? Research on the Functional Relationship of Descartes' Analytic Geometry Description: Based on Descartes' motion and geometric curve, Descartes analyzed geometry and found a new way out under the guidance of Newton's teacher Barrow. At any time, when you see the average speed of a small time limit, it is a small distance, and this small time interval is infinitely smaller than the time interval, which is the exact value. This is the concept of difference.
? Distinguishing the relationship between time and distance is equivalent to finding the tangent of a point. The speed change of a moving object in a certain time can be regarded as walking in a very small time interval, which is a comprehensive concept. Orthogonal search for the following areas is equivalent to? The relationship curve between time and speed. These basic concepts of Newton's calculus.
? Calculus established Newton's most outstanding mathematical achievement. Newton established a mathematical theory directly related to physical concepts before solving exercises. Newton's theory is called "the number of mobile patients". It deals with some specific problems, such as "tangent problem, quadrature problem, the highest and lowest instantaneous speed of problems and functions?" The problem that Newton has been studying. Newton surpassed his predecessors. From a higher point of view, the conclusion of scattered ancient Greek synthesis solved the unification of two kinds of universal algorithms of infinitesimal technology-differential and integral, and established the inverse ratio relationship between these two kinds of businesses, thus completing the most critical step in the invention of calculus and opening up a new era of mathematics for the development of the most effective modern tool science.
? Newton did not publish the research results of calculus in time. He studied calculus earlier than Leibniz, but Leibniz adopted a more reasonable form, and the book of calculus was published earlier than Newton.
? The argument between Newton and Leibniz, the founder of the subject, actually caused an uproar. The quarrel lasted for a long time among his students, supporters and mathematicians, which led to the long-term opposition between mathematicians in continental Europe and Britain. For a period of time, the British mathematics community was closed to the outside world. Because of racial prejudice, the number of patients still standing in Newton's "stream" was too rigid, which led to the development of mathematics falling behind for a hundred years.
? It should be said that the establishment of a science is not a personal performance, it must be based on the efforts of many people and the accumulation of a large number of achievements, and finally completed by 1 person or several people. The same is true of calculus. Newton and Leibniz are independent of the former group.
1707 Newton's lecture notes in algebra class were later called "Arithmetic of Everything" and edited and published. ? He focuses on algebraic basis (group) to solve various problems in application. The book lists the basic concepts and operations of algebra, and illustrates how to discuss the roots and properties of algebraic equations in depth with a large number of examples, which has achieved fruitful results in the world. For example, he has learned to distinguish the relationship between the roots of an equation, and can use the coefficients of the equation to determine the number of roots and the power of Newton's power formula.
? Newton's analytic geometry and its contribution. 1736, he published the concept of closed circular straight line (or circular curve) in Analytic Geometry, put forward the curvature formula, and introduced the curvature and curvature center of calculation curve. And many research results, published in 1704. In addition, his mathematical work involves numerical analysis, probability theory and elementary number theory.
1665, Newton, who was just 22 years old, discovered the binomial theorem, which is an indispensable calculus for all-round development. The binomial theorem can be obtained directly by calculation.
? Promote simple results in the form of ...
? Influence of promotion form
? Binomial series expansion is a powerful tool to study series theory, function theory, mathematical analysis and equation theory. Today, we will find that this method is only applicable to the termination of a series of complete N+ 1 projects where n is a positive integer. If n is a positive integer, the series will not terminate, and this method is not applicable. However, we need to know when Leibniz introduced the function of 1694. Their level is in the early stage of calculus of transcendental function, which is the most effective method.
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Create calculus
Newton's greatest achievement in mathematics was to create a stone. The achievement beyond the past was solved by his special method in ancient Greece, because two commonly used algorithms were unified by infinite calculus, and the inverse relationship between these two types of business was established. For example, area calculation is regarded as the inverse process of tangent.
? As soon as Leibniz's research on calculus was put forward, it triggered the debate on the patent of calculus invention, and it didn't stop until the gap ceased. Later generations identified the micro-area, which was invented by them.
? Newton made an extremely important contribution to the calculus method. He not only saw this clearly, but also boldly used algebraic geometry to provide significant advantages over it. Algebraic method, which replaces the geometric method of cavalieri, Gallegor, Huygens and Barrow-integral algebra. Since then, mathematics has gradually changed from discipline consciousness to ideological discipline.
? In the early days of calculus, a solid theoretical foundation has not been established, and some people like to think. This led to the famous mathematical crisis. This problem was not solved until the limit theory of 19 century.
