Series of articles for disposition in pdf!
Imagine a little kid coming up to you saying he wasn't very good at school, couldn't possibly understand her, and was probably resigning. To such a confession, one might say that it is not so, that he will surely learn, if he persists in his diligence, that the endeavour will eventually bear fruit, and that he will grow to enjoy it. We live in an educated society where school is supposed to be a game and everyone has a chance at education. But what if the weeping child had come with a complaint about mathematics, not patriotism? In this situation, it has become commonplace to reply in the style of \u201cOh yeah, I couldn't do the math either, don't worry about that,\u201d or \u201cDon't worry, someone in math just doesn't have the cells, you'll get through somehow.\u201d It has simply become a common practice that math, a subject to be hated, has fallen to some and not others.
But I don't believe in such fateful mathematics, I think that - like Johann Strauss's \On the Beautiful Blue Danube\ or the painting \Starry Night\ - mathematics has a kind of universal beauty to it. But revealing this beauty requires the right frame of mind and the right insight. So I find it a shame that in our schools, teachers tend to focus on the practical, procedural part of mathematics that doesn't tell the real story.
It seems a pity, then, that the chilling elegence of mathematics, which transcends the world as we know it, yet our understanding ties it to human evolution and makes it human, remains undetected. So I decided to write about my favorite part of mathematics - differential calculus or mathematical analysis or calculus (calculus in English, Infinitesimalrechnung in German). I intend to make this subject accessible to other people and thereby bring about an understanding of the mathematics behind most of today's technology, which is at the beginning of the central European evolution of science as we know it.
You don't know what it's all about when you say differential calculus. In the introductory article, I will explain this concept while describing for which areas of human life it is appropriate. So you can test it to see if this series of articles is for you. However, a knowledge of mathematics at the level of the end of grade 2 in primary school is enough to understand the articles.
I'm a physics student at MFF UK, so naturally I have a slightly distorted view of mathematics. I'm not saying that everyone must like her or that you couldn't live without her, I understand when there are more interesting things in life for someone. On the other hand, I think that if one has any interest in mathematics, then one should not hesitate and laze. Yes, mathematics requires a comfort to absorb. It is, after all, a little bit about art, understanding mathematical notation, and looking at the world. And no art is free, either in the sense of art as an ability or the other.
There are plenty of popular media on which you can learn about mathematics and its interesting applications in physics, such as youtub channels or TED lectures. But will watching them really give you what you want (that is, knowledge [of mathematics])? Not from my point of view. By passively watching population lectures, you may learn the cool facts and feel you're in the loop when it comes to mathematical knowledge, but you never really try math, never really get into the really interesting stuff. I will endeavour to include what I find amusing and revealing. Problem solving and building thought structures. And hopefully you'll actually learn something in a series of articles.
I have broken down my article on calculus into several chapters. Each discusses one closed problem more or less from real life, but always an unusual one. To solve this problem, we need to build new mathematical tools, which form the main point of the chapter. The whole series then tends towards a poetic fundamental theorem of differential calculus, which shines a new light on previous articles and opens up a lot of new perspectives on problems from physics. I don't anticipate much mathematical knowledge throughout the series, so perhaps anyone can understand the explanation. Mathematical ignorance may also be an advantage, for one can accept new ideas without worrying about previous misconcepts. Then, at the end of the series, as an addition beyond the anticipated knowledge, I attach derivations and integrals of some functions (I'll explain what this is about) together with a clear table.
To understand abstract concepts of mathematics, it is sometimes important to link abstraction to familiar situations from everyday life. In particular, it is often most helpful to visualise mathematical ideas, that is to associate them with pictorial perception. As an example of this idea, we will now envision the Pascal Triangle, which will also be used for later interpretation. Later, and because of it, we deduce, for example, the sum of the first hundred natural numbers or the first ten second powers of natural numbers.
In the last chapter, we looked at noble mathematical ideas and famous people. Now let's look at the more practical side of mathematics. Have you ever had it rain on you, but when you moved on, did you get out of the rain? Probably not, but it would be magical if you could find yourself at the edge of the rain. Why doesn't this happen very often? Is it worth buying a bigger pizza at a pizzeria? I will try to answer these questions from everyday life in this section. What they have in common is that they both touch quadratic function. Perhaps you can get an intuition about this function, and I'll use that to introduce another of the core concepts of differential calculus -- derivative.
The calculus I'm aiming to build in this series has its origins in antiquity. Archimedes was one of the first people to brush up on some concepts of this discipline with his ideas. This Greek Sicilian philosopher, whom we would now consider more of a mathematician and physicist, has dealt with many problems. For example, he devised his own numerical system for counting large numbers, he designed the battle machines with which to sink enemy ships, he formulated the law of buoyancy, but what we'll be interested in is that he calculated the area under the segment of the parabola. Calculating the area under a curve is one of the fundamental functions of a differential count. Later, we will look at this more broadly and in more depth, showing what use we can find for this role. In the meantime, however, let's look at this role in a similar way to Archimedes -- purely as an interesting mathematical problem that has a different character than calculating the area of ordinary geometric formations as $n$-angles.
In past chapters, we have dealt with old knowledge from Pascal or Archimedes. Now we jump to more recent times and zoom in on how devices that no engineer can do without work: calculators. It is possible to imagine that calculators can easily handle addition and multiplication, but what if we want to calculate, for example, square roots or logarithms? We will at least partially answer that question and become familiar with the key notion of differential calculus: derivative.
In this series, I'm trying to convey an intuitive understanding of differential calculus, so I'm deliberately not completely mathematically accurate. Still, a little more pedantry is needed in this part, especially with regard to the notions of limit transition and derivative presented in the last chapter. So let's talk about them in more detail and back them up with more solid reasoning and logic so that we can really use them in square root estimates. Surely knowledge of these concepts will prove valuable in the way ahead.
We're in the final episode of the series' journey beyond calculus. It's time to look back at the mathematical concepts we've encountered along the way and meet them together. Specifically, we will reveal the relationship that exists between the recently introduced derivative and the integration that I roughly outlined in the chapter on Archimedes. It turns out that concepts as different as calculating the area under the curve and calculating the rate of change of function share very much, their relationship being mathematically called the fundamental theorem of the differential. Once illuminated, this sentence will also shine a number of physical applications of derivative and integration onto the surface. A door full of unprecedented possibilities will open, and we will, unfortunately, no longer be able to pass through it. But at least the right soil will be prepared for passage.
For all of those who are already a little bit oriented on the importance of derivative, integration, and how they relate to each other, I will address more practical matters in this appendix. I'm going to derive the derivatives of a few basic functions, and these are going to produce some integrals (because integration is the inverse of the derivative). All the results are then summarised in a clear table. At the same time, you can use derivation to verify that you understand everything correctly.