Calculus digital currency mining
Publish: 2021-04-29 20:11:06
1. Calculus is a branch of higher mathematics which studies the differentiation, integration, related concepts and applications of functions. It is a basic subject of mathematics. The content mainly includes limit, differential calculus, integral calculus and its application. Differential calculus, including the operation of derivative, is a set of theory about the rate of change. It makes function, velocity, acceleration and slope of curve can be discussed with a set of general symbols. Integral calculus, including the calculation of integral, provides a set of general methods for defining and calculating area and volume. With the development of science and technology, some calculators can also solve calculus (differential and definite integral). The following are function calculators that can solve calculus (the following models are for reference only): casioms series: fx-100ms fx-115ms fx-570ms fx-991ms es series (natural writing display): FX-115ES fx-570es fx-991es plus series (natural writing display): FX-115ES plus fx-570es plus fx-991es plus fx-991es plus C Programming Series: fx-3650p fx-3950p fx-4800P fx-5800p fx-7400g Fx-9750g fx-9860g and its upgrades
2. Calculus is a branch of higher mathematics which studies the differential and integral of functions and related concepts and applications. It is a basic subject of mathematics. The content mainly includes: limit, differential calculus, integral calculus and its application. Differential calculus, including the operation of derivative, is a set of theory about the rate of change. It makes function, velocity, acceleration and slope of curve can be discussed with a set of general symbols. Integral calculus, including the calculation of integral, provides a set of general methods for defining and calculating area and volume
what is the meaning of learning calculus from modern point of view< br />
1. Continuity and limit
by introcing topological space and metric space, the concepts of continuity, completeness, compactness, connectivity and limit are understood
2. Differentiation
do differentiation on Banach spcace, understand linear mapping and multilinear mapping, introce differential form, introce manifold (such as R ^ n's embedded submanifold), understand the relationship between analysis and geometry, and understand de Rham cohomology
3. Integral
measure and Lebesgue integral are introced
is it necessary? In addition to Lebesgue integral, they are all formalized things. When we understand these things, we still need to have excellent techniques to complete the series of function terms and integral with parameter variables. We don't understand these formalized things all at once. Modern people always hope to learn everything at a young age. It's too urgent. After the maturity of mathematical thinking, these so-called modern words are actually very natural. It's good to mention it appropriately, but there's no need to pursue it too much
first of all, we need to expand the domain of integral. In fact, measure theory has extended the domain of integral to any set, but what we need to do is to extend it to any n-dimensional Euclidean space. I believe the problem owner has learned this in multiple integrals of higher numbers. Then, we should reexamine the elements of integral. In ∫ YDX, DX can be called & quot; Dumb label;, It doesn't work. It doesn't matter if you change it into any letters. What really works is the integral range and the mapping In principle, we have to agree on an integration rule, but the method of multiple integration is sufficient to deal with physical problems). Therefore, we don't really need an "independent variable". The meaning of X only means that it is an element within the scope of integration. As mentioned earlier, the integral range can be Euclidean space, so the element is the point in Euclidean space. If you know this, you will know why DV can be used to replace dxdydz in triple integral. The last thing is very easy, is to promote "mapping". Whether it is a vector, a matrix or a tensor, the addition operation can be decomposed into the addition of real number components, and the integral is the same. Now let's sum up: the so-called integral is that given an n-dimensional Euclidean space (or its subset) as the integral range, there is a mapping that maps every element in the space to a real number or vector or matrix or tensor. With these conditions, an integral value can be uniquely determined. The principle of integral value is given and the method of multiple integral is followed. Let's look at the main problem: every point of a charged object is described by three coordinates, so the whole charged range is a subset of three-dimensional Euclidean space. Mapping is the electric field intensity caused by the amount of charge (charged volume density) mapped to each point. DQ is a "mmy mark", which only indicates a point on the charged body (which has been weighted by the charge density) and has no practical significance.
what is the meaning of learning calculus from modern point of view< br />
1. Continuity and limit
by introcing topological space and metric space, the concepts of continuity, completeness, compactness, connectivity and limit are understood
2. Differentiation
do differentiation on Banach spcace, understand linear mapping and multilinear mapping, introce differential form, introce manifold (such as R ^ n's embedded submanifold), understand the relationship between analysis and geometry, and understand de Rham cohomology
3. Integral
measure and Lebesgue integral are introced
is it necessary? In addition to Lebesgue integral, they are all formalized things. When we understand these things, we still need to have excellent techniques to complete the series of function terms and integral with parameter variables. We don't understand these formalized things all at once. Modern people always hope to learn everything at a young age. It's too urgent. After the maturity of mathematical thinking, these so-called modern words are actually very natural. It's good to mention it appropriately, but there's no need to pursue it too much
first of all, we need to expand the domain of integral. In fact, measure theory has extended the domain of integral to any set, but what we need to do is to extend it to any n-dimensional Euclidean space. I believe the problem owner has learned this in multiple integrals of higher numbers. Then, we should reexamine the elements of integral. In ∫ YDX, DX can be called & quot; Dumb label;, It doesn't work. It doesn't matter if you change it into any letters. What really works is the integral range and the mapping In principle, we have to agree on an integration rule, but the method of multiple integration is sufficient to deal with physical problems). Therefore, we don't really need an "independent variable". The meaning of X only means that it is an element within the scope of integration. As mentioned earlier, the integral range can be Euclidean space, so the element is the point in Euclidean space. If you know this, you will know why DV can be used to replace dxdydz in triple integral. The last thing is very easy, is to promote "mapping". Whether it is a vector, a matrix or a tensor, the addition operation can be decomposed into the addition of real number components, and the integral is the same. Now let's sum up: the so-called integral is that given an n-dimensional Euclidean space (or its subset) as the integral range, there is a mapping that maps every element in the space to a real number or vector or matrix or tensor. With these conditions, an integral value can be uniquely determined. The principle of integral value is given and the method of multiple integral is followed. Let's look at the main problem: every point of a charged object is described by three coordinates, so the whole charged range is a subset of three-dimensional Euclidean space. Mapping is the electric field intensity caused by the amount of charge (charged volume density) mapped to each point. DQ is a "mmy mark", which only indicates a point on the charged body (which has been weighted by the charge density) and has no practical significance.
