A. mathematical analysis, advanced mathematics, advanced algebra, linear algebra. What are the differences and connections between these courses?

Advanced mathematics is mainly calculus

Mathematical analysis is also calculus, but the explanation route is different.

These two books are basically the same, which is helpful for you to learn physics and finite element. study hard

There is basically no calculus in advanced algebra, and it is all about how to solve multivariate equations, and then it is extended to matrices, and how to solve matrices.

The same is true of linear algebra, and then there are some things like probability theory and computational mathematics.

These two books are helpful to programming, probability and computer language.

What do you think of advanced mathematics?

In order to make everyone understand the position of "advanced mathematics" in mathematics, let's briefly introduce the history of mathematics.

From the most general point of view, the history of mathematics can be divided into four basic and different stages. Of course, it is impossible to accurately divide these stages, because the essential characteristics of each successive stage are gradually formed, and in each "early stage", the content of "late stage" is pregnant, even budding; And each "later stage" is a stage of the continuous development of its "earlier stage" content. However, the differences between these stages and the transition between them can be clearly shown.

The first stage: the embryonic stage of mathematics

This period began in ancient times and ended in the 5th century BC. During this period, human beings accumulated a lot of mathematical knowledge in long-term production practice, gradually formed the concept of number and produced the operation method of number. Due to the need of field measurement and astronomical observation, the initial development of geometry has been caused. This period is the period of the formation of arithmetic and geometry, but they are not separated, but closely intertwined with each other. It has not formed a strict and complete system, and more importantly, it lacks logic, and basically can't see the proof, deductive reasoning and axiomatic system of propositions.

The second stage: the period of constant mathematics

That is, the "elementary mathematics" period. This period began in the 6th and 7th centuries BC and ended in the middle of17th century, which lasted for more than 2,000 years. During this period, mathematics changed from concrete stage to abstract stage, and gradually formed an independent and deductive science. During this period, arithmetic, elementary geometry, elementary algebra and trigonometry all became independent branches. The basic achievements of this period constitute the main content of middle school mathematics textbooks.

The third stage: the period of variable mathematics

That is, the period of "advanced mathematics". This period began with the birth of Cartesian analytic geometry in the middle of17th century and ended in the middle of19th century. The difference between this period and the previous period is that in the previous period, the individual elements of the objective world were studied by static methods, while in this period, the laws of things' change and development were explored by the viewpoint of movement and change.

During this period, the concepts of variables and functions entered mathematics, and then calculus came into being. Although new branches of mathematics appeared in this period, such as probability theory and projective geometry, they all seemed to be covered by the excessive brilliance of calculus. The basic achievements of this period are analytic geometry, calculus and differential equations, which are the basic courses in colleges and universities today.

The fourth stage: modern mathematics stage.

This period began in the middle of19th century. This period is characterized by profound changes in algebra, geometry and mathematical analysis. Geometry, algebra and mathematical analysis become more abstract. It can be said that in modern mathematics, the concepts of "number" and "shape" have developed to a very high level. For example, many algebraic structures of non-number "numbers", such as groups, rings, fields and so on. Some intangible abstract spaces, such as linear space, topological space, manifold, etc.

There is also mathematics in the research field of human intellectual activities. Mathematical logic is a new subject that came into being at the end of 19 and has been widely developed. It studies the structure and reasoning process of propositions by mathematical methods.

With the development of science and technology, the basic disciplines of mathematics, mathematical physics, economics and other disciplines intersect and penetrate each other, forming many marginal disciplines and comprehensive disciplines. The emergence and development of * * * theory, computational mathematics and electronic computers. , which constitutes a colorful modern mathematics that permeates all scientific and technological departments.

The distinction between "elementary" mathematics and "advanced" mathematics is completely established. It can be pointed out that this kind of middle school curriculum, which is customarily called "elementary mathematics", has two inherent characteristics.

The first characteristic is that its research object is an invariant quantity (constant) or an isolated invariant regular geometry; The second characteristic lies in its research method. Elementary algebra and elementary geometry are constructed according to independent paths that are not related to each other, so we can neither state geometric problems in algebraic terms nor solve geometric problems by algebraic methods through calculation.

