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Mongolia invented descriptive geometry (not to be confused with projective geometry); Fourier's classical study of the theory of heat conduction opened up the modern stage of mathematical physics. Without Monge's geometry, mass production of machines in the 19th century might not have been possible. Descriptive geometry is the root of all mechanical drawing and illustration methods that make mechanical engineering a reality. The method pioneered by Fourier in his work on heat conduction is of equal importance in boundary value problems. Therefore, a large part of our civilization should be attributed to Monge and Fourier: Monge in practical and industrial terms, Fourier in pure science.

Monri Gaspar Monge was born on May 10, 1746, in Bona, France. At the age of 14, Mengri showed his special talents in designing a fire engine. He is a born geologist and engineer with a gift for visualizing complex spatial relationships.

At the age of 16, Monge was appointed a professor of physics for drawing an excellent map of Bona. Later, Monge was sent to a military school to study. In the military school, he was keen on surveying and mapping, which left him a lot of time to study mathematics and successfully solved an important problem.

This is the beginning of descriptive geometry. Monge taught this new method to future military engineers. The problem that used to be as annoying as a nightmare is now very simple. Monge kept descriptive geometry as a military secret for 15 years. It was not until 1794 that he was allowed to teach this method publicly at a normal school in Paris. After listening to a speech, Lagrange said, "I don't know the geometry of painting before listening to Monge's speech."

Descriptive geometry (or perspective geometry) is a technique and theory used to draw and depict objects in three-dimensional space. The basic principle of descriptive geometry is to present three-dimensional objects in two-dimensional form by simulating the visual effects of objects in the real world in the human eye. This technique can help painters create more realistic and realistic pictures in painting.

The three-dimensional or other graphics of the space are now drawn on the same plane by two projections. For example, a plane is represented by its intersection; a solid, such as a cube, is represented by the projection of its edges and vertices. The surface intersects the vertical plane and the horizontal plane; these curves, or the intersection of the surface, represent the surface on a plane.

In this way, we have a method of painting what we usually see in three-dimensional space on a piece of paved paper. It was this simple invention that revolutionized military engineering and mechanical design. Its most obvious feature is simplicity. The discipline is now so well established that professional mathematicians have little interest in it.

In addition, Monge's name is associated with surface geometry, which is an important part of calculus. Mongolia systematically uses calculus to study the curvature of surfaces. In his general theory of curvature, Monge paved the way for Gauss, Gauss inspired Riemann, and Riemann developed Riemannian geometry, and finally provided theoretical support for the establishment of the theory of relativity.

A letter from Napoleon in 1796 began a long and intimate relationship with Mongolia. Napoleon appreciated Monge's conversation and endless interesting information, and Monge was amused by the commander-in-chief's cordial humor. At that time, there was Fourier beside Napoleon.

Fourier Jean-Baptiste Joseph Fourier was born on March 21, 1768 in Auxerre, France. He was a problem child at the age of 13, stubborn and bad-tempered. Then, when he first came into contact with mathematics, he suddenly changed as if he were possessed. In order to study math, he went to the kitchen to collect candles. His secret study was by the fire behind a screen.

In December 1789, Fourier (then 21) went to Paris to send his research paper on the solution of numerical equations to the Academy of Sciences. This work surpasses Lagrange and is still valuable to this day. But since Fourier's method in mathematical physics has surpassed it, we will not discuss it any further.

Napoleon saw that ignorant soldiers would not be enough to fail. To this end, Napoleon founded the higher normal School in 1794. Fourier was hired as a professor of mathematics. This appointment ushered in a new era of mathematics teaching in France. The success of the project exceeded expectations and led to one of the most glorious periods in the history of mathematics and science in France. In the comprehensive engineering school, he livened up the math class with a rare method of citing history, and he skillfully used some interesting practical applications to illustrate abstract problems.

Fourier's "Thermal Analytic Theory" is a milestone in mathematical physics. He submitted his first paper on heat conduction in 1807. Laplace, Lagrange and Legendre are reviewers. While acknowledging the innovation and importance of Fourier's work, they pointed out that there are still shortcomings in mathematical processing and much room for improvement in rigor. Lagrange himself found some special cases of Fourier's main theorem, but he failed to get general results.

By the way, there is a basic difference between pure mathematicians and mathematical physicists. The only weapon that pure mathematicians can use is accurate and strict proof. Unless the theorem cited can withstand the strictest criticism of its time, pure mathematicians will hardly use it. On the other hand, applied mathematicians and mathematical physicists will imagine that the infinitely complex physical world can be completely described by any mathematical theory simple enough for human beings to understand.

This indifferent attitude of scientists towards mathematics for the sake of science itself infuriates one kind of pure mathematicians, just as it angers another kind of pedant by omitting a suspicious small subscript. The result is that only a few pure mathematicians have made important contributions to science. Lord Kelvin ignored the apparent lack of rigour in Fourier's masterpiece on thermal analysis and called it "a great mathematical poem".

