Among all Newton's scientific contributions, mathematical achievements occupy a prominent position. The first creative achievement of his mathematical career was the discovery of the binomial theorem. According to Newton's own recollection, he discovered this theorem in the winter of 1664 and 1665 when he was studying Dr. Wallis's "Infinite Arithmetic" and trying to modify his series for finding the area of ??a circle.
Descartes' analytic geometry corresponds the functional relationships describing motion to geometric curves. Under the guidance of his teacher Barrow, Newton found a new way out based on studying Descartes' analytic geometry. The speed at any moment can be regarded as the average speed in a small time range. This is the ratio of a small distance to the time interval. When this small time interval shrinks to infinitesimal, it is the exact value of this point. . This is the concept of differentiation.
Finding the differential is equivalent to finding the tangent slope at a certain point based on the relationship between time and distance. The distance traveled by a variable-speed moving object in a certain time range can be regarded as the sum of the distances traveled in tiny time intervals. This is the concept of integration. Finding the integral is equivalent to finding the area under the curve that relates time to velocity. Starting from these basic concepts, Newton established calculus.
The creation of calculus is Newton's most outstanding mathematical achievement. In order to solve the problem of motion, Newton created this mathematical theory that is directly related to physical concepts. Newton called it "fluid mathematics". Some of the specific problems it handles, such as tangent problems, quadrature problems, instantaneous velocity problems, and maximum and minimum value problems of functions, have been studied before Newton. But Newton surpassed his predecessors. He stood at a higher perspective, synthesized the scattered conclusions in the past, and unified the various techniques for solving infinitesimal problems since ancient Greece into two common types of algorithms-differential and integral. The reciprocal relationship between these two types of operations was established, thereby completing the most critical step in the invention of calculus, providing the most effective tool for the development of modern science, and opening up a new era in mathematics.
Newton did not publish the results of his research on calculus in time. He may have studied calculus earlier than Leibniz, but the expression form adopted by Leibniz was more reasonable, and his works on calculus were published. It's also earlier than Newton.
When Newton and Leibniz argued about who was the founder of this discipline, a violent commotion was caused. This quarrel was widely discussed among their respective students, supporters and mathematics The family feud lasted for a long time, resulting in a long-standing antagonism between continental European mathematicians and British mathematicians. For a period of time, British mathematics was closed to the country, limited by national prejudices, and stagnant in Newton's "fluid mathematics". As a result, the development of mathematics lagged behind for a hundred years.
It should be said that the creation of a science is by no means the performance of one person. It must be the result of the efforts of many people and the accumulation of a large number of results, and finally summarized by one or several people. Completed. The same is true for calculus, which was independently established by Newton and Leibniz on the basis of their predecessors.
In 1707, Newton's algebra lectures were compiled and published as "Universal Arithmetic". He mainly discusses the basics of algebra and its application (by solving equations) to solving various types of problems. The book states the basic concepts and basic operations of algebra, uses a large number of examples to illustrate how to turn various problems into algebraic equations, and conducts in-depth discussions on the roots and properties of equations, leading to fruitful results in equation theory, such as: He derived the relationship between the roots of an equation and its discriminant, and pointed out that the coefficients of the equation can be used to determine the sum of the powers of the roots of the equation, which is the "Newton Power Sum Formula".
Newton contributed to both analytic geometry and synthetic geometry. He introduced the center of curvature in "Analytical Geometry" published in 1736, gave the concept of a close line circle (or curved circle), and proposed a curvature formula and a method for calculating the curvature of curves. He summarized many of his research results into a monograph "Enumeration of Cubic Curves", published in 1704. In addition, his mathematical work also involves many fields such as numerical analysis, probability theory and elementary number theory.
In 1665, Newton, who was just twenty-two years old, discovered the binomial theorem, which was an essential step for the full development of calculus. The binomial theorem generalizes simple results such as those found by direct calculation
to the following form
Extended form
The binomial series expansion is A powerful tool for studying series theory, function theory, mathematical analysis, and equation theory.