16th century, due to the direct promotion of the industrial revolution, the study of sports became the central issue of natural science at that time, which was different from the previous mathematical problems in principle. To solve them, elementary mathematics is not enough. It is necessary to create brand-new concepts and methods, and create new mathematics to study the changes between various quantities in phenomena. The new concepts of variables and functions came into being in time, which led to the transition from elementary mathematics to advanced mathematics.

Higher mathematics, contrary to elementary mathematics, is developed on the basis of the close combination of algebraic method and geometric method. This combination first appeared in analytic geometry founded by Descartes, a famous French mathematician and philosopher. Descartes introduced variables into mathematics and created the concept of coordinates. With the concept of coordinates, on the one hand, the geometric theorem can be successfully proved by the operation of algebraic operators, on the other hand, because of the obvious concept of geometry, new analytical theorems can be established and new arguments can be put forward. Descartes analytic geometry has made epoch-making changes in the history of mathematics. Engels once spoke highly: "The turning point in mathematics is Descartes' variable. With variables, movement enters mathematics, with variables, dialectics enters mathematics, with variables, differentiation and integration become necessary ... "

Someone made a shallow analogy: if the whole mathematics is compared to a big tree, then elementary mathematics is the root, all branches of mathematics are branches, and the trunk is "advanced analysis, advanced algebra and advanced geometry" (they are collectively called advanced mathematics). This simple metaphor vividly illustrates the position and function of "three highs" in mathematics, and calculus has a more special position in "three highs". Of course, learning calculus should have the foundation of elementary mathematics, and learning any modern mathematics or engineering technology must first learn calculus.

Newton, a British scientist, and Leibniz, a German scientist, independently founded calculus on the basis of summarizing the previous work, which is a major event in the history of science rather than mathematics. Engels pointed out: "Among all the theoretical achievements, nothing is necessarily regarded as the highest victory of human spirit like the invention of calculus in the second half of17th century." He also said; "Only calculus can make it possible for natural science to express not only the state, but also the process and movement with mathematics." Today, calculus has always been listed as an important basic theory course among all the majors of economics and science and engineering in universities.

The main learning content and teaching purpose of higher mathematics

The course "Advanced Mathematics" we are going to study includes limit theory, calculus, infinite series theory and primary differential equations, and the most important part is calculus.

The research object of calculus is function, and limit is the basis of calculus (and the basis of the whole analysis). Through the study of advanced mathematics, students should obtain:

(1) function, limit and continuity;

(2) Calculus of unary function;

(3) Calculus of multivariate functions;

(4) Infinite series (including Fourier series);

(5) Ordinary differential equations.

Basic concepts, basic theories and basic operational skills. To lay the necessary mathematical foundation for the following courses. It is necessary to cultivate students' abstract generalization ability, logical reasoning ability and self-study ability through various teaching links, and pay special attention to cultivating students' more skilled computing ability and comprehensive application of learned knowledge to analyze and solve problems.

How to learn advanced mathematics well?

1. To learn advanced mathematics well, we must first understand its characteristics.

Advanced mathematics has three remarkable characteristics: high abstraction; Strict logic; Widely used.

(1) is highly abstract

The abstraction of mathematics has been shown in simple calculations. We use abstract numbers, but we don't always associate them with concrete objects. In the abstraction of mathematics, only the relationship between quantity and spatial form is left, and everything else is abandoned. Its abstraction greatly exceeds the general abstraction in natural science.

(2) strict logic

Every theorem in mathematics, no matter how many examples are verified, can only be established in mathematics if it is strictly proved logically. To prove a theorem in mathematics, we must start from conditions and existing mathematical formulas and use strict logical reasoning methods to deduce conclusions.

(3) wide applicability

Advanced mathematics has a wide range of applications. For example, if we master the concept of derivative and its algorithm, we can use it to describe and calculate geometric quantities such as tangent slope and curvature of curve; It can be used to describe and calculate physical quantities such as speed, acceleration and density. It can be used to describe and calculate economic quantities such as product output growth rate and cost reduction rate; …… 。 After mastering the concept and algorithm of definite integral, we can use it to describe and calculate geometric quantities such as arc length of curve, area of irregular figure and volume of irregular solid. It can be used to describe and calculate the travel, variable force work, center of gravity and other physical quantities of variable speed objects. It can be used to describe and calculate economic quantities such as total output and total cost; …… 。

Advanced mathematics not only provides convenient calculation tools and mathematical methods for other disciplines, but also is the necessary mathematical foundation for learning modern mathematics.

