1. Differential calculus of multivariate functions

1) Understand the concept of multivariate function and the geometric meaning of binary function.

2) Understand the concepts of limit and continuity of binary functions and the properties of continuous functions on bounded closed fields.

3) To understand the concepts of partial derivative and total differential, we will seek total differential, understand the necessary and sufficient conditions for the existence of total differential, and understand the invariance of differential form.

4) Understand the concepts of directional derivative and gradient, and master their calculation methods.

5) Mastering the solution of the first-order partial derivative of the composite function, we can get the second-order partial derivative of the composite function.

6) Find the partial derivative of implicit function (including implicit function determined by equation).

7) Understand the concepts of tangent and normal plane of curve and tangent and normal plane of surface, and find their equations.

8) Understand Taylor formula of binary function.

9) Understand the concepts of multivariate function extremum and conditional extremum, grasp the necessary conditions of multivariate function extremum, and understand the sufficient conditions of binary function extremum, so as to find the binary function extremum. Using Lagrange multiplier method will solve the simple maximum and minimum problems.

2. Multivariate function integral calculus

1) Understand the concept, properties and mean value theorem of double integral.

2) Master the calculation methods of rectangular coordinates and polar coordinates of double integral, and master the calculation methods of rectangular coordinates, cylindrical coordinates and spherical coordinates of triple integral.

3) Understand the concepts, properties and relationships of two kinds of curve integrals.

4) Two kinds of curve integrals will be calculated.

5) Be familiar with Green's formula, be able to use the condition that the plane curve integral has nothing to do with the path, and be able to find the original function of total differential.

6) Understand the concepts, properties and relationships of two kinds of surface integrals, master the calculation methods of two kinds of curve integrals, and calculate surface and curve integrals by using Gaussian formula and Stokes formula.

7) The concepts of dissolution and rotation are introduced and calculated.

8) Some geometric physical quantities (such as volume, surface area, arc length, mass, center of gravity, moment of inertia, flow, work, gravity, etc.). ) can be obtained by using multiple integral, curve integral and surface integral.

3. Infinite series

1) Understand the concept of convergence and divergence of infinite series, understand the concept of sum of infinite series, and master the basic properties of series and the necessary conditions for convergence.

2) Familiar with geometric series and convergence of series.

3) Master the comparison trial convergence method of positive series, the ratio trial convergence method of positive series and the reuse radical trial convergence method.

4) Master Leibniz theorem of staggered series.

5) Understand the concepts of absolute convergence and conditional convergence of series, and the relationship between absolute convergence and convergence.

6) Understand the convergence domain of function term series and the concept of function.

7) Understand the concept of convergence radius of power series and master the solution of convergence radius, convergence interval and convergence domain of power series.

8) To understand some basic properties of power series in its convergence interval, we will find the sum function of some power series in the convergence interval, and then we will find the sum of some polynomial series.

9) Understand the necessary and sufficient conditions for the function to expand into Taylor series.

10) will be used to expand some simple functions into power series.

1 1) Understand the simple application of power series in approximate calculation.

12) Understand the concept of Fourier series, Dirichlet's theorem, the sufficient conditions for functions to expand into Fourier series, expand the functions defined above into sine and cosine series, and write the expressions of Fourier series and functions.

4. Ordinary differential equations

1) Understand the concepts of differential equations, solutions, general solutions, special solutions and initial conditions.

2) Master the solutions of equations with separable variables and first-order linear equations.

3) Be able to solve homogeneous equation and Bernoulli equation, understand the idea of variable substitution to solve equations, and be able to solve fully differential equations.

4) Solve the equation by order reduction method.

5) Understand the properties and structure of solutions of linear differential equations.

6) Master the solution of second-order homogeneous linear differential equation with constant coefficients and understand the solution of higher-order homogeneous linear equation with constant coefficients.

7) The second-order non-homogeneous linear differential equation with constant coefficients can be obtained.

8) Will use differential equations to solve some simple geometric and physical problems.

Fourth, focus, depth and breadth

Chapter 8 focuses on the calculation of partial derivative, geometric application of partial derivative and conditional extreme value. It is required to calculate the second partial derivative of composite function skillfully and accurately.

The ninth and tenth chapters mainly introduce the calculation and application of double integral, triple integral, curve integral and surface integral, Green formula and Gaussian formula.

Chapter 11 focuses on the convergence method of positive series, the solution of convergence domain of power series, and the expansion of function into power series by indirect method.

Chapter 12 focuses on the solution of first-order constant coefficient differential equations and second-order non-homogeneous linear differential equations.

Verb (short for verb) others

1. Features and some special requirements of this course

To learn advanced mathematics well, we must do a considerable number of exercises, especially derivatives and indefinite integrals. There are many methods and difficulties, so we must concentrate our time and energy on repeated practice. Ask the teacher to arrange appropriate exercises and review them in time. It is suggested that teachers adopt heuristic teaching methods, concentrate more on teaching and practice, and avoid analogy.

2. Use teaching materials:

Advanced Mathematics published by Higher Education Press (edited by Tongji University, fifth edition);