Today we will find that this method is only applicable when n is a positive integer. When n is a positive integer 1, 2, 3,..., the series terminates at exactly n+1 terms. If n is not a positive integer, the series will not terminate and this method will not be applicable. But we must know that at that time, Leibniz only introduced the word function in 1694. In the early stages of calculus, dealing with their stages was the most effective method when studying transcendental functions.
Creating calculus
Newton's most outstanding achievement in mathematics was the creation of calculus. His achievement that surpassed his predecessors was that he unified various special techniques for solving infinitesimal problems since ancient Greece into two general types of algorithms - differential and integral, and established the reciprocal relationship between these two types of operations, such as: area The calculation can be viewed as the inverse process of finding the tangent line.
At that time, Leibniz happened to present a research report on calculus, which triggered a controversy over the patent rights of the invention of calculus, which lasted until Leibniz's death. Later generations have determined that calculus was invented by them at the same time.
In terms of calculus methods, Newton's extremely important contribution was that he not only clearly saw, but also boldly used the methodology provided by algebra that was far superior to geometry. He replaced the geometric methods of Cavalieri, Gregory, Huygens and Barrow with algebraic methods and completed the algebraization of integrals. Since then, mathematics has gradually shifted from a subject of feeling to a subject of thinking.
In the early days of calculus, because a solid theoretical foundation had not yet been established, it was studied by some people who loved thinking. This triggered the famous second mathematical crisis. This problem was not solved until the establishment of limit theory in the nineteenth century.
Equation Theory and Method of Variations
Newton also made classic contributions in algebra. His "General Arithmetic" greatly promoted equation theory. He discovered that the imaginary roots of real polynomials must appear in pairs, and found the rules for the upper bound of the roots of polynomials. He expressed the formula of the sum of the roots of polynomials to nth power using the coefficients of the polynomial, and gave Cartesian formula for the limit of the number of imaginary roots of real polynomials. A generalization of children's symbolic rules.
Newton also designed a method for finding the real root approximation of numerical equations that is suitable for both logarithms and transcendental equations. The modification of this method is now called Newton's method.
Newton also made great discoveries in the field of mechanics, which is the science of explaining the movement of objects. The first law of motion was discovered by Galileo. This law states that if an object is at rest or moving in a straight line at a constant speed, it will remain at rest or continue to move in a straight line at a constant speed as long as there is no external force acting on it. This law is also called the law of inertia, which describes a property of force: force can make an object move from rest to motion and from motion to rest, and it can also cause an object to change from one form of motion to another. This is called Newton's first law. The most important question in mechanics is how objects move under similar circumstances. Newton's second law solves this problem; it is considered the most important fundamental law of classical physics. Newton's second law quantitatively describes how force can change the motion of an object. It explains the time rate of change of velocity (that is, acceleration a is directly proportional to force F and inversely proportional to the mass of the object, that is, a=F/m or F=ma; the greater the force, the greater the acceleration; the greater the mass, The smaller the acceleration. Both force and acceleration have magnitude and direction. Acceleration is caused by force, and the direction is the same as force. If there are several forces acting on an object, the acceleration is caused by the resultant force. The second law is the most important. , from which all the fundamental equations of power can be derived by calculus
Furthermore, Newton formulated his third law based on these two laws. Newton's third law states that the interaction of two objects always occurs. The magnitude is equal and the direction is opposite. For two objects in direct contact, this law is easier to understand. The downward pressure of the book on the table is equal to the upward force of the table on the book, that is, the action force is equal to the reaction force, and the same is true for flight. The force of the plane pulling the Earth upward is numerically equal to the force of the Earth pulling the plane downward.
Newton's laws of motion
Newton's laws of motion. The laws of motion are the collective name for the three laws of motion in physics proposed by Isaac Newton, and are known as the foundation of classical physics.
It is "Newton's first law (law of inertia): all objects move in the same direction. Without the influence of any external force, it always maintains a state of uniform linear motion or a state of rest until an external force forces it to change this state.
