Elementary mathematics studies constants and advanced mathematics studies variables.

Advanced mathematics (also known as calculus, which is the general name of several courses) is an important basic subject in science and engineering colleges. As a science, advanced mathematics has its inherent characteristics, namely, high abstraction, strict logic and wide application. Abstract is the most basic and remarkable feature of mathematics-high abstraction and unity, which can profoundly reveal its essential laws and make it more widely used. Strict logic means that in the induction and arrangement of mathematical theory, whether it is concept and expression, or judgment and reasoning, we must use the rules of logic and follow the laws of thinking. Therefore, mathematics is also a way of thinking, and the process of learning mathematics is the process of thinking training. The progress of human society is inseparable from the wide application of mathematics. Especially in modern times, the appearance and popularization of electronic computers have broadened the application field of mathematics. Modern mathematics is becoming a powerful driving force for the development of science and technology, and it has also penetrated into the field of social sciences extensively and deeply. Therefore, it is very important for us to learn advanced mathematics well. However, many students are confused about how to learn this course well. If you want to learn advanced mathematics well, you must at least do the following four things:

First, understand the concept. There are many concepts in mathematics. Concepts reflect the essence of things. Only by figuring out how it is defined and what its essence is can we really understand a concept.

Secondly, master the theorem. Theorem is a correct proposition, which is divided into two parts: condition and conclusion. In addition to mastering its conditions and conclusions, we should also understand its scope of application and be targeted.

Third, do some exercises on the basis of understanding the examples. Especially remind learners that the examples in the textbook are very typical, which is helpful to understand concepts and master theorems. Pay attention to the characteristics and solutions of different examples, and do appropriate exercises on the basis of understanding examples. When writing a topic, you should be good at summing up-not only the methods, but also the mistakes. You will gain something after doing this, so you can draw inferences from others.

Fourth, clear the context. We should have an overall grasp of the knowledge we have learned and summarize the knowledge system in time, which will not only deepen our understanding of knowledge, but also help us to further study.

Advanced mathematics includes calculus and solid analytic geometry, series and ordinary differential equations. Calculus is the most systematic and widely used in other courses. The theory of calculus was completed by Newton and Leibniz. (Of course, calculus has been applied before them, but it is not systematic enough. ) The basic concepts of calculus and extreme calculus are difficult to understand.

Advanced mathematics is divided into several parts:

First, the function limit continuity.

Second, the differential calculus of unary function

3. Integral calculus of unary function

4. Vector Algebra and Spatial Analytic Geometry

Verb (abbreviation of verb) Differential calculus of multivariate functions

Six, multivariate function integral calculus

Seven, infinite series

Eight, ordinary differential equations

The high figures mainly include

First, function and limit are divided into

Constants and variables

function

Simple behavior of function

inverse function

Elementary function

Sequence limit

functional limit

Infinite quantity and infinite quantity

Comparison of infinitesimal quantities

Functional continuity

Properties of continuous functions and functional continuity of elementary functions

Second, derivative and differential

The concept of derivative

Derivation rule of sum and difference of functions

Derivation rule of product sum quotient of functions

Derivation rule of compound function

Inverse function derivation rule

higher derivative

Implicit function and its derivative rule

Differential of function

Third, the application of derivatives.

Differential mean value theorem

Uncertain problem

Determination of monotonicity of function

Extreme value of function and its solution

Maximum and minimum values of functions and their applications

Concave direction and inflection point of curve

Fourth, indefinite integral

Concept and properties of indefinite integral

The method of finding indefinite integral

Examples of several special function integrals

Five, definite integral and its application

The concept of definite integral

Integral formula of calculus

Partial substitution integral method for definite integral

Generalized integral

Six, spatial analytic geometry

Space rectangular coordinate system

Direction cosine and direction number

Plane and spatial straight line

Surfaces and space curves

Eight, multivariate function differential calculus

The concept of multivariate function

Limit and continuity of binary function

partial derivative

complete differential

Derivation method of multivariate composite function

Extreme value of multivariate function

Nine, multivariate function integral calculus

The Concept and Properties of Double Integral

Calculation method of double integral

The concept of triple integral and its calculation method

Ordinary differential equation

Basic concepts of differential equations

Differential equation and homogeneous equation of separable variables

linear differential equation

Degradable higher order equation

Structure of solutions of linear differential equations

Solution of second-order homogeneous linear equation with constant coefficients

Solution of second-order non-homogeneous linear equation with constant coefficients

XI。 infinite series

The concept of derivative

Before learning the concept of number, let's discuss the instantaneous speed of linear motion with variable speed in physics.

