Differential calculation force

1. The history of computer

the birth and development of modern computer before the advent of modern computer, the development of computer has gone through three stages: mechanical computer, electromechanical computer and electronic computer

as early as the 17th century, a group of European mathematicians began to design and manufacture digital computers that perform basic operations in digital form. In 1642, Pascal, a French mathematician, made the earliest decimal adder by using a gear transmission similar to clocks and watches. In 1678, Leibniz, a German mathematician, developed a computer to further solve the multiplication and division of decimal numbers

British mathematician Babbage put forward an idea when he made the model of difference machine in 1822. One arithmetic operation at a time will develop into a certain complete operation process automatically. In 1884, Babbage designed a program-controlled universal analyzer. Although this analyzer has described the rudiment of the program control computer, it can not be realized e to the technical conditions at that time< During the more than 100 years since Babbage's idea was put forward, great progress has been made in electromagnetics, electrotechnics and electronics, and vacuum diodes and vacuum triodes have been successively invented in components and devices; In terms of system technology, wireless telegraph, television and radar were invented one after another. All these achievements have prepared technical and material conditions for the development of modern computer< At the same time, mathematics and physics are developing rapidly. In the 1930s, all fields of physics experienced the stage of quantification. The mathematical equations describing various physical processes, some of which were difficult to solve by classical analysis methods. As a result, numerical analysis has been paid attention to, and various numerical integration, numerical differentiation, and numerical solutions of differential equations have been developed. The calculation process has been reced to a huge amount of basic operations, thus laying the foundation of modern computer numerical algorithm

the urgent need for advanced computing tools in society is the fundamental driving force for the birth of modern computers. Since the 20th century, there have been a lot of computational difficulties in various fields of science and technology, which has hindered the further development of the discipline. Especially before and after the outbreak of the Second World War, the need for high-speed computing tools in military science and technology is particularly urgent. During this period, Germany, the United States and the United Kingdom started the research of electromechanical computer and electronic computer almost at the same time<

Giuseppe in Germany was the first to use electrical components to make computers. The fully automatic relay computer Z-3, which he made in 1941, has the characteristics of modern computer, such as floating-point counting, binary operation, instruction form of digital storage address and so on. In the United States, the relay computers mark-1, mark-2, model-1, model-5 and so on were made successively from 1940 to 1947. However, the switching speed of the relay is about one hundredth of a second, which greatly limits the computing speed of the computer

the development process of electronic computer has experienced the evolution from making components to whole machine, from special machine to general machine, from "external program" to "stored program". In 1938, the Bulgarian American scholar atanasov first made the computing unit of the electronic computer. In 1943, the communications office of the British Foreign Office made the "giant" computer. This is a special cryptanalysis machine, which was used in the Second World War< In February 1946, ENIAC, a large-scale electronic digital integrator computer, was developed by Moore College of the University of Pennsylvania in the United States. At first, ENIAC was also specially used for artillery trajectory calculation. Later, it was improved many times and became a general-purpose computer capable of various scientific calculations. This computer, which uses electronic circuit to perform arithmetic operation, logic operation and information storage, is 1000 times faster than relay computer. This is the first electronic computer in the world. However, the program of this kind of computer is still external, the storage capacity is too small, and it has not fully possessed the main characteristics of modern computer

the new breakthrough was completed by a design team led by mathematician von Neumann. In March 1945, they published a new general electronic computer scheme of stored program - electronic discrete variable automatic computer (EDVAC). Then in June 1946, von Neumann and others put forward a more perfect design report "preliminary study on the logical structure of electronic computer devices". From July to August of the same year, they taught a special course "theory and technology of electronic computer design" for experts from more than 20 institutions in the United States and Britain at Moore college, which promoted the design and manufacture of stored program computers< In 1949, the Mathematics Laboratory of Cambridge University in England took the lead in making EDSAC; The United States made the eastern standard automatic computer (SFAC) in 1950. At this point, the embryonic period of the development of electronic computer came to an end, and the development period of modern computer began

at the same time of creating digital computer, we also developed another kind of important computing tool analog computer. When physicists summarize the laws of nature, they often use mathematical equations to describe a process; On the contrary, the process of solving mathematical equations may also adopt the physical process simulation method. After the invention of logarithm, the slide rule made in 1620 has changed multiplication and division into addition and subtraction for calculation. Maxwell skillfully transformed the calculation of integral (area) into the measurement of length, and made the integrator in 1855< Fourier analysis, another great achievement of mathematical physics in the 19th century, played a direct role in promoting the development of simulators. In the late 19th century and the early 20th century, a variety of analytical machines for calculating Fourier coefficients and differential equations were developed. However, when trying to popularize the differential analysis machine to solve partial differential equations and use the simulator to solve general scientific calculation problems, people graally realize the limitations of the simulator in the aspects of universality and accuracy, and turn their main energy to the digital computer

after the advent of electronic digital computer, analog computer still continues to develop, and hybrid computer is proced by combining with digital computer. Simulators and mixers have become special varieties of modern computers, that is, efficient information processing tools or simulation tools used in specific fields
since the middle of the 20th century, the computer has been in a period of high-speed development. The computer has developed from a hardware only system to a computer system which includes three subsystems: hardware, software and firmware. The performance price ratio of computer system is increased by two orders of magnitude every 10 years. The types of computers have been divided into microcomputers, minicomputers, general-purpose computers (including giant, large and medium-sized computers), and various special computers (such as various control computers and analog-to-digital hybrid computers)
computer devices, from electron tubes to transistors, from discrete components to integrated circuits to microprocessors, have made three leaps in the development of computers< In the period of electron tube computer (1946-1959), computers were mainly used for scientific calculation. Main memory is the main factor that determines the appearance of computer technology. At that time, the main memory included mercury delay line memory, cathode ray oscilloscope electrostatic memory, magnetic drum and magnetic core memory, which were usually used to classify computers.

