The significance of calculus can be seen from the following aspects.
The influence of (1) on mathematics itself
Mathematics inherited from ancient Greece is constant mathematics and static mathematics. Since the advent of analytic geometry and calculus, the era of variable mathematics has opened, and variable mathematics is dynamic mathematics. Mathematics began to describe changes and movements, which changed the whole mathematical world. Mathematics has also entered the era of analysis from the era of geometry.
Calculus has injected great vitality into mathematics, which has greatly developed and reached unprecedented prosperity. Such as the establishment of differential equations, infinite series, variational methods, and the generation of complex variable functions and differential geometry. The rigorous logical basic theory of calculus further shows its universal significance in the field of mathematics.
(2) the role of other disciplines and engineering technology
With calculus, human beings have mastered the process of movement, and calculus has become the basic language of physics and a powerful tool for finding answers to questions. With calculus, there will be an industrial revolution, with large-scale industrial production, there will be a modern society. Modern means of transportation, such as the space shuttle and spacecraft, are the direct result of calculus. With the help of calculus, Newton discovered the law of universal gravitation, and found that there is no corner in the universe that is not included in these laws, which strongly proved the mathematical design of the universe.
Now chemistry, biology, geography and other disciplines have to deal with calculus.
(3) the impact on human material civilization
Modern engineering technology directly affects people's material production, and the basis of engineering technology is mathematics, which can not be separated from calculus. Nowadays, calculus not only becomes the foundation of natural science and engineering technology, but also permeates people's extensive economic and financial activities, which means that calculus has a wide range of applications in the humanities and social sciences.
(4) Influence on human culture
Now we can't study the laws of nature and society without calculus, because calculus is the science of studying the laws of motion.
The foundation of modern calculus theory is a leap in understanding. The concept of limit reveals the dialectical unity of opposites between variable and constant, infinite and finite. From the point of view of limit, infinitesimal is just a variable with zero limit. That is, in the process of change, its value can be "non-zero", but its trend is "zero", which can be infinitely close to "zero". Therefore, the establishment of modern calculus theory, on the one hand, eliminated the "mystery" of calculus for a long time, and made the attacks of theologians such as Bishop Becquerel on calculus completely bankrupt, which profoundly influenced the development of modern mathematics in thought and method. This is the enlightenment of calculus to philosophy, and also the enlightenment and influence to human culture.
Extension and continuation
What is the definition of calculus?
Calculus is divided into differential calculus and integral calculus.
Differential calculus mainly studies how to determine the instantaneous change rate (or differential) of the function value when the independent variable of the function changes. In other words, the method of calculating the derivative is called differential calculus. Another calculation method of differential calculus is Newton method, also called applied geometry method, which mainly calculates the slope of points through the tangent of function curve.
Integral calculus is the inverse operation of differential calculus, that is, the original function is calculated from the derivative. Divided into definite integral and indefinite integral. The definite integral of a unary function can be defined as the sum of the areas of infinite small rectangles, which is approximately equal to the actual area contained under the function curve. According to the above knowledge, we can use integral to calculate the area contained in a curve on a plane, the surface area or volume of a sphere or cone, etc. However, indefinite integral is less useful, mainly used to solve differential equations.
The History of Calculus (I) The Background of Its Origin
1 secant circle method
Archimedes (287 BC-2 BC12) was an ancient Greek mathematician, physicist, inventor, engineer and astronomer. He once said, "Give me a fulcrum and I can lift the whole earth."
Archimedes, drawing the inscribed polygon and circumscribed polygon of a circle, estimated by the perimeter of the polygon.
(this is also called "tangent circle method", which is an "exhaustive method");
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Archimedes also realized that the area of a polygon can be infinitely close to the area of a circle, which shows that there is no infinitesimal number.
Ludolph van ceulen, a martyr, used the method of "cutting circles" all his life to make polygons accurate to 35 decimal places, and he was proud of it. Even when he died, he carved these numbers on his tombstone. Now we just need to drag the slider above to calculate easily.
2 calculate the area under the parabola
17th century, under the guidance of "exhaustive method", the area under parabola can be calculated in this way.
A key step in this calculation is to divide the bottom indefinitely until it is divided into the smallest unit, which is the same mistake as keeping the arrow still.
Bonaventura Francesco cavalieri (1598- 1647), Italian geometer.
Cavalieri defended it and said, "This method is not strict, but isn't it very useful? Whether it is strict or not is a matter for philosophers. Aren't other geometricians as strict as me? "
Three sporadic calculus results
At that time, Fermat and cavalieri also gave them respectively (in the present writing), but only used the geometric method (that is, finding the area under the curve).
4 abstract
With the demand and ideological origin, at this time, someone needs to sum it up and make it develop into a discipline, which often requires a master. History has given two places. Maybe calculus is too important to be afraid of problems.