——It clarifies the relationship between force and motion and proposes the concept of inertia)", "Newton's second law (the acceleration of an object is directly proportional to the net external force F on the object, and inversely proportional to the mass of the object. The direction of acceleration is The direction of the resultant external force is the same. ) Formula: F=kma (when the unit of m is kg and the unit of a is m/s2, k=1), Newton’s third law (the action force and reaction force between two objects, in On the same straight line, the magnitude is equal and the direction is opposite)"
Newton's method
Newton's method is also called Newton-Raphson method. , it is a method proposed by Newton in the 17th century to approximately solve equations in the real number field and the complex number field. Most equations do not have root finding formulas, so it is very difficult or even impossible to find the exact roots. Therefore, it is necessary to find the approximate roots of the equations. It is particularly important. The method uses the first few terms of the Taylor series of the function f(x) to find the roots of the equation f(x) = 0. Newton's iteration method is one of the important methods for finding the roots of the equation. Its greatest advantage is in the equation. There is square convergence near the single root of f(x) = 0, and this method can also be used to find the multiple roots and complex roots of the equation. In addition, this method is widely used in computer programming. Let r be the root of f(x) = 0. , select x0 as the initial approximation of r, and draw the tangent line L of the curve y = f(x) through the point (x0, f(x0)). The equation of L is y = f(x0)+f'(x0)(x-x0 ), find the abscissa x1 = x0-f(x0)/f'(x0) of the intersection of L and the x-axis, and call x1 a first-order approximation of r. Draw the curve y = f through the point (x1, f(x1)). (x), and find the abscissa x2 = x1-f(x1)/f'(x1) of the intersection of the tangent line and the x-axis. Call x2 the quadratic approximation of r. Repeat the above process to obtain the sequence of approximate values ??of r. , where x(n+1)=x(n)-f(x(n))/f'(x(n)), is called the n+1 approximation of r, and the above formula is called Newton's iteration formula. Newton's method for the nonlinear equation f(x)=0 is an approximate method for linearizing the nonlinear equation. It expands f(x) into a Taylor series f(x) = f(x0)+( near the x0 point. x－x0)f'(x0)+(x－x0)^2*f''(x0)/2! +... Take its linear part as the approximate equation of the nonlinear equation f(x) = 0, that is, Taylor The first two terms of the expansion are f(x0)+f'(x0)(x－x0)=f(x)=0. Assuming f'(x0)≠0, the solution is x1=x0－f(x0) /f'(x0) In this way, an iteration sequence of Newton's method is obtained: x(n+1)=x(n)-f(x(n))/f'(x(n))
<. p>Optical ContributionNewton's Telescope
Before Newton, Mozi, Bacon, Leonardo da Vinci and others had studied optical phenomena. The law of reflection is one of the optical laws that people have known for a long time. When modern science was on the rise, Galileo shocked the world by discovering the "new universe" through his telescope. Dutch mathematician Snell first discovered the law of refraction of light. Descartes proposed the particle theory of light...
Newton and his almost contemporaries such as Hooke and Huygens, like Galileo, Descartes and other predecessors, used great interest and Passionate about researching optics. In 1666, while on vacation at home, Newton obtained a prism, which he used to conduct his famous dispersion experiment. After a beam of sunlight passes through the prism, it is decomposed into several color spectral bands. Newton then uses a baffle with a slit to block the other colors of light, allowing only one color of light to pass through the second prism. The result is are just the same color of light. In this way, he discovered that white light is composed of light of various colors. This was his first major contribution.
In order to verify this discovery, Newton tried to combine several different monochromatic lights into white light, and calculated the refractive index of different colors of light, accurately explaining the dispersion phenomenon. The mystery of the color of matter is revealed. It turns out that the color of matter is caused by the different reflectivity and refractive index of light of different colors on objects. In 1672 AD, Newton published his research results in the "Philosophical Journal of the Royal Society". This was his first publicly published paper.
Many people study optics to improve refracting telescopes.
Because Newton discovered the composition of white light, he believed that the dispersion phenomenon of refracting telescope lenses could not be eliminated (later, some people used lenses composed of glasses with different refractive indexes to eliminate the dispersion phenomenon), so he designed and built a reflecting telescope.