2. Teaching characteristics of advanced mathematics.

For university courses, especially advanced mathematics as a basic theoretical course, classroom teaching is an important link. Compared with the classroom teaching of middle school mathematics, the classroom teaching of advanced mathematics has the following three significant differences.

(1) Class size

The advanced mathematics class is a big class of one or two hundred students, so it is impossible to ask students questions often in this big class. There must be differences between students in learning, level, understanding and acceptance, but the basic point of teachers' teaching can only be to take care of the majority, and it is impossible to give detailed and repeated lectures to a few students who can't keep up and don't fully understand.

(2) Time is long

Two classes at a time, *** 100 minutes.

(3) rapid progress

Advanced mathematics is extremely rich in content and relatively few in class hours (compared with middle school mathematics classes). On average, one or two (or even more) textbooks are taught in each class. In addition, the teaching requirements of universities and middle schools are quite different. Teachers mainly focus on key points, difficulties and doubts, methods of analyzing problems, and ideas of explaining problems. Examples are much less than those in middle schools. Unlike math classes in middle schools, teachers should repeat an important theorem carefully, and then give a lot of typical examples after the lecture.

3. Pay attention to the six links of learning.

Advanced mathematics is the first course and the most important basic course that students meet after entering the university. Because there is a great difference between the teaching method from middle school and the training goal of students' ability, students will feel uncomfortable at first. In order to adapt to this environment as soon as possible, we should pay attention to the following six learning links.

(1) preview

In order to improve the effect of lectures, teachers should preview what they want to say before each class. The focus of preview is to read the definitions, theorems and main formulas to be talked about. The main purposes of preview are as follows: 1. When listening to the class, I have a bottom in my heart and don't passively follow the teacher's "heel"; Second, know what the key points are, what your difficulties and doubts are, so as to improve efficiency in class; Third, it can make up for the difficulties in attending classes caused by differences in foundation and understanding. Vividly speaking, preview is like buying a tourist map and its description before going to a scenic spot, so that you can be more active and gain more when traveling.

(2) Attend lectures

Listening to lectures is the main link to acquire knowledge in universities. Therefore, we should be full of energy, have a strong interest in acquiring new knowledge, preview the doubts and difficulties in the preparation, and listen attentively to how teachers ask questions, analyze problems, solve problems and think positively.

When listening to the class, I often encounter some problems that I don't understand. At this time, don't dwell on these issues, which will affect the continuation of the class. You have to admit it and mark it in your textbook or notes to keep up with the teacher's teaching. Problems and questions left over will be considered and studied after class review, or discussed with classmates, or answered by teachers, or read reference books.

(3) Take notes

The teacher's lecture is not "scripted". The teacher mainly focuses on the key points, difficulties, doubts, ideas and methods, and also puts forward some problems that should be paid attention to, supplementing some contents and examples that are not in the textbook. Therefore, taking good class notes is an important part of learning advanced mathematics well. However, it should be noted that the central task of classroom learning is listening, watching and thinking, and the purpose of taking notes is to facilitate after-class review and digest what is said in class. So take notes and don't take up too much class time. Notes don't need to be neat, comprehensive and coherent, but leave more blanks for supplement, writing experience and taking notes after class.

Step 4 review

Learning includes two aspects: learning and not learning. "Learning" means acquiring knowledge, while "learning" means digesting, mastering and consolidating knowledge. On the second day after each class, we should review the class content in time in combination with the class notes and teaching materials. However, before you open your textbooks and notes, you should review the main contents of the class. In addition, we should review what we have said before frequently and repeatedly, so that on the one hand, we can avoid forgetting when studying, on the other hand, we can deepen our understanding of what we have learned before and raise our knowledge level to a higher level.

Do your homework.

