The practical purpose of mathematics is to measure. One of the oldest examples is triangulation, and another is differential equations. What is the latter for? All it does is the sum of a series of triangulations. Therefore, if you know triangulation, you will be able to know differential equations, which is the so-called learning from the old. However, the complexity of the two is slightly different: the former only takes one measurement, and the latter requires a series of measurements.
Since the differential equation was created by Newton and Leibniz (Figure 0.1), it has been inherited and used by many scientists, and even every discipline corresponds to a differential equation.
Figure 0.1
For example, electromagnetism corresponds to Maxwell's equations, quantum mechanics corresponds to Schrödinger's equation, and even population theory corresponds to Malthus' equation.
In the summer of 2002, several experts from the West came to China to visit and give lectures. What happened was that their lecture topics were either differential equations in electromagnetic waves or differential equations in quantum mechanics. why is that? They replied: Whether it is a mobile phone manufacturing company or a nano research company, they are required to solve these differential equations.
Differential equations also have a personal impact on the lives of the public, such as mobile phones or nanometers, and differential equations are involved in related research.
Some major issues related to the national economy and the people's livelihood, such as population prediction, can be solved by differential equations within a few minutes. Even in the humanities, such as Tolstoy's novel "War and Peace", the interpretation of the concept of history also embodies the idea of calculus. It can be said that natural sciences, engineering technology, social sciences, and humanities all use calculus or differential equations.
Middle school only talks about algebraic equations and trigonometric functions, so what is a differential equation? Although the university has taught it, the general public has only a half-knowledge of differential equations and feels that it is "bottomless". Until one day, when I heard a discussion about "how to measure the height of a tree", it suddenly dawned on me, and a new understanding of differential equations also surfaced. Readers and I are invited to experience this comprehension process together (Figure 0.2).
Figure 0.2
The calculus of Newton, Leibniz or Barrow has long been written in textbooks, but what they write is not equal to what they think. Only they know what they think, and future generations can only talk about their experiences based on their own experience.
— One day, I was walking under an old tree and heard the discussion below.
Tour guide: This old tree grows taller every year, and surveyors and mappers come to measure the height of the tree every year.
Tourist: How to measure the height of a tree? Want to chop down a tree or climb it?
I think: Middle school students know that if they have trigonometry, they don’t need to cut down or climb trees, and they can measure the height of trees only by the slope of a virtual hypotenuse (Figure 0.3)!
Figure 0.3
But at the same time, I also had an epiphany: this is also what a differential equation has to do.
In fact, if we are faced with a mountain, it also corresponds to a "right triangle", but it has a curved hypotenuse, or hillside (Fig. 0.4).
Figure 0.4
We are at a point on a hillside and cannot see far because of limited vision.
At this time, its slope is no longer constant. If it is assumed that the slope of each point (this only involves the local properties of the curved hillside around this point) is known, then the same measurement problem will arise here: can the height of the mountain be measured without going through the mountain, but only by these slopes? ?
This belongs to the hypotenuse trigonometry (because it is based on the hypotenuse triangle), and it actually solves the simplest differential equation: the slope (or slope curve) of each point on the hillside is known, and the height of the mountain (or height curve, Figure 0.5 ).
Figure 0.5
The slope curve and the height curve are combined into one picture (since the slope is the most important quantity in triangulation, record them one by one to become the slope curve).
Figure 0.6
Let the left graph in Figure 0.6 be shrunk into a section, and on a section of the curve, the slopes of all points are almost the same. If we take the slope of the starting point as the slope of this section, and then use it to measure, the height increment of this section ≈ the slope of the starting point × the length of the bottom ≈ the area enclosed by the slope curve after shortening. The sum of the individual measurements is
Total mountain height = area enclosed by the slope curve.
This is the Newton-Leibniz formula.
So comparing differential equations to hypotenuse triangulation, its complexity can be compared with elementary triangulation: they are both triangulations, but the number of measurements is different.
This differential equation is simple (sometimes called the simplest differential equation), but extremely useful. For example, to measure the area of some curved sides, it only takes a few minutes to solve a differential equation. Otherwise, if there is no differential equation or Newton-Leibniz formula, it needs to do countless calculations, which can never be finished, and the efficiency is very different. This is the necessity of inventing differential equations.
In short, the measurement of the height of a tree leads to trigonometry, and the measurement of the height of a mountain leads to a differential equation. It can be seen that reality (measurement) will promote the evolution of mathematics from elementary to advanced.
There are many similar examples in reality. For example, in my country's census in 2000, the entire population was directly counted from door to door, and it took a year to count 1.266 billion. Using differential equations to calculate the predicted value, it only takes a few minutes for a college student to calculate 1.345 billion, which is almost the same. This is a practical example of the necessity of inventing differential equations (Fig. 0.7).
Figure 0.7
That's what mathematics is all about, finding another way, (using slope or growth rate, for example) to get efficiency.
College students solve the differential equation (
) to calculate the predicted value of the census.
We can now explain to the public what differential equations, originally developed in middle school triangulation, do, but go beyond measuring tree heights and can measure the area of many curved surfaces, calculate population projections, and more. To deal with these problems, there is no need to do countless arithmetic directly, and it can be calculated in a few minutes by using differential equations. So if you want to be efficient, you need to learn calculus or differential equations.
recommended reading
"Differential Equations and Triangulation"
Author: Lin Qun
Lin Qun, academician of the Chinese Academy of Sciences, used an example to explain differential equations thoroughly, and appreciate the freedom and charm of mathematics in the transformation of thinking in measuring the height of trees and mountains
