Life's Hard. But Here's Calculus Made Easy
Considering how many fools can calculate, it is surprising that it should be thought either a difficult or a tedious task for any other fool to learn how to master the same tricks.
Some calculus-tricks are quite easy. Some are enormously difficult. The fools who write the textbooks of advanced mathematics — and they are mostly clever fools — seldom take the trouble to show you how easy the easy calculations are. On the contrary, they seem to desire to impress you with their tremendous cleverness by going about it in the most difficult way.
Being myself a remarkably stupid fellow, I have had to unteach myself the difficulties, and now beg to present to my fellow fools the parts that are not hard. Master these thoroughly, and the rest will follow. What one fool can do, another can.
Thus begins the book:
CALCULUS MADE EASY
By Silvanus P. Thompson
If I define my affinity towards mathematics as a function f(x), then f(x) drawn on a graph will begin from the origin with a steady negative slope: math was waterloo for a major part of my schooldays. Then around my grade X and XII, when I actually put in effort to try and understand heavy-weight math textbooks, f(x) attains a stationary point - a local minima, you could say. After that, I really tried to change my attitude towards math - "Why can't I do it? There must be a better way to understand this." A positive slope finally!
I gave up on math textbooks in college and found help from YouTube channels like 3Blue1Brown, Tibees and ZedStatistics.
But unlikely help came from the Mathematics shelf in the Allied Sciences section of the TNAU Library - a small hardbound book CALCULUS MADE EASY by Silvanus P. Thompson (139375). Written in 1910, even before the First World War, this old-timer has always been in vogue. Its hyperlinked version is freely available to read on https://calculusmadeeasy.org/ and on Project Gutenberg (oldest digital library) too. The copy I borrowed from the library was printed in Great Britain in 1961 and has been available for loan since 7/7/1993.
Now we all know the basic rules of differential calculus and integral calculus. We memorised them by heart in school. You probably will be able to say with zero hesitation that d/dx of sin(x) is cos(x) in the middle of the night if someone asks you.
But sometimes, it feels good when a teacher or a book considers you a total beginner or even a complete idiot and explains things from the scratch. You suddenly grasp the roots of things you had taken for granted. It also gives you extra confidence to teach the same stuff to a peer. That's what Mr. Thompson does to you in CALCULUS MADE EASY!
Being bite-sized and accessible in writing, this book is a great read for anyone who has got anything (or even nothing) to do with mathematics. It has 23 chapters and here, I have picked out my takeaways from each one.
I. To deliver you from the preliminary terrors
Deals with the "dreadful symbols" d (a little bit of) and ∫ (the sum of).
II. On different degrees of smallness
Provides intuition to the differentiation rules by identifying the values that are so small that they can be ignored.
III. On relative growings
Introduces vital concepts jovially: "differential coefficient of y wrt x" is dy/dx, implicit vs. explicit functions and dependent vs. independent variables. SPT goes to extreme lengths, look below!
IV. Simplest cases
Why is d(x^n)/dx = nx^(n-1)?
V. Next stage. What to do with constants
This mystery is solved here and I'm not giving any spoilers! Who even would have thought that a calculus text can creep into the mystery genre!
VI. Sum, differences, products and quotients
Reveals the cogs behind the obvious operations we use obliviously.
VII. Successive differentiation
VIII. When time varies
Introduces "rate" and velocity and acceleration in calculus terms. More like a Physics chapter.
IX. Introducing a useful dodge
Put ugly function of x = t and solve! Everybody lived happily ever after!
X. Geometrical meaning of differentiation
Here we see curves of different functions and how differentiating the function gives its slope or gradient.
XI. Maxima and minima
XII. Curvature of curves
The rate of change of the slope itself is explained as f''(x).
XIII. Other useful dodges
Partial fractions.
XIV. On true compound interest and the law of organic growth
Simple interest vs. compound interest. e = 2.71828..... "a number never to be forgotten". Logs and laws.
XV. How to deal with sines and cosines
You wouldn't find a more eloquent explanation for why d/dx of sin(x) is cos(x).
XVI. Partial differentiation
XVII. Integration
Sum of infinite series.
XVIII. Integrating as the reverse of differentiating
Why do we put that +C? Multiple integrals.
XIX. On finding areas by integrating
Definite integrals.
XX. Dodges, pitfalls and triumphs
Integration by parts.
XXI. Finding solutions
"Integration is an art. As in all arts, so in this, facility can be acquired only by diligent and regular practice."
XXII. A little more about curvature of curves
Curvature is inversely proportional to the radius.
XXIII. How to find the length of an arc on a curve
Last chapter, with a farewell message to the reader along with a tabulated gift:
All of these chapters have exercises and all those problems have answers at the back.
One thing that absolutely stood out for me in CALCLULUS MADE EASY is that it talks to you. Mr. Thompson literally addresses you, and his voice has been teaching calculus for more than a 100 years now!
So go ahead and grab this book! Life is hard but understanding calculus doesn't need to be!
