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In the previous series, Reviving Geometry we introduced differential calculus to the world: a powerful theoretical weapon that succumbs to every mathematical and physical problem. However, you may feel in relation to the differential count as if you were given the keys to a helicopter after watching training videos. Therefore, we will no longer focus so much on mathematical finesse and formality, and instead look to the field where you will be able to feel and become familiar with the differential number.
From the training hall, we're going to very specific places. In the following chapters, we will address one important science each time that uses differential calculus. Let's start with economics, through physics or biology to... let's be surprised! In addition to problem solving, you should get some insight into the frequent application of differential calculus, through which you can appreciate its usefulness.
Algebra is a word of Arabic origin derived from the word al-jabr: the union of broken parts.
In this series of texts, you often encounter algebra, or mathematical formalism, which is a way to write down and conceptualize known ideas. For example, addition is formalized using the $+$ symbol and multiplication, as a repeat addition, is formalized using the $\cdot$ symbol. With a knowledge of formalism, we can effectively do abstract operations and only then insert a specific meaning into them, e.g. we can quickly calculate $2+4 and only then assume that this was, for example, a financial transaction.
The algebra in our series is not laid bare because it is something vulgar. On the contrary, we carefully avoid working with previously undefined variables, so we do not fall into the frequent trap of using so-called differentials without intuition. The nakedness here is berme in the sense of transparency, of being able to see the true nature of algebraic manipulations, for they are no longer hiding anything.
We've shown the concept of derivatives in Living Geometry on several functions, we even know how to calculate them. By calculating the derivative, we've figured out how big the growth rate of these functions is. In some cases, however, it is not just this rate of growth that is important to us, but also other rates: e.g. the average of the function at a certain interval. We'll show how the average calculation in different situations can be done through integration, thereby giving us a powerful tool to solve all kinds of problems - average sea level per day, average age-related pay, average car consumption, all of which and much more we can calculate through integration. But most of the averages will be used in statistics and probability theory: to do this at the end, we give a short example, but we don't go deeper because of the complexity.
We will also show here what some integration is, which we will need later.
They say that he who treats what also misses. This is certainly not the case in the financial sector, for perhaps no other economic sector has as much money moving around as here. Investors who manage portfolios, banking executives, or scientists who make up the nation's economic models treat unimaginable financial flows. They have such a big responsibility, and that's why they can't shield themselves from differential calculus. Indeed, it provides powerful tools for estimating the future or maximizing profit. So let's try to uncover some of the tricks used in dealing with money.
The concepts of differential calculus that we've been building in previous chapters and in Living Geometry perhaps make sense. For their practical use, however, they need not only to make sense, but to be as easy to use as possible. We need to develop labeling, algebra. So, going back in history for a moment, we look at how the signposting of differential calculus was conceived by the two ``founders'' of differential calculus, I. Newton and G. Leibniz.
The two thinkers have long argued over which of them first conceived the theory, but by today's consensus both developed their theory independently and independently. But because each was working on the theory without the other's knowledge, they both developed their own markings for the same mathematical concepts. The markings we use today combine elements from both origins.
Although the two creators couldn't stand each other, today we can be thankful that there were two somewhat independent views of the same thing. In fact, the mathematical markings we use to make the calculations affect how we think about the issue. If we choose good markings, things will go smoothly and gracefully, if our markings contain ambiguities or we don't fully understand them, we won't get very far. A proper understanding of the handling of mathematical markings and the conformity of those markings with the described reality, that is the essence of algebra that we are dealing with here. And in few places can this be seen as well as the so-called chain rule, which is the subject of this chapter.
In the previous two chapters, we dealt with two specific topics on derivatives and integrals. We took this opportunity to look at some concrete examples, but it was not conceptually revolutionary - we imagine that now. In fact, we will continue our trip to the differential analysis with a field trip to the differential equations. These equations combine both derivative and integration tools, so we're probably going to sweat a little, but for that we can apply our knowledge to hitherto elusive problems.
In the last chapter, we watched a falling stone to sample a differential equation. We describe the movement of the stone in a foreign word as translational, to highlight that it is moving on some trajectory. In contrast, the subject of this chapter will be rotational motion and the related question of coordinate systems.
We mentioned that differential equations are used to introduce new functions. In this chapter, we'll talk about two new, so-called goniometric functions. We will present them independently of the differential equations, but we will clarify the context later.
The concepts we present in this chapter are not modern at all. All the examples could have been solved, or solved, by Newton himself when he started the differential. These are somewhat outdated problems in terms of the evolution of mathematics: while other areas of mathematical analysis are still in bloom, the problems presented here have long been solved by humans and are now taken for granted. But it is ironic that the same problems, solved on paper with sweat on their faces by 18th-century European mathematicians, are now being exploited in a very different field: the development of computer games.