Newton was not only good at mathematical calculations, but he was also able to make various experimental equipment and conduct precise experiments by himself. In order to manufacture telescopes, he designed his own grinding and polishing machine and experimented with various grinding materials. In 1668 AD, he made the first prototype of a reflecting telescope, which was his second greatest contribution. In 1671 AD, Newton presented his improved reflecting telescope to the Royal Society. Newton became famous and was elected as a member of the Royal Society. The invention of the reflecting telescope laid the foundation for modern large-scale optical astronomical telescopes.
At the same time, Newton also conducted a large number of observation experiments and mathematical calculations, such as studying the abnormal refraction phenomenon of glacial rocks discovered by Huygens, the color phenomenon of soap bubbles discovered by Hooke, and the "Newton Rings" Optical phenomena and so on.
Newton also proposed the "particle theory" of light, believing that light is formed by particles and takes the fastest linear motion path. His "particle theory" and later Huygens' "wave theory" constitute the two basic theories about light. In addition, he also produced various optical instruments such as Newton's color disk.
Constructing the Building of Mechanics
Newton is the master of classical mechanics theory. He systematically summarized the work of Galileo, Kepler, Huygens and others, and obtained the famous law of universal gravitation and Newton's three laws of motion.
Before Newton, astronomy was the most prominent subject. But why do planets must orbit the sun according to certain rules? Astronomers can't explain this problem satisfactorily. The discovery of universal gravitation shows that the motion of stars in the sky and the motion of objects on the ground are governed by the same laws - the laws of mechanics.
Long before Newton discovered the law of universal gravitation, many scientists had seriously considered this issue. For example, Kepler realized that there must be a force at work to keep the planets moving along the elliptical orbit. He believed that this force was similar to magnetism, just like a magnet attracting iron. In 1659, Huygens discovered from studying the motion of a pendulum that a centripetal force is needed to keep an object moving in a circular orbit. Hooke and others thought it was gravity, and tried to deduce the relationship between gravity and distance.
In 1664, Hooke discovered that the orbital bending of comets when approaching the sun was due to the gravitational effect of the sun; in 1673, Huygens derived the law of centripetal force; in 1679, Hooke and Halley derived the law of centripetal force from the Puller's third law states that the gravitational force that maintains planetary motion is inversely proportional to the square of the distance.
Newton himself recalled that around 1666, he had already considered the issue of gravity when he lived in his hometown. The most famous saying is that during holidays, Newton often sat in the garden for a while. Once, as happened many times in the past, an apple fell from the tree...
The accidental falling of an apple was a turning point in the history of human thought. It made the man sitting in the garden The man's mind was enlightened, causing him to ponder: What is the reason why all objects are almost always attracted towards the center of the earth? Newton thought. Finally, he discovered gravity, which was of epoch-making significance to mankind.
The brilliance of Newton is that he solved the mathematical argumentation problem that Hooke and others could not solve. In 1679, Hooke wrote to ask Newton if he could prove that the planets move in elliptical orbits based on the law of centripetal force and the law that gravity is inversely proportional to the square of the distance. Newton did not answer this question. When Halley visited Newton in 1685, Newton had already discovered the law of universal gravitation: there is a gravitational force between two objects, and the gravitational force is inversely proportional to the square of the distance and directly proportional to the product of the masses of the two objects.
At that time, accurate data such as the radius of the earth and the distance between the sun and the earth were already available for calculation. Newton proved to Halley that the earth's gravity is the centripetal force that causes the moon to move around the earth. He also proved that under the influence of the sun's gravity, the motion of the planets complies with Kepler's three laws of motion.
Under Halley's urging, Newton wrote the epoch-making great work "Mathematical Principles of Natural Philosophy" at the end of 1686. The Royal Society lacked funds and could not publish this book. Later, with Halley's funding, this one of the greatest books in the history of science was able to be published in 1687.
In this book, Newton started from the basic concepts of mechanics (mass, momentum, inertia, force) and basic laws (the three laws of motion), and used the sharp mathematical tool calculus he invented. , not only mathematically demonstrated the law of universal gravitation, but also established classical mechanics as a complete and rigorous system, unified the mechanics of celestial bodies and the mechanics of objects on the ground, and achieved the first major synthesis in the history of physics.