Fundamentals of calculus Chapter 2: Integration

Fundamentals of calculus

Chapter 2: Integration

Integration is the converse of differentiation. To understand this better, it must be known that during differentiation an important information is lost. Let $\frac{dz(x)}{dx} = z’(x)$. Then, we cannot get back $z(x)$ from $z’(x)$ because, while taking the difference $z(x+\Delta x) – z(x)$ only the information regarding the difference between a point in Z axis and a neighbourhood of that point is retained. But the actual orientation is lost. Throughout this article, unless stated $\Delta x$ is limited to zero. To see how this could cause the loss of a piece of information, let me elaborate this with a simple algebraic example. Consider $a-b = c$ to be true. Then we may write $a – (a-c) = c$, here $b=a-c$.

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.

Calculus Vs algebra:-

-By K Sreram

This article was first written on 2013

Introduction:-

This article was written to show the difference between arithmetic algebra and calculus. Differentiation and integration is far different from arithmetic algebra (though arithmetic algebra is used to represent the process of differentiation, differentiated equations, integrated equations or expressions) because, arithmetic algebra does not support the results provided by differentiation or integration process. Arithmetic algebra shows that the result provided by differentiation or integration process is approximated and the error will never be zero; but calculus says exact values.

I have proved all this in this article. This article also shows us what actually calculus is and how to apply them practically. Differential and integral calculus is a modification done to the mathematics at the time when arithmetic (and some geometric) algebra was prominent. It defines a wrong result (according to the ordinary algebra) to be correct (with proper proofs).

How to make a good chess move

How to calculate chess positions?

“The grandmaster's secret” is what every average chess player will want to know. But unfortunately there is no "secret" for evaluating chess positions with high accuracy. We just need to change the view by which we look at the board while evaluating. But that's also not going to be easy. Only extensive practical experience on applying the methods described in this article will improve the player's playing strength. This article is not written by an experienced rated chess player of any sort, the author just play chess and analyses chess to some degree. And the author does not claim that the methods presented in this article are standardized theoretical methods. 

Analysis on shell sorting algorithm's worse case

Analysis on shell sorting algorithm's worse case

Dear reader,

Before we get to the main part of this post, I would like to share my experience learning this algorithm. I am a student doing my computer science engineering, and this concept of shell sorting is a part of my first year syllabus. I couldn't initially understand why shell sort runs faster than insertion sort, given that shell sort tends to insertion sort algorithm if the separation by which the system does the sorting procedure reaches one. On all cases, it does reach one and the same kind of iterations as of insertion sorting algorithm is found to take place. So this question had bothered me for a two days after since I had come across it. But now, I am completely clarified with that doubt. Because I found it difficult to understand why shell sorting algorithm proceeds faster than insertion sorting algorithm when I was learning this algorithm, I would be able to give a better view on shell sorting algorithm, in a way that you don't face the same problem I had faced.
Yours,
 K Sreram.

Time travel

If time travel to the past were possible (A good solution to the grandfather paradox)

Traveling back in time is one of the most important science fictional concept that most science fiction author wish to exploit. Any story that involves time travel, will always forget to answer certain questions that stand vital for the reader’s understanding of time travelling concept. The main reason to this is time travel concept gives raise to many unique paradoxes that are hard to solve. Solving such paradox might not always seem easy. One such paradox is the grandfather paradox.

Maximize your creativity

Steps to maximize your Creativity easily

Of course, keeping the title of this blog as "Creativity Maximized" and not talking about maximizing your creativity seems a bit offbeat. So in this article I have shared my views on how to maximize your creativity; what I say here are purely out of my experience and does not go beyond that. Usually when we talk about "creativity", we refer to doing something "different" or "out of the box". When do we say something is "out of the box?" Have you ever thought about it? When we feel something seems different, we call it creative.