3.
Calculus is a mathematical concept. It is a branch of higher mathematics that studies the differentiation and integration of functions as well as related concepts and applications. It is a basic subject of mathematics, including limit, differential, integral and its application
integral is the inverse operation of differential, that is to know the derivative function of function and reverse the original function. In application, definite integral is more than that. It is widely used in summation. Generally speaking, it is used to find the area of triangle with curved sides. This ingenious solution is determined by the special property of integral
extended data:
since the establishment of the calculus system for hundreds of years, great achievements have been made in the application of methods. However, there are many imperfections and inaccuracies in the current principles of calculus. This is not only because:
1
There are many subtle problems There are many logical errors in this calculus principle. What's more, the principle of calculus hardly works as a principle. Therefore, it is an important step in the development of mathematics to correct the mistakes of the current calculus principle, establish a new number shape model, and reconstruct the new calculus principle that meets the requirements of the development of mathematics Engels pointed out: "in all theoretical progress, compared with the invention of calculus in the second half of the 17th century, nothing else may be regarded as such a noble victory of the human spirit." Von Neumann also pointed out: "calculus is the highest achievement of modern mathematics, and its importance can not be overestimated."source of reference: network calculus (mathematical concepts)
4.
Calculus is a branch of higher mathematics which studies the differentiation, integration, related concepts and applications of functions. It is a basic subject of mathematics. The content mainly includes limit, differential calculus, integral calculus and its application. Differential calculus, including the operation of derivative, is a set of theory about the rate of change. It makes function, velocity, acceleration and slope of curve can be discussed with a set of general symbols. Integral calculus, including the calculation of integral, provides a set of general methods for defining and calculating area and volume< please accept if you are satisfied. If you have any questions, please ask. Thank you
5. Hello
calculus is a branch of higher mathematics which studies the differentiation, integration and related concepts and applications of functions. It is a basic subject of mathematics. The content mainly includes limit, differential calculus, integral calculus and its application. Differential calculus, including the operation of derivative, is a set of theory about the rate of change. It makes function, velocity, acceleration and slope of curve can be discussed with a set of general symbols. Integral calculus, including the calculation of integral, provides a set of general methods for defining and calculating area and volume.
calculus is a branch of higher mathematics which studies the differentiation, integration and related concepts and applications of functions. It is a basic subject of mathematics. The content mainly includes limit, differential calculus, integral calculus and its application. Differential calculus, including the operation of derivative, is a set of theory about the rate of change. It makes function, velocity, acceleration and slope of curve can be discussed with a set of general symbols. Integral calculus, including the calculation of integral, provides a set of general methods for defining and calculating area and volume.