As already said, the main research direction of Fourier is the boundary value problem-the solution of the differential equation is suitable for the specified initial conditions, which may be the central problem of mathematical physics. Since Fourier applied this method to the mathematical theory of heat conduction, highly talented people have gone further than he had ever dreamed of, but his step was decisive. One or two of the things he does are very simple and can be described here.

In algebra, we need to draw a graph of simple algebraic equations. But in turn, what kind of equation leads to the pattern of infinitely repetitive segments like the one shown below?

Such figures consisting of disconnected straight or curved segments are repeated in physics, such as the theories of heat, acoustics, and fluid motion. It can be proved that it is impossible to express them with limited and accurate mathematical expressions. But the Fourier Theorem provides a way to express and study this kind of graph mathematically: it is continuous in a certain interval, or there are only a finite number of discontinuous points in the interval, and a given function with only a finite number of turning points in the interval can be expressed as the infinite sum of sine or cosine functions, or the infinite sum of sine and cosine functions.

Now that we have mentioned sine and cosine, let's recall their most important property, that is, periodicity. Let the radius of the circle in the following figure be 1 unit length, and draw the axis at right angles to each other through the center 0, as in Cartesian geometry, marking that AB is equal to 2 π unit length, so that AB is equal to the circumference of the circle in length.

Let the point P start from An and trace the trajectory of the circle in the direction shown by the arrow. Draw PN perpendicular to OA. So, the length of AOP NP anywhere is called the sine of angle AOP, and ON is called the cosine of angle AOP; NP and ON take their symbols as they do in Cartesian geometry (NP takes a positive sign above OA and a negative sign below OA; ON is positive on the right side of OC and negative on the left side of OC).

No matter where P is, the angle AOP is part of four right angles (360 °), corresponding to the part of the arc AP on the whole circumference. So we can mark the corresponding proportion of the arc AP in 2 π along the AB to represent these angles AOP. In this way, when P is at C, it passes through the entire circumference of 1x4; therefore, corresponding to the angle AOC, we have a bit K at the distance from point A to 1/4AB.

At each point on the AB, we draw a vertical line segment whose length is equal to the sine of the corresponding angle, and determine whether the vertical line segment is above or below the AB according to whether the sine is positive or negative. The end of these vertical segments that is not on the AB falls on the continuous curve shown in the figure, that is, the sine curve. When P returns to point An and starts to rotate again along the circle, the curve repeats after B, so that it is infinite. If P rotates in the opposite direction, the curve repeats to the left. The curve repeats after a 2 π interval: the sine of the angle (in this case AOP) is a periodic function with a period of 2 π.

It is not difficult to find that sin2x is twice as fast as sinx through a full cycle, so the graph of a full cycle is half as long as sinx. Similarly, the complete period of sin3x is 2 π / 3, and so on. For cosx,cos2x,cos3x, … It's the same thing.

It is now possible to roughly describe Fourier's main mathematical achievements. Within the limitations related to "discontinuous" graphics that have been mentioned, any function with a clearly defined graph can be expressed by the following type of equation:

The ellipsis indicates that the two series continue indefinitely according to a certain rule, and when any known function y of x is known, the coefficient

That's for sure. In other words, any known function of x, such as f (x), can be expanded into a series of the above type, namely trigonometric series or Fourier series. All of these are true only under certain constraints, which are fortunately not very important in mathematical physics; the exceptions are cases that have little or no physical meaning. Once again, the work of Fourier series is the first great achievement of boundary value problems.

The concept of cycle described above is of obvious importance to natural phenomena. Tides, lunar phases, seasons and many other things that people are familiar with are periodic in nature. Sometimes a periodic phenomenon, such as the recurrence of sunspots, can be well approached by the superposition of a certain number of simple periodic patterns. Therefore, these phenomena can be reduced to some independent and simple periodic phenomena, and the original periodic phenomena are composed of them.

In the real world, many natural phenomena are periodic, such as:

Astronomical phenomena: lunar phases, solar eclipses and annual seasons are all periodic, which can be described and predicted by periodic functions.

Mechanical vibration: vibration refers to the periodic reciprocating motion of an object around a fixed point. The study of mechanical vibration can help people understand the motion law and stability of mechanical equipment, and how to adjust and control mechanical vibration.

Electrical signals: electrical signals are also periodic in the process of transmission, such as alternating current and digital signals. Because of the periodicity, the signal can be divided into different time periods in the process of transmission, thus improving the transmission efficiency of the signal.

Biorhythm: many physiological phenomena in the human body, such as heartbeat, breathing, sleep and so on, are periodic. Understanding the laws of these biological rhythms can help people better maintain their health.

When he finished the work he began in 1807 in 1822 and collected it in monographs on heat conduction, it was found that the stubborn Fourier had not changed a word of the paper he had originally submitted. He died of a heart attack on May 16, 1830 at the age of 63.

This article comes from the official account of Wechat: Lao Hu Shuo Science (ID:LaohuSci). Author: I am Lao Hu.

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