For example, suppose a particle moves along the X axis, and its position X is a function of time T, and y=f(x), and find the instantaneous velocity of the particle at t0?

We know that when the time increases from t0 to δt, the position of particles increases.

This is the displacement of the particle in the time period △ t, so the average velocity of the particle during this period is;

If the particle moves at a uniform speed, this is the instantaneous speed at t0; If the particle moves along a non-uniform straight line, this is not the instantaneous velocity at t0.

We think that when the time period △t is infinitely close to 0, the average velocity will be infinitely close to the instantaneous velocity of the particle t0.

Namely: instantaneous velocity of particles at t0 =

To this end, the definition of derivative is produced as follows:

Definition of derivative

Let the function y=f(x) be defined in the neighborhood of point x0. When the independent variable x has an increment △ x at x0 (x+△ x is also in this neighborhood), correspondingly,

Function has increment

If the ratio of △y to △x has a limit when △x→0, it is called the derivative of y=f(x) at x0.

Remember as follows:

It can also be recorded as:

The function f(x) has a derivative at point x0. The function f(x) is simply derivable at point x0, otherwise it is not derivable.

If the function f(x) is differentiable at every point in the interval (a, b), it is said that the function f(x) is differentiable in the interval (a, b). At this time, the function y=f(x) is different for this region.

Each definite value of x in (a, b) corresponds to a definite derivative, which constitutes a new function.

Let's call this function the derivative of the original function y=f(x).

Note: the derivative is also the limit of the difference quotient.

Left and right derivatives

We have the concept of left and right limit, and the derivative is the limit of difference quotient, so we can give the concept of left and right derivative.

Ruoji

Existence, we call it the left derivative of the function y=f(x) at x=x0.

Ruoji

Existence, we call it the right derivative of the function y=f(x) at x=x0.

Note: The existence and equality of the left and right derivatives of the function y=f(x) at x0 is a necessary and sufficient condition for the function y=f(x) to be derivable at x0.

Discrete mathematics

[Edit this paragraph] Introduction

Discrete mathematics is a mathematical discipline that studies the structure and relationship of discrete quantities and an important branch of modern mathematics. It is widely used in various disciplines, especially in computer science and technology. At the same time, discrete mathematics is also a necessary prerequisite for many professional courses of computer specialty, such as programming language, data structure, operating system, compilation technology, artificial intelligence, database, algorithm design and analysis, theoretical computer science foundation and so on. Through the study of discrete mathematics, we can not only master the descriptive tools and methods for dealing with discrete structures, but also create conditions for subsequent courses, improve abstract thinking and strict logical reasoning ability, and lay a solid foundation for participating in innovative research and development in the future.

With the advent of the information age, the mainstream position of continuous mathematics represented by calculus has changed in the era of industrial revolution, and the importance of discrete mathematics has been gradually recognized by people. The ideas and methods taught in discrete mathematics are widely reflected in various fields of computer science and technology and related majors, from scientific calculation to information processing, from theoretical computer science to computer application technology, from computer software to computer hardware, from artificial intelligence to cognitive system, which are closely related to discrete mathematics.

Because the digital electronic computer is a discrete structure, it can only deal with discrete or discrete quantitative relations. Therefore, both computer science itself and modern scientific research fields closely related to computer science and its application are faced with the problem of how to establish corresponding mathematical models for discrete structures. How to discretize the mathematical model established by continuous quantitative relationship so that it can be processed by computer.

Discrete mathematics is a comprehensive subject which integrates traditional logic, set theory (including function), number theory, algorithm design, combinatorial analysis, discrete probability, relation theory, graph theory and tree, abstract algebra (including algebraic system, group, ring, field, etc.). ), Boolean algebra, computational models (languages and automata) and so on. The application of discrete mathematics covers many fields of modern science and technology.

