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Simplicity is the ultimate sophistication.” — Leonardo da Vinci
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Wednesday, 15 July 2015

Fundamentals of calculus Chapter 1: Differentiation.

Fundamentals of calculus
Chapter 1: Differentiation.

Can we really call calculus an art? Moreover, can we call mathematics an art? Most mathematicians will agree to the answer: yes. We express ourselves, our creativity through mathematics. It’s not always about calculation. Read my article why do we need mathematics?’ to better understand what I believe mathematics is really. I will not agree that the explanation I had given there was enough. But anyway, I believe mathematics is an art because it is not based on mere calculations but based on creating new rules, using them to express what is around us in a more precise and defined manner. Most of the time, mathematicians would define these rules not for the sake of answering questions that follow up with our reality. But they just do it, in hope that their discovery might come in handy to some scientist in the near future. Some don’t even care about its necessity and for the sake of not expressing themselves to be someone ignorant of the reality most people scramble through physics equations or some other kind of scientific material to find the right place for their equations or mathematical rules to fit in and prove to be beneficial.

So why would anyone want to create mathematical formulations that are seldom being useful practically. Not just creating, most spend a large part of their life for the sake of mathematics. The answer is simple, because it’s an art. An artist does his work for the sake of expressing himself and for nothing else. If you have gone through my article ‘why do we need mathematics?’ you may feel that what I say here is contrary to what I had said there. But always remember, the view point matters. There in that article, I wanted to show how mathematics can be useful practically. But here in this introduction, I want to tell you why many consider mathematics to be an art. Mathematics is beneficial, but it’s not necessary that it was created for the sake of being beneficial!   

Now let’s go to calculus. The first part of calculus is differentiation. Every good textbook that teaches calculus start with differentiation; the reason to this will seem obvious as we go on. But to quench the curiosity of the readers, let me tell you why we start with differentiation. It is because “It’s easier to analyse what we have at hand and to know its working mechanism rather than to plan about building something bigger with it and trying to predict how it would behave once it’s built”. Differentiation and integration are the two operations which calculus is all about. Differentiation is the art of splitting the equation we have at hand into a smaller unit to study a particular property of it. This “particular property” is the rate at which the original equation changes for each change in its dependable variable. Confused? Don’t worry, I’ll elaborate further.   

Basics of differentiation:
Let’s say we have an expression $z = x^2$. It is obvious that $z$ is a function of $x$. We can infer one more property, $z$ never becomes negative when we deal only with the real numbers. So how fast does $z$ change as $x$ changes? We can plot a graph from it, taking $z$ as the Z axis and $x$ as the X axis. Now let’s say we want to find the value $z(b)$. The value $z(b) = b^2$. Now assume there to be a number $c$ such that $c > b$. So imagine $b$ gradually increasing to reach $c$. So at each point of time how do we express how fast $b$ moves to reach $c$? One obvious method is the difference method. As $b$ increases, let’s say that at each instant $b$ changes from being $b_1$ to $b_2$ and to $b_3$ and so on. So we may write, $b_1>b_2>b_3> … >b_n$. Each time we would want to find how closer it is to $c$. So we may write $z(c) – z(b)$ at any instant gives the distance between $a$ and $b$. For simplicity, let’s consider $b_2 – b_1 = b_3 – b_2 = b_4 – b_3 = … = \delta x$. So the distance at any instant $i$ can be written as, $z(c) – z(b_1 + \delta x \times i)$. So if for $i$ instances the distance covered can be written as $z(c) – z(b_1 + \delta x \times i)$ then the average speed can be said to be $\frac {z(c) – z(b_1 + \delta x \times i)}{\delta x\times i}$. Try working around with the $x^2$ example. $\frac {(x+\delta x \times i)^2 – x^2}{\delta x \times i}$ is the average rate of change for the duration $\delta x \times i $. We can find that for different durations we have different results.  

So what is the shortest possible such duration? For every value $j$, it is obvious that there is another value $j/2$ which is lesser than $j$, assuming $j$ to be positive and belonging to the set of real numbers. So the shortest possible duration is zero and we have no other choice! So how do we find the rate of change at a point? If we use zero, then our calculations would become undefined. If we use a real number, there should always be a number lesser than one. So let’s extend the existing rules to create a new rule which breaks all the old ones! I have talked about this “extending the old rule-set to a newer ruleset while coming to calculus from algebra” in the article: calculus vs algebra. Ok the newly extended rule is, lets define $\lim_{\delta x \to 0}{\delta x}$ to hold the property of both the property of zero and a property of a finite real number! If you are a physics student, you would get remained of the particle-wave duality. This kind of necessity to introduce the “hybrid property” is prominent in most areas, especially if the older rules fails to explain something. It’s more like an “exceptional case”. So how do we implement this rule? We all know that $\frac {0}{0}$ is undefined. So now our first extension is to define $\lim_{\delta x \to 0}{\frac {\delta x}{\delta x}}$. Now we use the property of any common real number to cancel off the $\delta x$ in both the numerator and the denominator to get 1. But the other interesting part is $\lim_{\delta x \to 0}{\delta x} = 0$.

So for a case where substituting zero in place of $\delta x$ causes the expression to be undefined, we “solve” the expression first and see if it comes to a definable point. If it does come, like the above example, we simply assign the value zero. So this way, $\delta x$ inherits the property of both zero and a perfect real number. So why is this kind of a definition necessary?  

The most common reason is, we actually choose $\Delta x$ smaller than any system can measure. If the least count of a measuring rod is $l$, then it is obvious that $\Delta x << l$ but $\Delta x > 0$. So how does such a value behave with respect to the measuring system? $\lim_{\Delta x\to 0}{\frac{a\Delta x}{\Delta x}} = a$. Now if we put $\Delta x$ as absolute zero, we have both the numerator and the denominator to be zero and ultimately an undefined expression. Why is $\frac{0}{0}$ undefined? We define division of two numbers the following way. If $\frac{a}{b} = c$ then, $c$ is the total count of the number of $b$’s that can be subtracted from $a$. Here, $a$ and $b$ have well defined magnitude. So when we talk about length, it should be the distance from a point on the number line to another point. So this “first point” is zero and the second point is the value $x$. Now the quantity $x$ has some magnitude.

Now what happens if $x$ is zero? We have a magnitude less quantity. So obviously dividing that by itself is not defines. What about multiplying by it? That’s defined! It only means that we want an expression that is “no magnitude” times “some magnitude”. So this is obviously again “no magnitude”. If we write $a\times b$, then if $b$ is one we get $a$ as the result. If $b$ is something lesser than 1 then we get a value lesser than $a$ as the result. A value $b$ lesser than 1 is a fraction that looks like this: $\frac{b}{1}$. So if  $b$ becomes zero, then we retain no part of $a$. Because zero has no magnitude, it can’t divide any quantity.


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