6. Calculus is a branch of mathematics that studies the differential and integral of functions, as well as related concepts and applications. Calculus is based on real numbers, functions and limits. The most important idea of calculus is to use & quot; Microelement & quot; And & quot; Infinite approximation;, It seems that a thing is changing all the time. It's hard for you to study it, but if you divide it into small pieces by infinitesimal, it can be regarded as constant processing, and finally add it up
calculus is the general term of differential calculus and integral calculus. It is a kind of mathematical thought, "infinite subdivision" is differential, and "infinite summation" is integral. Infinity is the limit. The idea of limit is the basis of calculus. It is to treat problems with the idea of a kind of movement. For example, the instantaneous velocity of the bullet flying out of the gun is the concept of differential, and the sum of the distances traveled by the bullet at each instant is the concept of integral. If the whole mathematics is compared to a big tree, then elementary mathematics is the root of the tree, many branches of mathematics are branches, and the main part of the trunk is calculus. Calculus is one of the greatest achievements of human intelligence
the concepts of limit and calculus can be traced back to ancient times. In the second half of the 17th century, Newton and Leibniz completed the preparatory work of many mathematicians, and established calculus independently. Their starting point of calculus is intuitive infinitesimal, and their theoretical basis is not solid. It was not until the 19th century that Cauchy and wilstras established the limit theory and Cantor established the strict real number theory that the discipline became more rigorous
calculus is a branch of mathematics that studies the differential and integral of functions, as well as related concepts and applications. Calculus is based on real numbers, functions and limits. The most important idea of calculus is to use & quot; Microelement & quot; And & quot; Infinite approximation;, It seems that a thing is changing all the time. It's hard for you to study it, but if you divide it into small pieces by infinitesimal, it can be regarded as constant processing, and finally add it up
calculus is the general term of differential calculus and integral calculus. It is a kind of mathematical thought, "infinite subdivision" is differential, and "infinite summation" is integral. Infinity is the limit. The idea of limit is the basis of calculus. It is to treat problems with the idea of a kind of movement. For example, the instantaneous velocity of the bullet flying out of the gun is the concept of differential, and the sum of the distances traveled by the bullet at each instant is the concept of integral. If the whole mathematics is compared to a big tree, then elementary mathematics is the root of the tree, many branches of mathematics are branches, and the main part of the trunk is calculus. Calculus is one of the greatest achievements of human intelligence
the concepts of limit and calculus can be traced back to ancient times. In the second half of the 17th century, Newton and Leibniz completed the preparatory work of many mathematicians, and established calculus independently. Their starting point of calculus is intuitive infinitesimal, and their theoretical basis is not solid. It was not until the 19th century that Cauchy and wilstras established the limit theory and Cantor established the strict real number theory that the discipline became more rigorous
calculus has been developed in connection with practical application. It has been more and more widely used in natural sciences, social sciences and applied sciences such as astronomy, mechanics, chemistry, biology, engineering and economics. In particular, the invention of the computer is more concive to the continuous development of these applications
everything in the objective world, from particles to the universe, is always moving and changing. Therefore, after introcing the concept of variable in mathematics, it is possible to describe the phenomenon of motion with mathematics
e to the deepening of the concept of function and the need of the development of science and technology, a new branch of mathematics came into being after analytic geometry, which is calculus. Calculus plays a very important role in the development of mathematics. It can be said that it is the biggest creation in all mathematics after Euclidean geometry
the basic method of calculus is to study the function and the movement of things from the aspect of quantity. This method is called mathematical analysis< In a broad sense, mathematical analysis includes calculus, function theory and many other branches, but now it is used to equate mathematical analysis with calculus. Mathematical analysis has become a synonym of calculus. When we talk about mathematical analysis, we know that it means calculus. The basic concepts and contents of calculus include differential calculus and integral calculus< The main contents of differential calculus include: limit theory, derivative, differential, etc
the main contents of integral science include definite integral, indefinite integral and so on< In order to derive three laws of Kepler's planetary motion from the law of universal gravitation, Newton applied calculus and differential equations. Since then, calculus has greatly promoted the development of mathematics, but also greatly promoted the development of astronomy, mechanics, physics, chemistry, biology, engineering, economics and other natural sciences, social sciences and applied sciences. And there are more and more extensive applications in these disciplines, especially the emergence of computer is more concive to the continuous development of these applications.
calculus is the general term of differential calculus and integral calculus. It is a kind of mathematical thought, "infinite subdivision" is differential, and "infinite summation" is integral. Infinity is the limit. The idea of limit is the basis of calculus. It is to treat problems with the idea of a kind of movement. For example, the instantaneous velocity of the bullet flying out of the gun is the concept of differential, and the sum of the distances traveled by the bullet at each instant is the concept of integral. If the whole mathematics is compared to a big tree, then elementary mathematics is the root of the tree, many branches of mathematics are branches, and the main part of the trunk is calculus. Calculus is one of the greatest achievements of human intelligence
the concepts of limit and calculus can be traced back to ancient times. In the second half of the 17th century, Newton and Leibniz completed the preparatory work of many mathematicians, and established calculus independently. Their starting point of calculus is intuitive infinitesimal, and their theoretical basis is not solid. It was not until the 19th century that Cauchy and wilstras established the limit theory and Cantor established the strict real number theory that the discipline became more rigorous
calculus is a branch of mathematics that studies the differential and integral of functions, as well as related concepts and applications. Calculus is based on real numbers, functions and limits. The most important idea of calculus is to use & quot; Microelement & quot; And & quot; Infinite approximation;, It seems that a thing is changing all the time. It's hard for you to study it, but if you divide it into small pieces by infinitesimal, it can be regarded as constant processing, and finally add it up
calculus is the general term of differential calculus and integral calculus. It is a kind of mathematical thought, "infinite subdivision" is differential, and "infinite summation" is integral. Infinity is the limit. The idea of limit is the basis of calculus. It is to treat problems with the idea of a kind of movement. For example, the instantaneous velocity of the bullet flying out of the gun is the concept of differential, and the sum of the distances traveled by the bullet at each instant is the concept of integral. If the whole mathematics is compared to a big tree, then elementary mathematics is the root of the tree, many branches of mathematics are branches, and the main part of the trunk is calculus. Calculus is one of the greatest achievements of human intelligence
