Advanced mathematics

Chapter 1, Function, Limit and Continuity.

Examination content: the concept and expression of function: boundedness, monotonicity, periodicity and parity of function, the properties of basic elementary functions of composite function, inverse function, piecewise function and implicit function, and the establishment of functional relationship of graphic elementary function; The definitions and properties of sequence limit and function limit: the concepts of left limit, right limit and infinitesimal quantity of function and their relations; The nature of infinitesimal and four operational limits of the comparison limit of infinitesimal: two criteria for the existence of operational limits: monotone bounded criterion and pinching criterion; the types of discontinuous points of continuous function concept function; Properties of continuous functions on closed intervals of elementary functions.

Examination requirements:

1. Understand the concept of function and master the expression of function, and you will establish the functional relationship of application problems.

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

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

4. Grasp the nature and graphics of basic elementary functions and understand the concept of elementary functions.

5. Understand the concept of limit, the concept of left and right limit of function and the relationship between the existence of function limit and left 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 functions and the continuity of elementary functions. 1. Understand the properties of continuous functions on closed intervals (boundedness, maximum theorem, mean value theorem) and apply these properties.

Chapter 2: Differential calculus of unary function.

Examination content: the geometric meaning of derivative and differential concepts and the relationship between derivability and continuity of physical meaning function; Four operations of tangent, normal derivative and differential of plane curve: derivative compound function, inverse function, implicit function and differential method of function determined by parameter equation; first-order differential invariant differential mean value theorem of higher derivative; L'H?pital's law function monotonicity discriminates the convexity, inflection point and asymptote of extreme value function graph; graph depicts the maximum and minimum values of concept curvature circle and curvature radius of arc differential curvature.

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. The derivative of piecewise function, implicit function, function determined by parameter equation and inverse function can be obtained.

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

6. Master the method of finding the indefinite limit by L'H?pital method.

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. Will judge the concavity and convexity of the function graph by derivative (note: in the interval (a, b), let the function f(x) have the second derivative. When f``(x) >; 0, the graph of f(x) is concave; When f `` (x) < 0, the graph of f(x) is convex), you will find the inflection point and the horizontal, vertical and oblique asymptotes of the function graph, thus depicting the function graph.

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

Chapter 3: Integral of unary function.

Examination content

The concept of original function and indefinite integral, the basic properties of indefinite integral, the concept of basic integral formula and the basic properties of the mean value theorem of definite integral, the upper limit of integral and the function of its derivative Newton-Leibniz formula, the substitution integration method of indefinite integral and definite integral, and the rational formula of partial integral, rational function and trigonometric function and the application of 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. Can find the integral of rational function, rational formula of trigonometric function and simple unreasonable function.

4. If you understand the function of the upper limit of integral, you will 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 some geometric and physical quantities (the area of a plane figure, the arc length of a plane curve, the volume and lateral area of a rotating body, the area of a parallel section, the volume, work, gravity, pressure, center of mass, centroid, etc. of a known solid. ) and definite integral to find the average value of the function.

Chapter 4: Calculus of Multivariate Functions

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, derivative method of partial derivative of multivariate function and fully differential multivariate composite function and implicit function, concepts, basic properties and calculation of extreme value and conditional extreme value of second-order partial derivative multivariate function.

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. Knowing the concepts of partial derivative and total differential of multivariate function, we can find the first and second partial derivatives of multivariate composite function, total differential, existence theorem of implicit function and 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).

Chapter 5: Ordinary differential equations

Examination content

The basic concepts of ordinary differential equations are separated from variable differential equations, homogeneous differential equations, properties and structural theorems of solutions of first-order linear differential equations, which can be reduced to higher order. Some simple applications of second-order homogeneous linear differential equations with constant coefficients are higher than second-order homogeneous linear differential equations with constant coefficients.

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. The following differential equations will be solved by order reduction method: and

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

5. Grasp the homogeneity of second-order constant coefficient.

linear algebra

Chapter 1: Determinant

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.

Chapter 2: Matrix

Examination content

Concept of matrix, linear operation of matrix, multiplication of matrix, concept and properties of transposed inverse matrix of determinant matrix, necessary and sufficient conditions for matrix reversibility, equivalent block matrix of elementary transformation of matrix and rank matrix of elementary matrix and its operation.

Examination requirements

1. Understand the concepts and properties of matrix, identity matrix, quantitative matrix, diagonal matrix, triangular matrix, symmetric matrix and antisymmetric 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, grasp the properties of inverse matrix and the necessary and sufficient conditions of matrix reversibility, understand the concept of adjoint matrix, and use adjoint matrix to find 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.

Chapter 3: Vector

Examination content

The linear combination of concept vectors of vectors is equivalent to the linear representation of linear correlation of vector groups and the maximal linear independent group of linear independent vector groups. The orthogonal normalization method of inner product linear independent vector group between the rank of vector group and the rank of matrix.

Examination requirements

1. Understand the concepts of n-dimensional vectors, linear combinations of vectors and linear representations.

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.

Chapter four: linear equations.

Examination content

Cramer's Law of Linear Equations Necessary and Sufficient Conditions for Homogeneous Linear Equations to Have Non-zero Solutions Necessary and Sufficient Conditions for Non-homogeneous Linear Equations to Have Solutions Properties and Structures Basic Solution System of Homogeneous Linear Equations and General Solution of Non-homogeneous 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.

Chapter 5: Eigenvalues and eigenvectors of matrices.

Examination content

The concepts of eigenvalues and eigenvectors of matrices, the concepts of property similarity matrices and the necessary and sufficient conditions for similar diagonalization of property matrices, and the eigenvalues and eigenvectors of similar diagonal matrices and their real symmetric matrices.

Examination requirements

1. Understand the concepts of matrix eigenvalues and eigenvectors, master the properties of matrix eigenvalues, and master the methods of finding matrix eigenvalues and eigenvectors.

2. Understand the concept of matrix similarity, master the properties of similar matrix, understand the necessary and sufficient conditions for matrix similarity to diagonal, and master the method of transforming matrix into similar diagonal matrix.

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

Chapter six: quadratic form

Examination content

Quadratic form and its matrix represent contract transformation and rank inertia theorem of quadratic form of contract matrix. The canonical form and canonical form of quadratic form are transformed into canonical quadratic form and the positive definiteness of its matrix by orthogonal transformation and matching method.

Examination requirements

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

2. Understand the concept of rank of quadratic form, the concepts of standard form and standard form of quadratic form, and inertia theorem, and transform quadratic form into standard form by orthogonal transformation and collocation method.

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