Infinitesimal | Mathewelten | ARTE

The world of mathematics is exotic and extremely strange. A strange language is spoken there. There is talk of home office collectors or transferred numbers, but there are also epic kingdoms, impressive and sometimes even very useful ideas. So let's go on a little trip, all you need is a tour guide, it's me and a brain ready to go and don't worry, we won't actually do any calculations, just look at the scenery a little. Velocity is such a common term that we have almost forgotten a mathematical concept in it. You see, until three or four centuries ago there was no such thing as speed at all.

At least it was very poorly defined. Take ancient Greece as an example. Back then, mathematics was still part of philosophy. The world described by the Greek philosophers was geometrically perfect and completely static, it existed From regular bodies to absolute constants and eternal truths, movement did not fit into the picture, it went so far that the philosopher 10 and Elijah in the 5th century BC. They invented a series of paradoxes that were supposed to prove that movement is impossible. Zeno said that with a flying arrow we have the impression that it is moving but when we observe it at a certain moment in time we notice that it occupies a very specific position in space and becomes motionless.

A period of time is made up of a sequence of individual points in time at which the arrow stops and the arrow moves forward. Not really, but the movement is an illusion that only takes place in the eye of the beholder. This paradox shows that the concept of speed is not self-evident, so be careful. Even in the Middle Ages, speeding could be cited as ten, but since the Renaissance, the concept of movement has been changing little by little, the idea that this is possible has been introduced into the world of mathematics, which is called Infiniti calculation of times and we owe this innovation to a certain Isaac Newton on this occasion for Toni I who Newton never had an apple fall on his head but he said that An apple falling from a tree served as a source of inspiration for his theory of gravity.

In the 1600s and 1960s, as a student at Cambridge, Newton learned everything that was then called natural philosophy, mathematics, physics, life sciences and astronomy. He was particularly interested in the motion of the planets and suspected that A A deeper understanding of the motion of an apple thrown into the air might be a first step toward understanding the motion of celestial bodies, but before we understand something we have to describe it. How can this complex movement be explained mathematically? To do this, Newton invented a new mathematical concept that was later called derivative. Let's look at this apple from a mathematical point of view.

We draw a graph that assigns the height of the apple to each time t. The movement of the apple is described by this parable. , which represents the height of our apple as a function of time and its speed. To calculate it, we write down its position at a time t1 and then at a time t2. Speed ​​is the relationship between the distance traveled and the time elapsed. It corresponds to the slope of this line that connects our two points on the graph. The horizontal line is the velocity equal to zero the more steyler it becomes, the greater the velocity, if t1 and t2 are far apart from each other, we obtain a velocity average but we are looking for the instantaneous velocity, the closer t1 and t2 are to each other the closer our measurement will be, but as soon as the two points merge the velocity is meaningless like in Zenon so it is about getting infinitely close to a moment without ever reaching it.

What is obtained from the infinite approach of our two points in time is the tangent to our curve and the slope of this tangent is what is obtained under the current speed. the velocity starts out very high, hence the steep slope, then steadily decreases until it reaches zero and finally changes direction and becomes negative, since we now know the velocity of the apple at any time, we can calculate the graph of this velocity of drawing at the beginning. is maximum then decreases uniformly, passes through the zero point at the moment the apple's trajectory reaches its peak, and increases negatively, meaning it continues to decrease.

We note that the velocity curve is simpler than the position curve. describes a parabola, the speed a straight line. That is exactly the concept of the derivative. If it is the function that describes the movement of the apple, we assign a new function f dash to its derivative, which at any time is the instantaneous rate of change of ms and the best part is that this concept of differentiation can be applied not only to apples and pillars but also to many other things. We can calculate the derivative for a variety of functions. Differential calculus is now used everywhere from physics to economics to aviation to epidemiology.

To understand the movement of the planets, Newton was inspired by apples. Apples led him to invent a new area of ​​mathematics that would later be called analysis. It is thanks to him that Science took a big step forward at that time, but unfortunately no one knew about it except a few friends. Newton waited more than 20 years until he published his mathematical results and when he finally did so in 1693 he discovered that someone else had already beaten him to it. and that other person was Gottfried Wilhelm Leibniz, a German philosopher and mathematician. In short, the greatest scientists of the time corresponded with a competitor.

The controversy between the two lasted for years. The English accused Leibnitz of plagiarism and the Germans considered Hughton. , who had arrived late, he was a sore loser. But the dispute also had a philosophical dimension: Leibnitz believed that celestial bodies move because they are driven by an invisible but material current. Newton was the movement of stars and apples. as a result of a mysterious force of gravity independent of the world of matter and mechanical connections. Leibnitz was actually the material. Like Newton, who despite his revolutionary ideas remained anchored in the ancient world with one foot, this also demonstrates his passion for alchemy, to which he dedicated much of his life without obtaining significant results.

In short, there are two candidates for the fatherhood of differential calculus. We would have to describe them both in detail to do them justice, but I don't have time for that because our current topic is speed. Thanks to derivation, we can calculate the speed of apples and convertibles, but also the interest on our savings accounts, the speed of spread of an epidemic or how quickly our coffee cools down. and we owe all that to an apple and two intelligent minds. Speaking of apples, going back to our curves from the position curve, we have derived the velocity curve, but why should we leave it at that?

Why don't we form the derivation? of the derivation of it then the so-called second derivative, let's try it. The velocity curve is a straight line, so its tangent merges with it and the slope of the tangent also remains the same. The curve that represents the derivative of the velocity with respect to the current is a horizontal line that corresponds to a constant and The derivation of velocity also has a name in everyday language: acceleration. The complex motion of the apple is actually the result of a constant acceleration that all terrestrial objects are subject to gravity, the meanings of which the Jews should show in the following years.

That is the magic of derivation that allows us to discover the laws that hide behind the complexity of reality. Behind the movement of objects and trajectories, the always same gravity of the earth makes it visible. In a way, after all, Elijah's Zeno was right. If we look closely, the constant is hidden behind the movement.