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Equation variational method
In algebra, Newton also made a classic contribution, and his theory of major equations of generalized arithmetic contributed a lot. He must pair the imaginary roots of real polynomials to find the roots of regular constraints on polynomials. He said that the n-th power of polynomial gives the limit of the imaginary root number of Cartesian symbol of real number at the root of polynomial, which generalizes the rule.
Newton also designed the approximate application method and correction method of the real roots of numerical equations, logarithm and transcendental equations, which is now called Newton method. ?
? Newtonian mechanics has also made great discoveries, and it is a science that describes the motion of objects. The law of motion was discovered by Galileo. The law stipulates that an object is stationary or moving in a straight line at a constant speed, and as long as there is no external force, it will remain unchanged or continue to move in a straight line at a constant speed. This method, also known as the law of inertia, describes a natural force: power can be changed from static to moving and moving objects, and from another form of motion to static or moving objects. This is the so-called Newton's first law. The most important problem in mechanics is how to move an object under similar circumstances. Newton's second law is the most important basic law of classical physics. Newton's second law can make a difference by quantitatively describing the object of motion power. Indicates the time rate of change of speed (that is, the acceleration force F is directly proportional to and inversely proportional to the mass of the object F/A = m or F = ma, and the acceleration increases with the increase of force; Mass, the value of force and acceleration with smaller acceleration and the force, direction and direction of acceleration caused by force, if several forces act on an object and work together to produce acceleration, the second law is the most important one. It can be calculated by the basic equations of all power supplies.
? In addition, Newton's third law is also promulgated according to these two laws. Newton's third law points out that the interaction between two objects is always opposite in the same direction. For two objects in direct contact, it is easier to understand the law. This book is equal to the praise pressure in the book, that is, force equals reaction. This gravity, the plane is flying, and the power of the earth is numerically equal to the power of the earth to pull the plane down. Newton's laws of motion are widely used in science and dynamics.
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Newton's law
Newton isaac newton's laws of motion put forward three laws of motion. In physics, the collective is called classical physics.
"Newton's first law (law of inertia: all objects without any external force always keep moving in a straight line or at rest-obviously, force and motion until external force forces it to change this state. Relationship and the concept of forward inertia) Newton's second law (accelerating an object together means that the mass of the object is directly proportional to the external force F, except that the acceleration is inversely proportional to the direction and cooperation direction) formula F = KMA(M kg, a unit, m/s2, K = 1), Newton's third law (force and reaction force are on the same straight line between two objects,)
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newton method
? Newton method, also known as Newton-Raphson method, is a real number domain and a complex number domain, and it is an approximate method for solving Newton equations in17th century. Most equations have no formula for finding the root, so it is very difficult or even impossible to ask for the exact root. It is particularly important to find the approximate root of an equation. This method uses Taylor series of function f(x) to find the root of equation f(x)= 0. Newton iteration method is one of the important methods to find the roots of equations. Its greatest advantage is that the equation f(x)= 0 converges to one side, and the legal demand equation can be used for multiple roots. In addition, this method is widely used in computer programming. Set r is the root of F(X)= 0, and the tangent point of curve (X0, F(X0))X0 is selected. In the initial approximation, the equation of Y = F(X)L is y = F( X0)+ F'(X0)(X-X0). Calculate the abscissa of the intersection of the L and X axes, X 1, X 1 = X0-F(X0)/ F'(X0). Make the tangent of the curve Y = F(X) pass through the point (x 1, F(X 1)), and find the abscissa of the intersection of the tangent and the x axis x2 = x1-f (x1)/f' (x1. Repeat the above process. In the approximation sequence, r, x (N+ 1) = x (n)-f (x (n))/f' (x (n)) is called r, n+1approximation. The above equation is called Newton's iterative formula. The solution of nonlinear equation f(X)= Newton is an approximate method of linear nonlinear equation. F is f (x) (x) of Taylor series near X0 = f (x0)+(x-x0) f' (x0)+(x-x0) 2 * f' (x0)/2! +... Whether the linear part of the approximate equation of nonlinear equation f(x) is 0 and the first two Taylor expansions, then f(X0)+ F'(X0)(X-X0)= F(X)= 0 Let F'(X0)≠0, then the solution is x65438+.
Contribution of optics
Newton telescope
? Before Newton, Mo, Bacon and Da Vinci studied optical phenomena. Reflection method is one of the people who realized the law of light very early. With the rise of modern science, Galileo discovered a "new universe" through a telescope, which shocked the world. The Dutch mathematician Snell first discovered the law of refraction of light. Descartes proposed light particles. ...
? Newton and his predecessors such as Hooke, Huygens, Galileo and Descartes all studied optics with great interest and enthusiasm. 1666, Newton was on vacation at home, and he used the famous dispersion experiment. A beam of sunlight is divided into several color bands through a prism. Newton and slit baffles are used to block other colors of light, and only one color of light passes through a second prism to produce light of the same color. In this way, he found that different colors of light constitute white light, which is the first major contribution.