the concepts of limit and calculus can be traced back to ancient times. In the second half of the 17th century, Newton and Leibniz completed the preparatory work of many mathematicians, and established calculus independently. Their starting point of calculus is intuitive infinitesimal, and their theoretical basis is not solid. It was not until the 19th century that Cauchy and wilstras established the limit theory and Cantor established the strict real number theory that the discipline became more rigorous
calculus has been developed in connection with practical application. It has been more and more widely used in natural sciences, social sciences and applied sciences such as astronomy, mechanics, chemistry, biology, engineering and economics. In particular, the invention of the computer is more concive to the continuous development of these applications
everything in the objective world, from particles to the universe, is always moving and changing. Therefore, after introcing the concept of variable in mathematics, it is possible to describe the phenomenon of motion with mathematics
e to the deepening of the concept of function and the need of the development of science and technology, a new branch of mathematics came into being after analytic geometry, which is calculus. Calculus plays a very important role in the development of mathematics. It can be said that it is the biggest creation in all mathematics after Euclidean geometry
the basic method of calculus is to study the function and the movement of things from the aspect of quantity. This method is called mathematical analysis< In a broad sense, mathematical analysis includes calculus, function theory and many other branches, but now it is used to equate mathematical analysis with calculus. Mathematical analysis has become a synonym of calculus. When we talk about mathematical analysis, we know that it means calculus. The basic concepts and contents of calculus include differential calculus and integral calculus< The main contents of differential calculus include: limit theory, derivative, differential, etc
the main contents of integral science include definite integral, indefinite integral and so on< In order to derive three laws of Kepler's planetary motion from the law of universal gravitation, Newton applied calculus and differential equations. Since then, calculus has greatly promoted the development of mathematics, but also greatly promoted the development of astronomy, mechanics, physics, chemistry, biology, engineering, economics and other natural sciences, social sciences and applied sciences. And there are more and more extensive applications in these disciplines, especially the emergence of computer is more concive to the continuous development of these applications.
7. 1、 Differentiation concept
1. Calculus is a branch of higher mathematics that studies the differentiation, integration, related concepts and applications of functions. It is a basic subject of mathematics. The content mainly includes limit, differential calculus, integral calculus and its application. Differential calculus, including the operation of derivative, is a set of theory about the rate of change. It makes function, velocity, acceleration and slope of curve can be discussed with a set of general symbols. Integral calculus, including the calculation of integral, provides a set of general methods for defining and calculating area and volume
2. Mathematical analysis, also known as advanced calculus, is the oldest and most basic branch of analysis. Generally speaking, it refers to a relatively complete mathematical discipline with calculus and the general theory of infinite series as the main contents, and including their theoretical basis (the basic theory of real number, function and limit). It is also a basic course of university mathematics major. The branch of analysis in mathematics is a branch of mathematics specialized in the study of real numbers, complex numbers and their functions. Its development starts from calculus and extends to the continuity, differentiability and integrability of functions. These characteristics help us to apply in the study of the physical world, study and discover the laws of nature< Second, calculus:
(1) mutual solution of velocity and distance in motion
given that the distance table of an object moving is a function of time, the velocity and acceleration of an object at any time can be obtained; On the other hand, the acceleration meter of a known object is a function formula with time as a variable to calculate the velocity and distance. The difficulty is that the velocity and acceleration are changing all the time. For example, to calculate the instantaneous velocity of an object at a certain moment, we can't use the moving distance to remove the moving time just like to calculate the average velocity, because at a given moment, the moving distance and time of the object are meaningless. However, according to physics, every moving object must have a speed at every moment of its movement, which is no doubt. The same difficulty is encountered when the velocity formula is known. Because the speed is changing all the time, we can't multiply the time of motion by the speed at any time to get the distance of the object
(2) the problem of finding the tangent of a curve
the problem itself is pure geometry, and it is of great importance for scientific application. Due to the need of studying astronomy, optics is an important scientific research in the 17th century. In order to study the passage of light through the lens, the designer of the lens must know the angle of the light incident on the lens in the cyclotomy method of calculus, so as to apply the law of reflection. The important thing here is the angle between the light and the normal line of the curve, and the normal line is perpendicular to the tangent line, So it's always about finding normals or tangents; Another scientific problem related to the tangent of a curve appears in the study of motion, which is to find the direction of motion of a moving object at any point of its trajectory, that is, the tangent direction of the trajectory< These problems include the length of the curve (such as the distance of the planet moving in a known period), the area enclosed by the curve, the volume enclosed by the surface, the center of gravity of an object, and the gravity of a fairly large object (such as a planet) acting on another object. In fact, the problem of calculating the length of an ellipse baffled mathematicians, so that for a period of time, their further work on this problem failed, and new results were not obtained until the next century. Another example is the problem of area. In early ancient Greece, people used the method of exhaustion to calculate some areas and volumes. For example, they used the method of exhaustion to calculate the area bounded by the parabola, the axis and the straight line on the interval. When the number of parts is more and more, the result is closer to the exact value of the area. However, in the application of exhaustion method, many skills must be added, and it is lack of generality, so it is often impossible to get a numerical solution. When Archimedes' work became famous in Europe, his interest in length, area, volume and center of gravity revived. The method of exhaustion was first graally modified, and then it was fundamentally modified e to the establishment of calculus
(4) the problem of finding the maximum and minimum value (quadratic function, belonging to the category of calculus)
for example, when a projectile is fired in the barrel, its horizontal distance, that is, the range, depends on the inclination angle of the barrel to the ground, that is, the firing angle. A "practical" problem is to find the angle at which the maximum range can be launched. At the beginning of the 17th century, Galileo concluded that the launch angle reached the maximum range (in vacuum); He also got the different maximum heights of the shells after they were fired from different angles. The study of planetary motion also involves the problem of maximum and minimum
2. Mathematical analysis
the main content of mathematical analysis is calculus, the theoretical basis of calculus is limit theory, and the theoretical basis of limit theory is real number theory. The most important characteristic of real number system is continuity. With the continuity of real number, we can discuss limit, continuity, differential and integral. It is in the process of discussing the legitimacy of various limit operations of functions that people graally establish a rigorous theoretical system of mathematical analysis.