3. Operation ability is one of the basic components of mathematical ability, which refers to the ability to use the knowledge about operation to carry out operation and reasoning to obtain the result of operation
operation is actually a dective reasoning process, and operation is reasoning. In the primary mathematics stage, mathematical operation mainly includes four operations: integral, rational, radical, exponential, logarithmic and trigonometric functions. In the higher mathematics stage, there are limit operation, differential and integral operation, vector and matrix operation, data, information processing and probability operation, set, matrix operation, and so on Mathematical operation ability includes all these aspects of operation ability. To cultivate the above operation ability, students should first master the relevant knowledge of all kinds of operation (such as the concept and nature of the operation object, the definition and rules of operation, etc.) With the popularity of calculators, it is no longer necessary to learn four digit mathematical tables and do more training in multi digit numerical calculation in middle schools. However, it is necessary to develop estimation ability and the ability to process a large number of data. Furthermore, e to the use of program calculators and computers, more complex operations can be carried out by programs, At this time, students need to learn programming or algorithmic language to get the ability to use them.

4. In my opinion, the resistance to accelerating air is a variable force, which is why we consider it from the angle of energy rather than from the angle of resistance integral. According to your method, we know that P = FV is actually the instantaneous power when the force pulls a certain mass of object, but we should know that in different times, with the acceleration of speed, more air is compressed to do work per unit time, so this formula is not applicable. From the problem solution, in DT time, through the consideration of energy conversion, it is really accurate, which is the real embodiment of instantaneous power
I am about to graate from University, so I use differential to understand the infinitesimal method

5. The training ways of operation ability are as follows:

1. Accurately understand and firmly grasp the concepts, properties, formulas, rules and some commonly used data required by various operations; The deep understanding of concepts, properties, formulas and rules directly affects the choice of methods and the speed of operation. The concept is fuzzy, the formula and the rule are ambiguous, which will certainly affect the accuracy of the operation. In order to improve the speed of operation, it is necessary to memorize some commonly used data. For example, the square of natural number within 20, simple Pythagorean number, special trigonometric function value,,,,, LG2, Lg3, e accurate to approximate value of 0.001, etc< Second, master the general methods and rules of operation, and flexibly use concepts, properties, formulas and rules for operation. Teachers can combine the content of teaching materials, compile and collect some flexible exercises, cultivate the flexibility of students' calculation, and guide students to collect, summarize and accumulate experience to form skilled skills, so as to improve the simplicity and rapidity of calculation< Third, pay attention to the typical demonstration of teachers and examples, and make clear the goal of solving problems, the steps of calculation and the basis. Through the typical demonstration, the transition from understanding knowledge to application knowledge is relatively smooth, thus forming the computing ability< The essence of mathematical operation is to dece the result from the known data and formula according to the definition and properties of operation, which is also a process of reasoning. Whether the operation is correct or not depends on whether the reasoning is correct. If the reasoning is not correct, the operation will be wrong. We should pay special attention to the equivalent transformation in operational reasoning< 5. Pay attention to the training of the ability of identity transformation of number and form

1. Sign transformation, for example, sign transformation when removing brackets and adding brackets

2. Reciprocal transformation, such as addition and subtraction, multiplication and division, power and root, differential and integral, etc

3. For example, A2 + B2 = (a + b) 2-2ab, etc< For example, given X-Y = 3, Y-Z = 5, finding x-z can decompose x-z = (X-Y) + (Y-Z)< For example, introcing auxiliary elements, constructing auxiliary functions, adding auxiliary lines, adding parameters and so on< Sixth, strengthen the operation practice. Any ability is formed and developed in certain practical activities. In order to effectively improve the students' operation ability, we must strengthen the practice. The practice should be purposeful, systematic and typical. Through changing, changing, solving and using one question, we can cultivate the proficiency, accuracy, flexibility and organization of calculation. In the form of question group training

7. Too many ellipsis! You should state the problem in detail. If I understand correctly, are you considering the friction of a board rotating on a plane around an axis
you can divide the board into many small parts, calculate the moment of each part, and then integrate, which is OK in principle. But it seems that you can't understand the meaning of M = RXF
suppose that we set up a plane rectangular coordinate system on a plane. Suppose that a small part of the coordinates on the block are (x (T), y (T)), then their velocity is (X & # 39; X, y (T)) t),y'( t))，' It's derivative. So the vector r = (x (T), y (T)), the direction of friction is opposite to that of velocity, so the vector f = mg λ/ v*(-x'( t),-y'( t) Where V is the fulcrum velocity sqrt (X & # 39 t)*x'( t)=y'( t)*y'( t))
RXF is the outer proct of R and F, and the numerical value is mg λ/ v*(x'( t)y(t)-x(t)y'( t) (x, y, Z are right-handed systems). Then you need to calculate DM * (X & # 39 t)y(t)-x(t)y'( t) ) / V integral (represents the torque applied to a unit of mass differential), and then multiplied by G λ

the problem with your calculation method is that DMF = XDF = XK λ DXG, the infinitesimal of the moment is not equal to XDF! One of the errors is that x is not a constant and cannot be extracted directly; The second mistake is that the moment is a vector, and its size must not be the size of R multiplied by the size of F, but also multiplied by the sine value of the angle between R and F!

8. At the beginning of my study, I didn't feel that I needed much computing power. However, in the process of learning will graally have higher requirements for mathematical ability, this is a graal process. If you are weak in mathematics, you don't need to worry too much. Just keep learning and making progress. As for what level of mathematical ability, at least have to learn advanced mathematics, calculus to learn.