? In order to verify this discovery, Newton tried several different monochromatic lights to synthesize white, and calculated the accurate expressions of refractive index and dispersion phenomenon of different colors of light. Uncover the mysterious color of materials. It turns out that the color of materials is caused by the different light reflectivity and refractive index of different colors on objects. In A.D. 1672, Newton's research was published in the Journal of Philosophy of the Royal Society, which was his first paper.
? Many optical systems improve the refraction of the telescope. Newton discovered the composition of white light, and the chromatic aberration of refractive lens could not be eliminated (later, some people used glass lenses with different refractive indexes to eliminate chromatic aberration), and designed and manufactured reflecting telescope.
? Newton was not only good at mathematics, but also made various test equipment and excellent experiments for himself. In order to make a telescope, he designed a polishing machine and experimented with various grinding materials. This year 1668, he made the first prototype of reflective telescope, which was the second largest contributor. 167 1 year, Newton made great progress and devoted himself to the Royal Society. Newton became famous and was elected a member of the Royal Society. Reflecting telescope's invention laid the foundation for modern large-scale optical telescopes.
? Newton's observation experiments and mathematical calculations have studied the abnormal refraction of Huygens stone, such as Hooke's discovery of soap bubbles in glaciers and the color of Newton's ring optical phenomena.
? Newton also suggested that the "particles" of light are composed of particles, and light moves in the fastest straight line. After his "particle", Huygens' "wave theory" constitutes two basic theories of light. In addition, he also made Newton color wheel and other optical instruments.
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Construction machinery construction
? Newton's master of classical mechanics theory. This paper systematically summarizes the work of Galileo, Kepler and Huygens, the famous law of gravity and Newton's three laws.
? Before Newton, astronomy was one of the most prominent themes. But why are planets around the sun in some legal provisions? Astronomers are not satisfied with the explanation. It is found that the motion of the gravitating sky and stars is governed by the same law as that of the objects on the ground-mechanics.
? As early as before, isaac newton discovered the law of gravity. Many scientists think this problem is very serious. For example, Kepler realized that in order to keep moving in an elliptical orbit, the planet must be a working force, which he thought was similar to magnetic force and attracted iron like a magnet. 1659, Huygens found from his research that centripetal force is needed to make an object swing and keep moving in a circular orbit. Tick, who is gravity, and try to deduce the relationship between gravity and distance
1664, Hooke found that when the orbit is curved, the comet is close to the sun due to the gravity of the sun; 1673 huygens deduces the law of centripetal force; 1679, Hooke and Harley deduced the third law from the centripetal force of Kepler's law. Planetary motion, gravity and the square of distance are inversely proportional to each other.
? Newton himself recalled that gravity had been taken into account when he lived in his hometown around 1666. The most famous saying is that Newton often sat in the garden for a while during holidays. Once, as often happened in the past, an apple fell from the tree. ......
? The apple accidentally fell to the ground, but it was a turning point in the history of human thought. It makes a person sit in the human mind in the garden. They opened their eyes and made him think deeply: what makes all objects almost always face the center of the earth to attract it? Newton thought. Finally, he discovered a milestone of human gravity.
? Newton's cleverness lies in that he solved the mathematical argument that Hooke and others failed to solve. 1679, Hooke wrote to Newton, asking about the centripetal force method that can be inversely proportional to the square at the same distance, and the law of universal gravitation that proves the motion of planets in elliptical orbits. Newton didn't answer the question. 1685, Harley visited Newton, and Newton discovered the law of universal gravitation: the gravitation between two objects is inversely proportional to the square of the distance and directly proportional to the quality of the product of the two objects.
? At that time, radius of the earth, the distance between the sun and the earth was accurately calculated. Newton proved to Harley that the gravity of the earth is the centripetal force of the moon's motion around the earth, and also proved that the gravity of the sun and the motion of planets conform to Kepler's three laws of motion.
? At the urging of Harley, at the end of 1686, Newton wrote an epoch-making masterpiece, Mathematical Principles of Natural Philosophy. This book didn't come out. Later, the Royal Society published 1687, which was funded by Harley. The history of this great scientific project lacked funds.
It was in this book that Newton invented the mathematical tools of calculus, the law of gravitation and the law of classical mechanics, starting from the basic concepts of mechanics (mass, momentum, inertia and force) rather than just the basic laws of mathematical parameters (three laws of motion), and for the first time in history, he established a complete integrated ground object and a strict system of celestial mechanics and physical mechanics.