To many, integral and differential calculus may as well be a foreign language from an alien planet. Many people don’t grasp the fundamental concepts which drive the calculus, and consequently fail to derive the value they otherwise could from that knowledge. I’ve always found the key to making use of some bit of knowledge is to internalize it, to restate the concepts in terms that are familiar and comfortable to you, but to compare this restatement and internalization to the textbooks, data, etc. to make sure sure you’ve truly grasped the concept.

I’ll give an example of what I mean by “internalize and restate it”.

To understand orbital motion (things like planets, moons, satellites), the mathematics to get a precise understanding of the shape and speed of an object in motion (Kepler’s First and Second Laws) can be fairly intimidating. For me, the breakthrough in highschool Physics class was comparing the speed and motion of an orbiting body to paddle ball toy. If you sling the ball part of a paddle ball toy while fixing the paddle part flat on a table, the elastic in the string will cause the ball to move slower the further away from the paddle it gets, and as it returns the elastic causes it to move faster.

Now this example doesn’t compare apples to apples because the elastic force in my example and gravitational forces in the physical world mathematically do not work the same way, but it’s an easy way to visualize the concept. PS: for a neat visualization with computer graphics, see here.

Back to the calculus now: I have found a book from the early 1910s called *Calculus Made Easy*, by Silvanus Thompson. Thompson’s approach to demystifying the concepts in basic calculus are the most straightforward I’ve ever read. He uses simple, relatable concepts to help the reader internalize and restate the problems in a domain that most have a keen understanding of, and as a result, the concepts become much less scary and much more useful! This book is freely available online as a PDF from Project Gutenbergm and a dedicated website. Both linked below for convenience.

Whether you’re learning calculus for the first time, need a basic refresher, or just want to reinforce your understanding of the concepts, I highly recommend giving it a look.

## One comment

Love this! You are spot on about “internalize and restate” being a much more effective method for intuitive learning than memorizing equations. Thanks for the tip Jacob. I’ll be using this to help my kids understand calculus.