In the early morning, the rising sun rises in the east, and a ray of morning sun sets on Tsinghua University Garden.
Room No. 28, West Courtyard.
In the study room.
The windows were stained with white frost, and rays of sunlight shone helplessly through the windows. The house was quiet, and a wooden vertical blackboard was moved into the study.
"To learn calculus, you must first understand what calculus is. You cannot know what it is or why it is." Hua Luogeng stood next to the blackboard and wrote six words.
What is calculus.
"Let's start with the most basic calculation of area. In the ancient Greek period, people of Archimedes' time were in the initial stage of development of geometry. Mathematicians encountered a thorny and severe problem, that is, finding the area of a triangle.
Figures such as squares and squares have area formulas, so the solution is very simple, but the problem is, how to find the area of those irregular figures?"
"For example, the S-shaped curve I am drawing now needs to be solved for the area enclosed by this curve, but there is no formula. At this time, how to solve for the area enclosed by a curve became a problem studied by mathematicians at that time."
"Archimedes found a way, Yu Hua, do you know what it is?"
Hua Luogeng looked at Yu Hua.
"The exhaustive method uses familiar graphics to infinitely approximate the area enclosed by the curve." Yu Hua replied.
"Yes, the exhaustive method was proposed by Antiphon, improved by Eudoxus, and perfected by Archimedes. The idea of the exhaustive method is to use infinite familiar figures to find the area of a figure surrounded by a curve. In the history of mathematics, exhaustion
Calculus is regarded as the predecessor of calculus and its rigor is impeccable."
Holding the chalk in his right hand, Hua Luogeng drew the solution process of the exhaustive method, using triangles one by one to fill the area enclosed by the S-shaped curve, and finally calculated the area size.
The whole process is extremely complicated, but extremely rigorous.
After Hua Luogeng finished solving the problem, he immediately erased the formulas and graphics with a brush and wrote down a new concept to find the area through a rectangle:
"The exhaustive method was used until the seventeenth century. During this more than a thousand years of history, there was the Chinese method of calculating the area of a circle, but the calculation was too complex and was not applicable. The limitations of the exhaustive method have gradually become apparent. For different curves surrounded by
The area of needs to be approximated by using different graphics, and the proof techniques for different graphics are different and extremely cumbersome. During this period, "using rectangles to approximate the original graphics" appeared in the mathematical world. The idea is consistent with the exhaustive method and is simpler, but the rectangular solution
There is a problem and that is a loss of rigor, which is a very serious situation."
Rigor is the soul of mathematics.
Without simplicity, mathematics loses many fools.
Without rigor, mathematics will lose everything.
If a theorem, a formula, or a mathematical constant loses its rigor, it means the collapse of the entire mathematical edifice.
Yu Hua listened with all his attention. He memorized all the key points of Hua Luogeng's explanation in his mind and understood it very quickly.
"Newton and Leibniz attached great importance to the problems of solving rectangles. After the unremitting research of these two mathematicians, Newton and Leibniz accidentally discovered a key thing, which is the most basic and important aspect of calculus.
The core idea is the reciprocal operation between differential and integral, expressed in mathematical formulas as the fundamental theorem of calculus."
With a serious look on his face, Hua Luogeng wrote the basic theorem of calculus on the blackboard: "Before, differential calculus and integral calculus were two separate subjects. Calculation of derivatives through differential calculus and area calculation through integral calculus were unrelated to each other. In Newton and Leibniz's work,
Under the influence, a complete system of calculus was established.”
The reciprocal operation between differential and integral.
This is the core of calculus. At this point, calculus, which is extremely important in the history of human civilization, was born. The fundamental theorem of calculus is also known as the Newton-Leibniz formula.
What a genius...
Yu Hua listened to the historical process of the birth of calculus and sighed slightly in his heart. He connected two separate subjects and keenly discovered the reciprocal operation between differential and integral. He was worthy of being the two top experts in history.
What is the concept of reciprocal operation?
To put it simply, the problem of finding the area can be transformed into finding the derivative, and the problem of finding the derivative is transformed into finding the area, and they can be transformed into each other.
If the path of integration fails, then switch from low-dimensional research to high-dimensional research and use differential calculus to solve the problem.
If the path of differential calculus fails, then switch from high-dimensional research to low-dimensional research and use integrals to solve the problem.
In addition, you can also integrate backwards to find the area.
If you want to ask what is its meaning?
The significance is very important because it greatly reduces the tedious calculation process, simplifies the calculation difficulty, and greatly improves the development efficiency of various branches of mathematics.
There are so many things that calculus can find, such as the extreme values of differential derivatives.
Extreme values are very important, such as the maximum flight distance of cannonballs, the profit data of a ship of goods, the shortest route from a certain place to a certain place, etc.
This is the most important tool for scientific research and a mathematical weapon created by humans themselves.
"Of course, the calculus system at this time was not perfect. The problem of infinitesimal quantities made the foundation of calculus unstable. The problem of infinitesimal quantities was to define the limit in a dynamic way. In the process of a quantity approaching 0, there are countless
Real numbers, this is not feasible, which triggered the second mathematical crisis. Later, mathematicians Cauchy and Weierstrass redefined the limit. At this point, the foundation of calculus was finally solid. Later, the French mathematician Le Bay
The Lebesgue integral studied in lattice is the end of calculus."
Hua Luogeng slowly talked about the relationship between calculus and infinitesimal quantities, and then wrote a series of formulas on the blackboard. This is the Lebesgue integral:
"While I was studying at the University of Cambridge in England, I was fortunate enough to go to France and meet Mr. Lebesgue, which was of great benefit. However, I think there is still great research value in the field of infinitesimal calculus. In the future, you can
Try this field, calculus is not only the basis of mathematical research, but also a tool for scientific research, do you understand?"
"Understood." Yu Hua nodded after hearing this, and wrote down a mathematical research direction given to him by Hua Luogeng.
Hua Luogeng nodded and said seriously: "After we know what calculus is, it will be easier for us to learn. Next, we will talk about functions, derivatives and limits. How many of the first books have you read?"
"After reading one-third of the book, I understand both functions and derivatives." Yu Hua responded that he didn't study long last night and he only read one-third of "Derivatives and Limits."
"Okay, let's start from the limit."
When Hua Luogeng heard this, his eyes showed admiration. He paused and explained in detail: "The limit of calculus is defined as..."