To master advanced mathematics, it is necessary to finish homework in time and seriously. It is best to finish each homework on the same day, but it should be finished after reviewing the content of the day. Doing homework is not only a means to test the learning effect, but also an important means to cultivate and improve the comprehensive analysis, written expression and calculation ability.

It is especially emphasized that finishing homework carefully is a link to cultivate students' rigorous scholarship. Therefore, the homework is required to be "neat in handwriting, accurate in drawing, clear in organization and sufficient in arguments". Don't copy, try not to look at the answer first.

(6) Answer questions

Answering questions is an important link in higher mathematics learning. When in doubt, you should discuss with your classmates or ask the teacher in time, and never put the question aside. For example, if the university curriculum is compared to a group of buildings, then advanced mathematics is the tallest building that needs to be built first, not "a group of buildings". If the quality is not good in the construction process, then this building can't be built, and it is difficult to build the building behind it.

In addition to attaching importance to the above-mentioned learning links, there is another point that should be strongly advocated, that is, mutual assistance and cooperation, and discussion and improvement. Team spirit is equally important for learning advanced mathematics well.

C. Your real feelings and suggestions about the advanced mathematics course

This is quite difficult. In fact, you still have to brush the questions to learn well.

D. How to write a good course description of advanced mathematics

You transfer credits. You'd better find an experienced person with excellent English to write this. I'm looking for an elite team.

E. What is higher mathematics, and how is it different from junior high school mathematics?

Elementary copying mathematics studies constants and uniform variables, while advanced mathematics studies non-uniform variables.

Advanced mathematics (a general term for several courses) is an important basic subject in universities of science and engineering. As a science, advanced mathematics has its inherent characteristics, namely, high abstraction, strict logic and wide application. Abstract and calculation are the most basic and remarkable characteristics 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.

F. Advanced mathematics courses in universities

The first part of the first volume of freshman is similar to that of senior three, but those theorems and so on all use up the definition of inner product, which is completely different from that of senior three. The third point you mentioned, I guess you may have seen the front of integral, but you haven't seen the content of calculus yet. The emphasis of advanced mathematics is calculus, which is found in many later courses, especially specialized courses. In fact, not only advanced mathematics, but also college physics. The classical physics in front of me was studied in high school, but many contents are defined by calculus, which is different from high school. Study hard, the first semester of freshman is not difficult in itself. I'll tell you one more thing. When you go to college, you will learn a lot. If you really study, you will find yourself smaller and smaller with limited knowledge. Work hard!

G. What's the difference between advanced mathematics (1) and advanced mathematics (specialized subject)? How difficult are the two courses?

Hehe, girl. The outline of advanced mathematics (I) includes: 1. Function 12, limit 12, derivative and differential 24, definite integral and indefinite integral 24 The outline of advanced mathematics (specialty) includes: 0 1 function 02 limit and continuity 03 derivative and differential 04 differential calculus application 05 indefinite integral 06 definite integral and application 07 spatial analytic geometry 08 multivariate function differential 09 multivariate function integral/kloc-. If Shantou decides to study information processing, mathematical knowledge is very useful. Anyway, come on. Good luck! 1 1 grade number

H. What did you learn about advanced mathematics in college?

In China, students majoring in science and engineering (except mathematics, who study mathematical analysis) have difficulty in learning mathematics, which is often called "advanced mathematics" in textbooks; Students majoring in literature and history learn a little shallower mathematics, and their textbooks are often called "calculus".

Different majors in science and engineering, literature and history have different degrees of depth. It is advanced mathematics that studies variables, but advanced mathematics does not only study variables. As for the courses related to "advanced mathematics", there are usually: linear algebra (advanced algebra for mathematics majors), probability theory and mathematical statistics (some mathematics majors study independently).

The basic concepts and contents of calculus include differential calculus and integral calculus.

The main contents of differential calculus include: limit theory, derivative, differential and so on.

The main contents of integral include definite integral, indefinite integral and so on.

Generalized mathematical analysis includes calculus, function theory and many other branches, but now it is generally customary to equate mathematical analysis with calculus, and mathematical analysis has become synonymous with calculus. When it comes to mathematical analysis, you know that it refers to calculus.