1. Calculus is a branch of higher mathematics that studies the differentiation, integration, related concepts and applications of functions. It is a basic subject of mathematics. The content mainly includes limit, differential calculus, integral calculus and its application. Differential calculus, including the operation of derivative, is a set of theory about the rate of change. It makes function, velocity, acceleration and slope of curve can be discussed with a set of general symbols. Integral calculus, including the calculation of integral, provides a set of general methods for defining and calculating area and volume
2. Mathematical analysis, also known as advanced calculus, is the oldest and most basic branch of analysis. Generally speaking, it refers to a relatively complete mathematical discipline with calculus and the general theory of infinite series as the main contents, and including their theoretical basis (the basic theory of real number, function and limit). It is also a basic course of university mathematics major. The branch of analysis in mathematics is a branch of mathematics specialized in the study of real numbers, complex numbers and their functions. Its development starts from calculus and extends to the continuity, differentiability and integrability of functions. These characteristics help us to apply in the study of the physical world, study and discover the laws of nature< Second, calculus:
(1) mutual solution of velocity and distance in motion
given that the distance table of an object moving is a function of time, the velocity and acceleration of an object at any time can be obtained; On the other hand, the acceleration meter of a known object is a function formula with time as a variable to calculate the velocity and distance. The difficulty is that the velocity and acceleration are changing all the time. For example, to calculate the instantaneous velocity of an object at a certain moment, we can't use the moving distance to remove the moving time just like to calculate the average velocity, because at a given moment, the moving distance and time of the object are meaningless. However, according to physics, every moving object must have a speed at every moment of its movement, which is no doubt. The same difficulty is encountered when the velocity formula is known. Because the speed is changing all the time, we can't multiply the time of motion by the speed at any time to get the distance of the object
(2) the problem of finding the tangent of a curve
the problem itself is pure geometry, and it is of great importance for scientific application. Due to the need of studying astronomy, optics is an important scientific research in the 17th century. In order to study the passage of light through the lens, the designer of the lens must know the angle of the light incident on the lens in the cyclotomy method of calculus, so as to apply the law of reflection. The important thing here is the angle between the light and the normal line of the curve, and the normal line is perpendicular to the tangent line, So it's always about finding normals or tangents; Another scientific problem related to the tangent of a curve appears in the study of motion, which is to find the direction of motion of a moving object at any point of its trajectory, that is, the tangent direction of the trajectory< These problems include the length of the curve (such as the distance of the planet moving in a known period), the area enclosed by the curve, the volume enclosed by the surface, the center of gravity of an object, and the gravity of a fairly large object (such as a planet) acting on another object. In fact, the problem of calculating the length of an ellipse baffled mathematicians, so that for a period of time, their further work on this problem failed, and new results were not obtained until the next century. Another example is the problem of area. In early ancient Greece, people used the method of exhaustion to calculate some areas and volumes. For example, they used the method of exhaustion to calculate the area bounded by the parabola, the axis and the straight line on the interval. When the number of parts is more and more, the result is closer to the exact value of the area. However, in the application of exhaustion method, many skills must be added, and it is lack of generality, so it is often impossible to get a numerical solution. When Archimedes' work became famous in Europe, his interest in length, area, volume and center of gravity revived. The method of exhaustion was first graally modified, and then it was fundamentally modified e to the establishment of calculus
(4) the problem of finding the maximum and minimum value (quadratic function, belonging to the category of calculus)
for example, when a projectile is fired in the barrel, its horizontal distance, that is, the range, depends on the inclination angle of the barrel to the ground, that is, the firing angle. A "practical" problem is to find the angle at which the maximum range can be launched. At the beginning of the 17th century, Galileo concluded that the launch angle reached the maximum range (in vacuum); He also got the different maximum heights of the shells after they were fired from different angles. The study of planetary motion also involves the problem of maximum and minimum
2. Mathematical analysis
the main content of mathematical analysis is calculus, the theoretical basis of calculus is limit theory, and the theoretical basis of limit theory is real number theory. The most important characteristic of real number system is continuity. With the continuity of real number, we can discuss limit, continuity, differential and integral. It is in the process of discussing the legitimacy of various limit operations of functions that people graally establish a rigorous theoretical system of mathematical analysis.
