๐Ÿ”ญ The Art of Calculus: How Math Transforms Our World

๐Ÿ”ญ The Art of Calculus: How Math Transforms Our World
Photo by Graham Holtshausen / Unsplash

๐Ÿ‘‹ Hey Friends,

last week we have discovered how incredible talent, hard work and astonishing creativity can lead to changing the world by creating revolutionary products with the example of Steve Jobs.

And although I was originally planning to continue talking about lesson I have taken from his biography I have just recently finished reading "Infinite Powers" by Steven Strogatz and I have been completely fascinated by his writing and the history and concepts of Calculus. So sit back, relax and enjoy traveling to time and discovering the language of the universe with me.

๐Ÿ“– Introduction

Steven Strogatz is a brilliant and creative US American mathematician who studies and teaches about the mathematical concepts of chaos at Cornell University. He is known for his fabulous teaching skill, his very fitting humour and his bestselling writing. In this book, named โ€žInfinite Powersโ€œ Steven Strogatz brings us back to the beginning of mathematics, draws us through the history of development in both math and physics and dialects the language of mathematics to the incredible concepts of the universe. He does all this while specifically specialising and concerning us about one of maths most famous, most complex but probably also most helpful topics, Calculus.

๐Ÿงฎ Calculus and it's simplicity

Calculus itself, so the study of infinitesimal small values, began as a dialect of transferring classical concepts of known shapes such as triangles and squares to the concept of curved shapes such circles, ellipses and even higher polynomial functions but this happened much later.

Through his talented story telling, his brilliant explanation and his wonderful insightful description Steven Strogatz delivers us the incredible historical evolution of calculus. He uses his book not to concern the reader with Calculus as being a subject of endless complexity but rather a subject with brilliant simplicity because thatโ€™s what Calculus essentially is actually about.

๐Ÿง  The main concept of Calculus

The main concept of calculus, as described in the book is to chop a bigger problem in to many smaller infinitely small pieces, than solve all of these infinitesimal small problems to only put them together in the end which results in final solution for the renounced bigger problem. And this is completely analogous to nearly all problems included within the greater concepts of calculus. When evaluating either an derivative or an integration we approach the problem with infinitely smaller problems as the considered interval \( \Delta x\) approaches zero, also known as limits. These infinitely smaller problems are now more simple doe solve since there desired property is just based on linearity which is a concept mathematicians know how to deal with for multiple millennia now.

๐Ÿ•ฐ Differential Equations๏ธ

It also applies to different types of changing equations known as differential equations. Instead of letting some desired interval \( \Delta x\) approach zero we now take a look at a specific infinitely small time interval \( \Delta t\). When we let this interval approach to zero we have can take a look at the rate of change, otherwise identified as the derivative, at a single point in time. We discover how this is actually paradoxical later within this article. A famous well known example of a so-called ordinary differential equation would be the connection of force, matter and acceleration \( F = ma\).

But why even is this a differential equation? As said before differential equations are equations where we take into account the change of time. This means that there evaluation gives us some sort of calculated output mostly at some certain statements. The shown specific differential equation can be solved by taking into account that the acceleration \(a\) is just the derivative of velocity \(v\) and this again is just the derivative of distance also denoted as \(s\).
Expressing this in a formula we now that the vectorial force \(\vec F\) is equivalent to mass times the second derivative of distance with respect to time, formally known as \(\vec F= m \cdot \frac{\Delta^2s}{\Delta t^2}\). To solve this oddly looking equation we have to evaluate its derivative twice.

๐ŸŒฑ Calculus within nature

All of this already could frighten most of us since its oddly looking construction and formalism redirects us back to complexity instead of simplicity. But how is this actually simple you might ask. And of course the true answer probably is that it isnโ€™t that simple after all. But when putting the formalism aside and just thinking about that limiting values of either \( \Delta x\) or \( \Delta t\) calculus always comes pack to its greatest concepts, the idea of breaking down a problem into such infinitely more smaller and easier problems to eventually come up with a solution for the big problem from the start. This really is what sticks all the way at the very, very roots of calculus and its actually what defines most of reality. Its the study of the infinitely small and big which makes reality seem so incredibly wonderful, frightening but also fascinating at the time. What we defined as the rules of mathematics directly implies to the conducted rules of nature itself.

๐Ÿช Calculus as the language of the universe

Finding these connection through the study of the infinite seems paradoxical at first but makes total sense when you think about the weirdness of change. Describing the change at one instance in time is not only nonsense but also the solution to finding a right way of describing the laws of nearly anything around us. Nature, the universe and us within it is written in the language of change, its written in the language of calculus.

I wish you all the best and see you on sunday with episode two of this new series where we talk about paradoxes of calculus and a little about geometry...๐Ÿ‘‹


Victor Thierling (@observethecosmos)

๐ŸŽฅ Youtube Video

The world of complex analysis: Complex Numbers

Embark on a fascinating journey through the depths of Complex Analysis. Get ready to uncover the hidden secrets of complex functions, understand the power of holomorphic functions, and broaden your mathematical knowledge. Join me and explore the beauty, elegance, and utility of Complex Analysis, and gain new insights into mathematics and its applications. This first episode covers most of the basics but be ready to dive into the details very soon...

๐Ÿ“ธ Insta Post of the week

โœ๏ธ Quote of the week

"One of the pleasures of looking at the world through mathematical eyes is that you can see certain patterns that would otherwise be hidden."

-Steven Strogatz

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