Mathematical statistics is a branch of mathematics developed with the development of probability theory. It studies how to effectively collect, sort out and analyze the data affected by random factors, and make inferences or predictions about the problems considered, thus providing basis or suggestions for taking certain decisions and actions.

Probability theory is a branch of mathematics that studies the quantitative laws of random phenomena. Random phenomena are relative to decisive phenomena. The phenomenon that a certain result must occur under certain conditions is called decisive phenomenon.

For example, at standard atmospheric pressure, when pure water is heated to 100℃, water will inevitably boil. Random phenomenon means that under the same basic conditions, before each experiment or observation, it is uncertain what kind of results will appear, showing contingency. For example, when you flip a coin, there may be heads or tails.

The realization and observation of random phenomena are called random experiments. Every possible result of random test is called a basic event, and a basic event or a group of basic events is collectively called a random event, or simply called an event. Typical random experiments include dice, coins, playing cards and roulette.

Linear algebra is a branch of mathematics, and its research objects are vectors, vector spaces (or linear spaces), linear transformations and linear equations with finite dimensions. Vector space is an important subject in modern mathematics.

Therefore, linear algebra is widely used in abstract algebra and functional analysis; Through analytic geometry, linear algebra can be expressed concretely. The theory of linear algebra has been extended to operator theory. Because the nonlinear model in scientific research can usually be approximated as a linear model, linear algebra is widely used in natural science and social science.

(8) Extended reading of the characteristics of advanced mathematics courses:

Of the three branches of mathematics established before19th century, the first two were originally branches of elementary mathematics and later developed into a part of advanced mathematics, and only analysis belonged to advanced mathematics from the beginning. Calculus, the basis of analysis, is regarded as the beginning of "mathematics of variables", so studying variables is one of the characteristics of higher mathematics.

The original concept of variables is a direct abstraction of variables that change in the material world, and the concept of variables in modern mathematics contains a higher level of abstraction. For example, the research in mathematical analysis is limited to real variables, while other branches of mathematics study complex variables of complex vectors and complex tensors.

As well as various geometric quantities and generations, as well as random variables, fuzzy variables and changing (probability) spaces-categories and random processes with accidental values. The concept of describing the dependency between variables has developed from function to functional, transformation and even functor.

Higher mathematics, like elementary mathematics, also studies spatial forms, but it is more abstract and embodies the characteristics of change, or it is studied in change. For example, the concepts of curves and surfaces have developed into general manifolds.

According to Herun's program, geometry is a theory about the invariance of graphics under certain transformation groups, that is, geometry is studied by putting various spatial forms under transformation.

Infinitely entering mathematics is another feature of higher mathematics. Everything in the real world appears in a limited form, and infinity is a summary of their identity. Therefore, infinite entry into mathematics is the embodiment of high theorization and abstraction of mathematics. Infinity in mathematics appears in two forms: latent infinity and real infinity.

In the limit process, the change of variables is infinite and belongs to the form of infinite potential. The existence of limit value reflects the real infinite process. The most basic limit process is the limit of sequence and function. Based on it, mathematical analysis establishes various concepts and related theories to describe the local and global characteristics of functions, and initially successfully describes the uneven changes and movements in the real world.

Other more abstract limit processes also play a fundamental role in other mathematical disciplines. There are many disciplines that study infinite individuals, that is, infinite * * *, such as groups, rings, domains and various abstract spaces. This is the real infinity in mathematics. Being able to deal with this infinity is a manifestation of the improvement of mathematics level and ability.

In order to deal with this infinity, various structures such as algebraic structure, ordered structure and topological structure are introduced into mathematics. There are metric structures such as norm, distance and measure in abstract space, which make the relationship between individuals quantitative and digital and become a bridge between mathematical qualitative description and quantitative calculation. The above structure makes these infinite * * * have rich connotations and can be distinguished from each other, thus forming many mathematical disciplines.

The computational aspect of mathematics. Even occupies a dominant position in elementary mathematics. Its position in higher mathematics is also obvious. In addition to many theoretical subjects, advanced mathematics also has a large number of computational subjects, such as differential equations, computational mathematics, statistics and so on. With highly abstract theoretical equipment, it is possible for these disciplines to deal with complex computing problems in modern science and technology.