my scientific age

Chapter 95 The Story of Calculus!

The next day.

In the early morning, the rising sun rises in the east, and a touch of morning sun falls on Tsinghua Garden.

Room 28 in the West Courtyard.

Inside the study.

The window was stained with a layer of hoarfrost, and a ray of sunlight shone through the window helplessly. The house was quiet and silent, and a wooden vertical blackboard was moved into the study.

"To learn calculus, you must first understand what calculus is. You can't know what it is or why it is." Hua Luogeng stood beside the blackboard and wrote six words.

What is calculus.

"Let's start with the most basic area. In ancient Greece, in the time of Archimedes, geometry was in the initial stage of development. Mathematicians encountered a difficult and severe problem, that is, to find area, triangle Figures such as squares and squares have area formulas, so the solution is very simple, but the question is, how to find the area of ​​those irregular figures?"

"For example, in this S-shaped curve I'm drawing now, the area enclosed by this curve needs to be solved, but there is no formula. At this time, how to solve the area enclosed by a curve has become a problem for mathematicians at that time."

"Archimedes found a way, Yu Hua, do you know what it is?"

Hua Luogeng looked at Yu Hua.

"The exhaustion method is to use a familiar figure to infinitely approximate the area enclosed by the curve." Yu Hua replied.

"Yes, the method of exhaustion, proposed by Antiphon, improved by Eudox, and perfected by Archimedes, the idea of ​​exhaustion method is to use an infinite number of familiar graphics to find the area enclosed by a curve. In the history of mathematics, exhaustion The method is regarded as the predecessor of calculus with impeccable rigor."

Holding the chalk in his right hand, Hua Luogeng drew the solution process of the exhaustion method, filling the area enclosed by the S-shaped curve with triangles one by one, and finally obtained the size of the area.

The whole process is extremely cumbersome, but extremely rigorous.

After completing the solution, Hua Luogeng immediately erased the formulas and graphs with a board brush, and then re-written a new concept to find the area through a rectangle:
"The method of exhaustion has been used until the seventeenth century. In the history of more than 1000 years, there is the circle cutting method in my country to find the area, but the calculation is too complicated and not applicable. The limitations of the method of exhaustion are gradually obvious. For the area enclosed by different curves It is necessary to use different graphics to approximate, and the proof skills of different graphics are not the same, which is extremely cumbersome. In this period, "approximating the original graphics with rectangles" appeared in the mathematics world. The idea is consistent with the exhaustive method, and it is simpler. The problem, that's the loss of rigor, which is a very serious situation."

Rigorousness is the soul of mathematics.

Lose simplicity, and math loses a lot of idiots.

Without rigor, mathematics will lose everything.

If a theorem, a formula, a mathematical constant loses its rigor, it means the collapse of the entire mathematical edifice.

Yu Hua listened attentively, and remembered all the key points of Hua Luogeng's explanation in his mind, and his understanding was very fast.

"Newton and Leibniz attached great importance to the problem 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 thing in calculus. The core idea, that is, the reciprocal operation between differentiation and integration, is expressed in mathematical formulas as the Fundamental Theorem of Calculus."

Hua Luogeng had a serious face, and wrote the Fundamental Theorem of Calculus on the blackboard: "Before, differentiation and integration were two separate disciplines. Differential for derivative and integral for area, they are not related to each other. In Newton and Leibniz's work Under the action, a complete system of calculus was established."

Reciprocal operation between differentiation and integration.

This is the core of calculus. So far, the extremely important calculus 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, connecting two separate disciplines, and keenly discovering the reciprocal operation between differential and integral, worthy of being two of the top giants in history.

What is the concept of reciprocal operation?
Simply put, that is the problem of finding area, which can be transformed into finding derivatives, and the problem of finding derivatives can be transformed into finding areas and transforming each other.

If the road of integration does not work, then change from low-dimensional research to high-dimensional research, and use differentiation to solve problems.

If the road of differentiation does not work, then change from high-dimensional research to low-dimensional research, and use integration to solve problems.

In addition, reverse integration can be used to find the area.

If you want to ask what is the meaning of it?
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.

Calculus can find too many things, such as the extremum of differential derivatives.

The extreme value is very important, the flight limit of the shells fired by the cannon, the profit data of a shipment of goods, the closest route from a certain place to a certain place, and so on.

This is the most important tool for scientific research and a mathematical weapon created by humans.

"Of course, the calculus system at this time is not perfect. The infinitesimal problem makes the foundation of calculus unstable. The problem of infinitesimal is to define the limit in a dynamic way. In the process of approximating a quantity to 0, there are countless Real numbers, this would not work, which led to the second mathematical crisis. Later, mathematicians Cauchy and Weierstrass redefined the limit. At this point, the foundation of calculus was finally solid, and later by the French mathematician Lebey The Lebesgue integral of lattice studies, and the end of calculus."

Hua Luogeng slowly talked about the relationship between calculus and infinitesimals, and then wrote a series of formulas on the blackboard. This is the Lebesgue integral:
"When I was studying at Cambridge University in the UK, I was fortunate enough to go to France and met Mr. Lebesgue, which was very profitable. However, I think there is still great research value in the field of calculus in infinitesimal areas. In the future, you can Try this field, calculus is both the foundation of mathematical research and a tool for scientific research, understand?"

"Understood." Yu Hua heard this, nodded, and wrote down a mathematical research direction Hua Luogeng gave him.

Hua Luogeng nodded and said sternly: "After we know what calculus is, it will be easier for us to learn. Next, let's talk about functions, derivatives and limits. How much did you read in the first book?"

"After reading one-third of the part, I understand both functions and derivatives." Yu Hua responded that last night's study time was not long, and he only watched one-third of "Derivatives and Limits".

"Okay, let's start with the limit."

Hua Luogeng heard this, and his eyes showed admiration. He paused and explained in detail: "The limit of calculus is defined as..."

(End of this chapter)