8. Calculus is the study of calculus and calculus collectively, the English name is calculus, which means calculation. This is because the early calculus was mainly used in astronomy, mechanics and geometry. Later, calculus is also called analysis, or infinitesimal analysis, which refers to the knowledge of using infinitesimal or infinitesimal limit process analysis to deal with calculation problems. Limit is the foundation of calculus. Differential calculus includes the operation of derivation and differentiation, which is a set of theory about the rate of change. It makes function, velocity, acceleration and curve slope can be discussed with a set of general symbols. Integral science includes the concept and application of indefinite integral and definite integral. It provides a set of general methods for defining and calculating area and volume
since the 17th century, with the progress of society and the development of proctive forces, as well as many problems to be solved, such as navigation, astronomy, mine construction and so on, mathematics has also begun to study the changing quantity, and mathematics has entered the era of "variable mathematics". Throughout the 17th century, dozens of scientists did pioneering research for the establishment of calculus, but Newton and Leibniz made calculus an important branch of mathematics
(1) the mutual solution of velocity and distance in motion
given that the distance table of an object moving is a function of time, the velocity and acceleration of the object at any time can be obtained; On the other hand, the accelerometer of a known object is a function formula with time as a variable to calculate the velocity and distance. The difficulty is that the velocity and acceleration are changing all the time. For example, when calculating the instantaneous velocity of an object at a certain moment, we can't use the moving time to remove the moving distance just like calculating the average velocity, because at a given moment, the moving distance and time of the object are meaningless. However, according to physics, every moving object must have a speed at every moment of its movement, which is no doubt. The same difficulty is encountered when the velocity formula is known. Because the speed is changing all the time, we can't multiply the time of motion by the speed at any time to get the distance of the object
(2) the problem of finding the tangent of a curve
the problem itself is pure geometry, and it is of great importance for scientific application. Due to the need of studying astronomy, optics is an important scientific research in the 17th century. To study the passage of light through the lens, the designer of the lens must know the angle of light incident on the lens, so as to apply the law of reflection. The important thing here is the angle between the light and the normal of the curve, and the normal is perpendicular to the tangent, so it is always to find the normal or tangent; Another scientific problem related to the tangent of a curve appears in the study of motion, which is to find the direction of motion of a moving object at any point of its trajectory, that is, the tangent direction of the trajectory< These problems include the length of the curve (such as the distance of the planet moving in a known period), the area enclosed by the curve, the volume enclosed by the surface, the center of gravity of an object, and the gravity of a fairly large object (such as a planet) acting on another object. In fact, the problem of calculating the length of an ellipse baffled mathematicians, so that for a period of time, their further work on this problem failed, and new results were not obtained until the next century. Another example is the problem of area. In early ancient Greece, people used the method of exhaustion to calculate some areas and volumes. For example, they used the method of exhaustion to calculate the area bounded by the parabola, the axis and the straight line on the interval. When the number of parts is more and more, the result is closer to the exact value of the area. However, in the application of exhaustion method, many skills must be added, and it is lack of generality, so it is often impossible to get a numerical solution. When Archimedes' work became famous in Europe, his interest in length, area, volume and center of gravity revived. The method of exhaustion was first graally modified, and then it was fundamentally modified e to the establishment of calculus
(4) the problem of finding the maximum and minimum value (quadratic function, belonging to the category of calculus)
for example, when a projectile is fired in the barrel, its horizontal distance, that is, the range, depends on the inclination angle of the barrel to the ground, that is, the firing angle. A "practical" problem is to find the angle at which the maximum range can be launched. At the beginning of the 17th century, Galileo concluded that the launch angle reached the maximum range (in vacuum); He also got the different maximum heights of the shells after they were fired from different angles. The study of planetary motion also involves the problem of maximum and minimum< The generation of calculus can be divided into three stages: the concept of limit; Infinitesimal method of quadrature; The relationship between integral and differential
the thought of calculus has sprouted in ancient China for a long time. In the 7th century BC, there was infinite separability and limit in Laozi and Zhuangzi's philosophy
since the 17th century, with the progress of society and the development of proctive forces, as well as many problems to be solved, such as navigation, astronomy, mine construction and so on, mathematics has also begun to study the changing quantity, and mathematics has entered the era of "variable mathematics". Throughout the 17th century, dozens of scientists did pioneering research for the establishment of calculus, but Newton and Leibniz made calculus an important branch of mathematics
(1) the mutual solution of velocity and distance in motion
given that the distance table of an object moving is a function of time, the velocity and acceleration of the object at any time can be obtained; On the other hand, the accelerometer of a known object is a function formula with time as a variable to calculate the velocity and distance. The difficulty is that the velocity and acceleration are changing all the time. For example, when calculating the instantaneous velocity of an object at a certain moment, we can't use the moving time to remove the moving distance just like calculating the average velocity, because at a given moment, the moving distance and time of the object are meaningless. However, according to physics, every moving object must have a speed at every moment of its movement, which is no doubt. The same difficulty is encountered when the velocity formula is known. Because the speed is changing all the time, we can't multiply the time of motion by the speed at any time to get the distance of the object