Coordinate systems are absolutely crucial for a computer to know where a player, enemies and game objects are located. Moreover, things are moving, and so the movement equations need to be solved to make things realistic. Of course, computational methods have shifted somewhat over 300 years, and computers solve equations numerically, by brute force. Still, it's important for game developers to know concepts so they don't bring any glitches to the game.
The popular riddle is this: a lily pad appeared on the lake one morning. By the second morning it was two, the third four, etc., doubling each day. On the 10th day, there were already 512, on what day was half? The impetuous man briskly answering that half the water lilies were in the quill on the fifth day is mistaken. As the number of plants doubles each day, half the final count was on the penultimate, ninth day.
For the most part, this gimmick serves nothing but amusement, but from a mathematical point of view it describes relatively accurately the phenomenon of the spread and emergence of life. Imagine, for a moment, that we are in a kind of primordial quill in which, according to scientific theories, life originated, and let us concentrate on the first organism formed-how is this life likely to spread around the world?
In past chapters, we have devoted much time to presenting exponential functions. We found that it is related to the following differential equation: $$ f'(x) = k\cdot f(x) \,,$$
that has solutions in the shape of $$f(x) = C\cdot e^{kx} \,,$$
where $C$ and $k$ are some constants. In fact, we could define the $e^x$ function as a function solving this kind of equation, so much is the exponential and its equation is related. Let's see where this differential equation stands out. We'll also look at another way of constructing differential equations using differentials and Leibniz notation.
Why devote one comprehensive chapter to one differential equation? Indeed, there are an infinite number of possible differential equations that have many solutions. But it turns out that only certain shapes of differential equations exist in nature. Explaining this by hard mathematics requires a knowledge of very advanced differential geometry. But we get a better understanding if we look at the individual examples of differential equations and feel them out, which is the aim of this chapter.
Using exponential function, the growth of the organism population can be described. But there is a big limitation to this model, which is that the number of individuals is constantly increasing. While unsupervised growth can be seen for some limited time at the start of the process, it must stop after a while. In this chapter, we look at another population curve model that describes this behavior. This so-called harmonic oscillator, however, has applications mainly in physics, which we'll get to in the next chapter.
Let's see another significant equation here: the harmonic oscillator equation. It's the second equation and the last equation that we imagine here. But the goal of naked algebra is to show that these two equations are enough to describe a large class of phenomena that we can describe in life. The exponential equation describes unbridled growth (or decline), as wild as life itself. Unfortunately, however, it is so out of control that it has no limit. As an antidote, the harmonic oscillator describes a bound development that is calm, rocking, harmonious. It can also be used to describe small displacements from stability.
That's roughly the nature of the two equations that you're going to see here. From their description, you can sense the potential breadth of their use. As far as describing transition phenomena, between stability and unfettered, or if we want more precision in the description, we would need different differential equations. There is certainly a lot of literature on this, but it is beyond the scope of this text.
So we became familiar with a differential equation that models an oscillating quantity and looks like this. $$ f''(x) = - \omega^2 f(x)$$
has a solution $$ f(x) = A \sin(\omega \, x + \varphi ) \,,$$
where $\omega$ and $\varphi$ are some constants. We named the plot described by this equation the harmonic oscillations. Like with an exponential equation, we can use a harmonic oscillator to define the function sine (and cosine). Here, we'll show where we can meet such a harmonic oscillator. These will now mainly be purely physical, mechanical examples. It's a bit of a shame we don't welcome other industries, but we imagine a principle by which a harmonic oscillator can be found in almost any physical system.
With this chapter, the text on naked algebra comes to an end. Here we have managed to give concrete form to the doctrine of differential calculus that we presented in Living Geometry, on many examples from a variety of scientific disciplines. Although physics dominated the most, this is to be expected, as a differential number directly for it was developed, and many physical applications go beyond their field.
I hope you now have some more or less comprehensive idea of what derivatives and integration and differential equations are. When one becomes acquainted with a new subject, it is often the case that one understands a concept, but is not sure how it is connected to another concept, creating great uncertainty: this text tried to connect everything as much as possible.
I would like to end this text hopefully, which is why I am going to bring up all the topics that did not fit in the original text, albeit interesting extensions. It is no longer an objective for you to learn anything in particular, but rather an overview of all the strange corners through which the study of differential calculus can take place. Therefore, the level of rigorousness will drop a notch from the previous chapters; on the other hand, I will try to replenish everything with resources where everything is explained properly.
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