric function and simple unreasonable function.

4. Understand the function of the upper limit of integral, find its derivative and master Newton-Leibniz formula.

5. Understand the concept of generalized integral and calculate generalized integral.

6. Master the expression and calculation of the average value of some geometric and physical quantities (the area of plane figure, the arc length of plane curve, the volume and lateral area of rotating body, and the area of parallel section are known solid volume, work, gravity, pressure, centroid, centroid, etc.). ) and definite integral function.

Four, multivariate function calculus

Examination content

Concept of multivariate function, geometric meaning of bivariate function, concept of limit and continuity of bivariate function, properties of bivariate continuous function in bounded closed region, concepts, basic properties and calculation of partial derivative and total differential of multivariate function, derivation method of multivariate composite function, implicit function, second-order partial derivative, extreme value and conditional extreme value of multivariate function, maximum value and minimum value, double integral.

Examination requirements

1. Understand the concept of multivariate function and the geometric meaning of bivariate function.

2. Understand the concepts of limit and continuity of binary function and the properties of binary continuous function in bounded closed region. 3. Understand the concepts of partial derivative and total differential of multivariate function, and you will find the first and second partial derivatives of multivariate composite function and total differential, and understand the existence theorem of implicit function, and you will find the partial derivative of multivariate implicit function.

4. Understand the concepts of extreme value and conditional extreme value of multivariate function, master the necessary conditions of extreme value of multivariate function, understand the sufficient conditions of extreme value of binary function, find the extreme value of binary function, find the conditional extreme value by Lagrange multiplier method, find the maximum value and minimum value of simple multivariate function, and solve some simple application problems.

5. Understand the concept and basic properties of double integral, and master the calculation methods of double integral (rectangular coordinates and polar coordinates).

Verb (abbreviation of verb) ordinary differential equation

Examination content

Basic concepts of ordinary differential equation, separable variable differential, homogeneous differential equation, first-order linear differential equation, reducible higher-order differential equation, properties and structure theorem of solutions of linear differential equation, second-order homogeneous linear differential equation with constant coefficients, some homogeneous linear differential equations with constant coefficients higher than second-order, simple second-order nonhomogeneous linear differential equation with constant coefficients, and simple applications of differential equations.

Examination requirements

1. Understand differential equations and their concepts such as order, solution, general solution, initial condition and special solution.

2. Mastering the solutions of differential equations with separable variables and first-order linear differential equations can solve homogeneous differential equations.

3. Multi-order differential equations can be solved by order reduction method.

4. Understand the properties of the solution of the second-order linear differential equation and the structure theorem of the solution.

5. Master the solution of second-order homogeneous linear differential equations with constant coefficients, and be able to solve some homogeneous linear differential equations with constant coefficients higher than the second order.

6. Polynomials, exponential functions, sine functions, cosine functions and their sum and product can be used to solve second-order non-homogeneous linear differential equations with constant coefficients.

7. Can use differential equations to solve some simple application problems.

linear algebra

I. Determinants

Examination content

The concept and basic properties of determinant, the expansion theorem of determinant by row (column)

Examination requirements

1. Understand the concept of determinant and master its properties.

2. The properties of determinant and determinant expansion theorem will be applied to calculate determinant.

Second, the matrix

Examination content

Concept of matrix, linear operation of matrix, multiplication of matrix, power of matrix, determinant of matrix product, transpose of matrix, concept and properties of inverse matrix, necessary and sufficient condition of matrix reversibility, adjoint matrix, elementary transformation of matrix, elementary matrix, rank of matrix, equivalence of matrix, block matrix and its operation.

Examination requirements

1. Understand the concepts and properties of matrix, identity matrix, quantization matrix, diagonal matrix, triangular matrix, symmetric matrix, antisymmetric matrix and orthogonal matrix.

2. Master the linear operation, multiplication, transposition and its operation rules of matrix, and understand the determinant properties of square matrix power and square matrix product.

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 of adjoint matrix and use adjoint matrix to find the inverse matrix.

4. Understand the concept of elementary transformation of matrix, understand the properties of elementary matrix and the concept of matrix equivalence, understand the concept of matrix rank, and master the method of finding matrix rank and inverse matrix by elementary transformation.

5. Understand the block matrix and its operation.

Third, the vector

Examination content

Concept of vector, linear combination and linear representation of vector, linear correlation of vector group has nothing to do with linearity, maximal linearity of vector group has nothing to do with, equivalent vector group, rank of vector group, relationship between rank of vector group and rank of matrix, inner product of vector, orthogonal normalization method of linear irrelevant vector group.

Examination requirements

Understand the concepts of n-dimensional vector, linear combination of vectors and linear representation.

2. Understand the concepts of linear correlation and linear independence of vector groups, and master the related properties and discrimination methods of linear correlation and linear independence of vector groups.

3. Understand the concepts of maximal linearly independent group and rank of vector group, and find the maximal linearly independent group and rank of vector group.

4. Understand the concept of vector group equivalence and the relationship between the rank of matrix and the rank of its row (column) vector group.

5. Understand the concept of inner product and master the Schmidt method of orthogonal normalization of linear independent vector groups.

Fourth, linear equations.

Examination content

Cramer's rule for linear equations, necessary and sufficient conditions for homogeneous linear equations to have nonzero solutions, necessary and sufficient conditions for nonhomogeneous linear equations to have solutions, properties and structure of solutions of linear equations, basic solution system and general solution of homogeneous linear equations, and general solution of nonhomogeneous linear equations.

Examination requirements

1. Cramer's law can be used.

2. Understand the necessary and sufficient conditions for homogeneous linear equations to have nonzero solutions and nonhomogeneous linear equations to have solutions.

3. Understand the concepts of basic solution system and general solution of homogeneous linear equations, and master the solution of basic solution system and general solution of homogeneous linear equations.

4. Understand the structure of solutions of nonhomogeneous linear equations and the concept of general solutions.

5. We can solve linear equations with elementary line transformation.

Eigenvalues and eigenvectors of verb (abbreviation of verb) matrix

Examination content

Concepts and properties of eigenvalues and eigenvectors of matrices, concepts and properties of similar matrices, necessary and sufficient conditions for similar diagonalization of matrices, eigenvalues and eigenvectors of similar diagonal matrices, real symmetric matrices and similar diagonal matrices.

Examination requirements

1. Understand the concepts and properties of eigenvalues and eigenvectors of matrices, and find the eigenvalues and eigenvectors of matrices.

2. Understand the concept, properties and necessary and sufficient conditions of similar diagonalization of matrix, and transform the matrix into similar diagonal matrix.

3. Understand the properties of eigenvalues and eigenvectors of real symmetric matrices.

Sixth, quadratic form

Examination content

Quadratic form and its matrix representation, contract transformation and contract matrix, rank of quadratic form, inertia theorem, canonical form and canonical form of quadratic form. Quadratic form is transformed into canonical form, quadratic form and positive definiteness of its matrix by orthogonal transformation and collocation method.

Examination requirements

Understand the concept of quadratic form, express quadratic form in matrix form, and understand the concepts of contract transformation and contract matrix.

Understand the concept of rank of quadratic form, the concept of canonical form and canonical form of quadratic form, inertia theorem. We will use orthogonal transformation and collocation method to convert quadratic form into standard form.

3. Understand the concepts of positive definite quadratic form and positive definite matrix, and master their discrimination methods.

The above is the original text of the 2020 Postgraduate Mathematics Examination Outline, including line generation and advanced mathematics. Pay attention to the outline information and prepare for the exam more efficiently. Everyone must make rational use of the examination syllabus and master more information related to the examination. Welcome to continue to pay attention.