(2) the problem of finding the tangent of a curve
the problem itself is pure geometry, and it is of great importance for scientific application. Due to the need of studying astronomy, optics is an important scientific research in the 17th century. To study the passage of light through the lens, the designer of the lens must know the angle of light incident on the lens, so as to apply the law of reflection. The important thing here is the angle between the light and the normal of the curve, and the normal is perpendicular to the tangent, so it is always to find the normal or tangent; Another scientific problem related to the tangent of a curve appears in the study of motion, which is to find the direction of motion of a moving object at any point of its trajectory, that is, the tangent direction of the trajectory< These problems include the length of the curve (such as the distance of the planet moving in a known period), the area enclosed by the curve, the volume enclosed by the surface, the center of gravity of an object, and the gravity of a fairly large object (such as a planet) acting on another object. In fact, the problem of calculating the length of an ellipse baffled mathematicians, so that for a period of time, their further work on this problem failed, and new results were not obtained until the next century. Another example is the problem of area. In early ancient Greece, people used the method of exhaustion to calculate some areas and volumes. For example, they used the method of exhaustion to calculate the area bounded by the parabola, the axis and the straight line on the interval. When the number of parts is more and more, the result is closer to the exact value of the area. However, in the application of exhaustion method, many skills must be added, and it is lack of generality, so it is often impossible to get a numerical solution. When Archimedes' work became famous in Europe, his interest in length, area, volume and center of gravity revived. The method of exhaustion was first graally modified, and then it was fundamentally modified e to the establishment of calculus
(4) the problem of finding the maximum and minimum value (quadratic function, belonging to the category of calculus)
for example, when a projectile is fired in the barrel, its horizontal distance, that is, the range, depends on the inclination angle of the barrel to the ground, that is, the firing angle. A "practical" problem is to find the angle at which the maximum range can be launched. At the beginning of the 17th century, Galileo concluded that the launch angle reached the maximum range (in vacuum); He also got the different maximum heights of the shells after they were fired from different angles. The study of planetary motion also involves the problem of maximum and minimum< The generation of calculus can be divided into three stages: the concept of limit; Infinitesimal method of quadrature; The relationship between integral and differential
the thought of calculus has sprouted in ancient China for a long time. In the 7th century BC, there was infinite separability and limit in Laozi and Zhuangzi's philosophy
9. Division of Engineering Science at University of Toronto (engsci) belongs to the Faculty of Applied Science and engineering. It is also the most challenging undergraate program in the University of Toronto
according to the 2009 U.S. News World University engineering discipline ranking, the Engineering Department of the University of Toronto ranks eighth in the world and first in Canada. Engineering science is one of the best undergraate programs offered by the school of engineering of the University of Toronto, which is defined as "academically elite" by the University of Toronto. Professor Michael P. Collins, a professor at the University of Toronto and a world-famous bridge and architecture expert, even described engsci as an "honorary engineering major". The famous freshman and sophomore year of engsci is known as "the most difficult curriculum in North America", which mainly focuses on theoretical mathematics and basic science. The average classroom teaching time is more than 40 hours per week. In the junior year, engsci students will be able to choose one of the eight major engineering branches from aerospace engineering, biomedical engineering, digital and computer engineering, energy system engineering, infrastructure engineering, mathematics and financial engineering, robotics engineering and engineering physics as their future major. From touch screen system development to artificial graft design to space system construction, engsci covers almost all the most advanced and cutting-edge subjects in today's engineering field
as one of the flagship majors of the University of Toronto, engsci has no independent faculty. Engsci's teaching tasks are all undertaken by outstanding professors from various departments of the University of Toronto. As a result, students of engsci can always get the first-hand information of scientific research achievements from physics, mathematics, life sciences and many other departments
like other majors in Engineering Department, engsci students can choose to join the "professional experience year" program of the University of Toronto in their junior or senior year to gain 12-16 months of pre graation work experience. Employers include IBM, Microsoft, rim, EA games, TD Bank of Canada and many other well-known enterprises. Between 2008 and 2010, 296 engsci students chose to participate in the "professional practice year", with an average annual salary of $45000 (about RMB 300000). The pey project has laid a good foundation for students to enter the future research or instrial fields
Engineering Science at the University of Toronto attracts top students from Canada and around the world. After four years of fierce competition, only about half of the students survive and graate successfully. 40% of the students will be eliminated in the first two years, and the whole engineering major of the University of Toronto can only be transferred from engsci, but can hardly be transferred from other majors to engsci (some students are transferred out because they feel unsuitable, not eliminated). After completing their undergraate studies, about two-thirds of engsci graates will be enrolled in world-class engineering graate schools, including MIT, Caltech, Stanford, Princeton, Harvard, Toronto Cambridge University (Cambridge) and other outstanding universities in the world graate school further qualification<