The course of Discrete Mathematics mainly introduces the basic concepts, theories and methods of various branches of discrete mathematics. These concepts, theories and methods are widely used in digital circuits, compilation principles, data structures, operating systems, database systems, algorithm analysis and design, artificial intelligence, computer networks and other professional courses. At the same time, the training provided by this course is very beneficial to improve students' ability of generalization and abstraction, logical thinking and inductive structure, and to cultivate students' rigorous, complete and standardized scientific attitude.

The teaching purpose of discrete mathematics course is not only the theoretical basis and core main course of computer science and technology and related majors, but also to provide necessary theoretical support for subsequent courses. More importantly, it aims to cultivate and improve students' mathematical thinking ability and ability to solve practical problems by strengthening specialization, repetitive research, training and application in mathematical reasoning, combinatorial analysis, discrete structure, algorithm conception and design, model construction and so on. "

The fields that discrete mathematics usually studies are: mathematical logic, set theory, algebraic structure, relational theory, function theory, graph theory, combinatorics, number theory and so on. It is one of the important basic courses for computer and related majors in colleges and universities.

This course covers:

1. set theory: sets and their operations, binary relations and functions, natural numbers and natural number set, cardinality of sets.

2. Graph theory: basic concepts of graphs, Euler graphs and hamiltonian graph, matrix representation of trees and graphs, planar graphs, graph coloring, dominating sets, covering sets, independent sets and matching, weighted graphs and their applications.

3. Algebraic structure: the basic concepts of algebraic system, semigroup and singularity, group, ring and field, lattice and Boolean algebra.

4. Combinatorial mathematics: combinatorial existence theorem, basic counting formula, combinatorial counting method and combinatorial counting theorem.

5. Mathematical logic: propositional logic, first-order predicate calculus and resolution principle.

Discrete mathematics is divided into three courses, namely set theory and graph theory, algebraic structure and combinatorial mathematics, and mathematical logic. The teaching method is mainly classroom teaching, supplemented by written homework after class, courseware release through the school network teaching platform and teacher-student communication. [Edit this paragraph] Related literature

1 Geng suyun, qu wanling, discrete mathematics (tenth five-year textbook), higher education press, 2004.

2 Yuan Chongyi, Qu Wanling, Wang Hanpin, Liu Tian, Discrete Mathematics and Its Application (4th Edition, translated), Machinery Industry Press, 2002.

3 Lu Zhongwan, Mathematical Logic in Computer Science, Science Press, 2002.

Hamilton, translated by Zhu Shuilin, Mathematical Logic, East China Normal University Press, 1987.

5 Geng Suyun, Discrete Mathematics Exercise Set-Mathematical Logic and Set Theory, Peking University Publishing House, 1993.

6 Sean, Discrete Mathematics Problem Set-Abstract Algebra Volume, Peking University Publishing House, 1990.

7 Geng Suyun, Discrete Mathematics Problem Set-Graph Theory Volume, Peking University Publishing House, 1990.

8 discrete mathematics exercise counseling software

9 propositional logic teaching software

10 discrete mathematics course, Geng Suyun, Qu Wanling, Wang Hanpin, Peking University Publishing House, 2002.

1 1 Discrete Mathematics and Its Applications, 4th Edition, Kenneth H. Rosen.

The value of this book has been confirmed by hundreds of universities all over the world. As a classic textbook in the field of discrete mathematics, almost all famous universities in the world use this book as a textbook. From my personal point of view, this book is an encyclopedia of discrete mathematics. The book not only introduces the theory and method of discrete mathematics, but also has rich historical materials and related learning website resources. What's more exciting is that this book rarely combines discrete mathematics theory and application so well. We can see the application examples of discrete mathematics theory in logic circuits, programming, commerce, Internet and other fields. The English version of this book (the fifth edition) has added quite a few biographies of mathematics and computer scientists. This is a rare reference in the history of computer science. As a teaching material, this book is accompanied by a considerable number of exercises. There is also a group of topics at the back of each chapter, which combines the contents of calculation and discrete mathematics that students have learned to train. This book is also the only English textbook I read personally when I was studying discrete mathematics. It is really a good book worth recommending.