미분의 탄생과 활용

미적분의 탄생과 활용

 

 

내용 요약: 미적분의 탄생과 활용에 대하여 조사한 보고서이다. 미분의 탄생은 영국의 과학자 뉴턴과 라이프니츠의 표절 공방으로 유명하며, 적분은 고대 이집트부터 현대에 오기까지 수 차례 개정되고 발전해왔다. 독립적으로 발전되어 온 미분과 적붑은 아이작 뉴턴과 고트프리스 폰 라이프니츠에 이르러서야 유기적인 관계를 지님이 밝혀졌다. 미분은 실생활 속에서 무인 단속 카메라, 애니메이션, 건축 등에 활용된다.

 

1. Birth of Calculus

 

(1) Birth of Differentials

 

British scientist Newton and German philosopher and mathematician Leibniz have been engaged in a fierce plagiarism battle for decades over who first created differential calculus. The incident began with London publisher John Collins sending some of Newton's unpublished material to Leibniz. Newton claimed that Collins' "traitor" leaked his calculus idea.

 

While Newton accused Leifnitz of "stealing calculus that I had already discovered," Leifnitz reportedly responded by saying, "I only discovered it independently at a similar time." In addition to Newton and Leibniz, scientists from Britain and the continent joined forces to root for each other and stop the exchanges for a while. However, British scholars suffer quite a lot because Leibniz's method was more intuitive and convenient than Newton's.

 

Currently, the mathematics community accepts the view that "it was Newton who first discovered (or invented) the concept of differential calculus in chronological order, and Leibniz also discovered the division on its own before looking at the calculus data handed over by Collins." In other words, each acknowledges its own discovery.

 

(2) Birth of Integrals

 

The integration is known to be based on the separation of land surveying and geometry, which developed in ancient Egypt due to non-periodic variations in the size of farmland due to flooding of the Nile River.

Integral calculus, as outlined in the abstract, is the branch of learning that is gradually developed and refined by several mathematicians, with separation quadrature methods found from volume-finding problems. Archimedes of ancient

 

Greece continued to draw triangles that encircled the parabola and the area in a straight line, resulting in the sum of each triangle's width. In this way, called consumption, Archimedes also obtained the width of the circle and the volume of the sphere. Johannes Kepler was the first person to think of the quadrature method, considering that it was not by

 

consumption method, but by countless amounts of space or volume. Also, Italian mathematician Cavalieri considered width to be a group of lines with no width and volume to be a group of faces with no thickness.

 

It was not until Isaac Newton and Gottfried von Leibnitz that the differential and compositional methods developed independently were found to have an organic relationship between the two concepts.

 

내용 요약: 미적분의 탄생과 활용에 대하여 조사한 보고서이다. 미분의 탄생은 영국의 과학자 뉴턴과 라이프니츠의 표절 공방으로 유명하며, 적분은 고대 이집트부터 현대에 오기까지 수 차례 개정되고 발전해왔다. 독립적으로 발전되어 온 미분과 적붑은 아이작 뉴턴과 고트프리스 폰 라이프니츠에 이르러서야 유기적인 관계를 지님이 밝혀졌다. 미분은 실생활 속에서 무인 단속 카메라, 애니메이션, 건축 등에 활용된다.

 

The concept of today's integrals began to be created after Augustine Louis Corsi, who defined the static of continuous functions in the bounded lung section by mathematical extreme concepts. Cosy's concept of integration is easily extended to a bounded function with finite discontinuities, but in the event of infinite discontinuities, integration by his

 

concept becomes impossible. By extending this concept, Bernhard Riemann is the scholar who attempted to integrate a function of the Confucian line in the Confucian section. Riemann's strict definition of integrals leads to the definition of integrals in the same meaning as those used in modern times.

 

2. Real life example where differentiation is used

 

(1) Unmanned camera

 

Fixed unmanned camera is a loop method that is drawn 20 to 30 meters in front of the camera in a square shape. It is to install two rows of loops with sensors that read speed on the road, measure the "time" of a car passing between them, and convert them into "speed."

 

According to the formula "Speed = distance ÷ time," the average rate of change in sensor values is calculated and the camera flash flashes immediately after the speed is recognized.

 

(2) Animation

 

Simulations of water motion are based on hydrodynamic theory. The basis of the design is the Navier-Stokes equation, a type of differential equation that can explain air or water flow. The Navier-Stokes equation is one of the world's seven mathematical challenges with a million dollar prize at stake. The solution of the equation has yet to be found, and there are no exact solutions, so only approximate solutions with errors can be obtained. Because 'visual effects' are more important than 'accuracy' in CG production, it is enough to be approximated.

 

Parametric, one of the differential formulas in an animation, also uses several independent variables to process graphical data of straight lines, curves, or surfaces. For example, in order to make a smooth curved surface, a polygon can be split into smaller triangles or squares to adjust the position of points that occur when the vertices of triangles or squares meet.

 

(3) Architecture

 

In architecture, you can see many cases of differential application in real life. This is because the tangent of the curve can be used to build on a safe road design. When a car exits a curved road and enters a straight road, a road design based on mathematical principles is required for the driver to enter safely.

Cars that run on curved roads tend to go in the direction of tangents on curves. A straight road that connects to the end of the curved road must be a tangent to the curved road to ensure safe entry. The formula used in the design is differential.