Engineering Science first semester:
Calculus I
structures & Materials
engsci praxis I
Engineering Mathematics and computation
classical mechanics
computer programming
second semester :
Calculus II
engsci Praxis II
computer programming
molecules & Materials
electric circuits
linear algebra
Engineering Science sophomore year courses include:
first semester:
calculus III
vector calculus Vector calculations
digital and computer systems
particles & waves
thermodynamics
Engineering, Second semester:
engineering design
probability and statistics
Modern Physics
electromagnetism
System Biology
according to the 2009 U.S. News World University engineering discipline ranking, the Engineering Department of the University of Toronto ranks eighth in the world and first in Canada. Engineering science is one of the best undergraate programs offered by the school of engineering of the University of Toronto, which is defined as "academically elite" by the University of Toronto. Professor Michael P. Collins, a professor at the University of Toronto and a world-famous bridge and architecture expert, even described engsci as an "honorary engineering major". The famous freshman and sophomore year of engsci is known as "the most difficult curriculum in North America", which mainly focuses on theoretical mathematics and basic science. The average classroom teaching time is more than 40 hours per week. In the junior year, engsci students will be able to choose one of the eight major engineering branches from aerospace engineering, biomedical engineering, digital and computer engineering, energy system engineering, infrastructure engineering, mathematics and financial engineering, robotics engineering and engineering physics as their future major. From touch screen system development to artificial graft design to space system construction, engsci covers almost all the most advanced and cutting-edge subjects in today's engineering field
as one of the flagship majors of the University of Toronto, engsci has no independent faculty. Engsci's teaching tasks are all undertaken by outstanding professors from various departments of the University of Toronto. As a result, students of engsci can always get the first-hand information of scientific research achievements from physics, mathematics, life sciences and many other departments
like other majors in Engineering Department, engsci students can choose to join the "professional experience year" program of the University of Toronto in their junior or senior year to gain 12-16 months of pre graation work experience. Employers include IBM, Microsoft, rim, EA games, TD Bank of Canada and many other well-known enterprises. Between 2008 and 2010, 296 engsci students chose to participate in the "professional practice year", with an average annual salary of $45000 (about RMB 300000). The pey project has laid a good foundation for students to enter the future research or instrial fields
Engineering Science at the University of Toronto attracts top students from Canada and around the world. After four years of fierce competition, only about half of the students survive and graate successfully. 40% of the students will be eliminated in the first two years, and the whole engineering major of the University of Toronto can only be transferred from engsci, but can hardly be transferred from other majors to engsci (some students are transferred out because they feel unsuitable, not eliminated). After completing their undergraate studies, about two-thirds of engsci graates will be enrolled in world-class engineering graate schools, including MIT, Caltech, Stanford, Princeton, Harvard, Toronto Cambridge University (Cambridge) and other outstanding universities in the world graate school further qualification<
Engineering Science first semester:
Calculus I
structures & Materials
engsci praxis I
Engineering Mathematics and computation
classical mechanics
computer programming
second semester :
Calculus II
engsci Praxis II
computer programming
molecules & Materials
electric circuits
linear algebra
Engineering Science sophomore year courses include:
first semester:
calculus III
vector calculus Vector calculations
digital and computer systems
particles & waves
thermodynamics
Engineering, Second semester:
engineering design
probability and statistics
Modern Physics
electromagnetism
System Biology
10. Most of the domestic mathematical analysis textbooks are rubbish. You them from me and I them from you. For example, Hua Normal University, Fudan Ouyang Guangzhong and others throw them away quickly, which will only mislead the children. I'm afraid Chang gengzhe's, Shi Jihuai's or Zhang Zhusheng's are suitable for teaching materials, while Xu Linlin's can only be used as reference books. For specific reasons, please refer to the book review I gave in Douban and Chen Tianquan's. I heard that they have a high point of view, And it takes a little bit of advanced mathematics to see. Fich's calculus course is very detailed. Some people even say that it is classical, wordy and has too many branches. The title of this book is calculus course, but it talks about the height of mathematical analysis. As for his principles of mathematical analysis, it is a reced version. You can see that the domestic teaching materials are not good, which means that you are still more insightful. I don't know if you can understand such difficult and profound books as zhuoliqi's and Herbert Amann's. Amann's three volumes of analysis are the most famous teaching materials in Germany. Starting from natural number, Tao zhexin's analysis is the most professional analysis teaching material I have ever seen, Also from the perspective of natural numbers, this book has errata and English versions. You can also read Apostol's or Rudin's. It's easy to read Apostol's and Dieudonne's analysis. It's very difficult to read. I appreciate your attitude. You should learn it well. In addition, to say that Chinese textbooks are poor is not to make a fuss with colored glasses, because Chinese professors are not pure in writing books, and even a lot of the content is not finished by themselves. They let their graate students do it, and the phenomenon of plagiarism is also very serious. Some professors really want to write books, but they don't have enough conditions, such as time and energy, As a result, there are also shortcomings in the books written. These are all the drawbacks of Chinese ecation. Of course, there are a few very good ones, such as Chen Tianquan's, which can be comparable with good foreign textbooks.
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