Introducing Calculus To My SS2 Son

By: Dr. Salihu Lukman

Introduction

As a parent, we should be following up on our children’s educational progress, especially their understanding of the fundamental scientific principles taught in Basic Science subjects at the junior levels or in Physics, Chemistry, and Further Mathematics at the senior levels. My first child, Muhammad, would ask me tons of questions related to the above subjects since when he was in the junior level, some of which I had to research before answering. Today, their Further Mathematics teacher started treating differential calculus. He asked me when he came back home from school, ‘What is this differential calculus all about and what are its applications?’ The popular way to refer to calculus is the notation dy by dx pronounced by reading the letters separately as dydx. I thought about sharing my responses with the public with the hope that someone with a similar question as Muhammad’s could benefit from my simple introduction to calculus. Happy reading!

How It All Began – The Falling Apple Observation

Most of us are aware of Newton’s falling apple story. Sir Isaac Newton (1642 – 1727), the greatest scientific genius the world has ever seen, was in a garden when he saw an apple fruit falling off from its tree. This simple observation sent the young Newton – who was not even up to the age of 26 at the time – to begin to wonder about the nature of gravity. He thought to himself, ‘Does the moon also fall toward the Earth in a similar way as the apple fruit fell toward the Earth?’. He answered in the affirmative after performing thought experiments in which he projected an object on the Earth’s surface with increasing velocities thereby picturing, for the first time, how an object can be placed in orbits around the Earth like satellites and how an object can escape the Earth’s gravitational influence like spaceship. In fact, Newton calculated for the first time the minimum Earth’s orbital velocity as 5 miles per second and escape velocity as 7 miles per second. Back to the falling moon problem, Newton did not have the mathematics to analyze it. So, he invented calculus, almost on a dare, as Neil dyGrasse Tyson (the most famous astrophysicist of our time) would put it.

Newton did not stop from just analyzing the falling moon problem, he went on further to predict the motion of the numerous moons of Jupiter with great accuracy. Newton became so fond of modeling the positions of celestial bodies that he would say, tell me the present state of a system and I will tell you how it will behave tomorrow and the day after tomorrow. Hence, calculus is the first mathematical tool invented for modeling physical systems and that is why it has applications in almost all engineering disciplines.

Gravity

From this apple fall story, Newton understood that the Earth pulled the apple toward it with a force of gravity. He came up with Newton’s Law of Universal Gravitation after formulating the 3 Newton’s Laws of Motion. When these laws were applied to predict the path of Mercury, the closest planet to the Sun, there were small deviations.

Fast forward to the 1900s when Albert Einstein (the greatest genius of the 20^th century) came up with his groundbreaking theories of relativity. The first one was in 1905, called the Special Theory of Relativity which modified Newton’s laws of motion to incorporate objects traveling at very high speeds, close to the speed of light, and introduced time as the 4^th dimension in what he described as spacetime. Einstein again formulated the 2^nd theory in 1915 called the General Theory of Relativity which modified Newton’s Law of Universal Gravitation to describe the motion of objects near a high-gravity source like the Sun and corrected Newton’s understanding of gravity as a pull by heavier objects on smaller ones with a simple space push on the smaller objects whose space is warped by the heavier objects. With Einstein’s theory of relativity, the path of Mercury was more accurately predicted without any deviations than with Newton’s laws because Mercury was near a high-gravity source, which was the Sun. In essence, when you plug in low gravity and low speeds into Einstein’s equations, they will reduce to Newton’s equations. This means that Einstein’s equations are special cases of Newton’s equations for a high-gravity source such as the stars and black holes and high speeds that approach the speed of light.

Conclusion

The foregoing introduction and applications of calculus with specific cosmic examples and historical perspectives could spark an excellent interest in any fresh student of calculus. Science and mathematics teachers should devote so much time and effort to introducing new topics to their students with a view to sparking a passion in the students and making them understand some of these complex principles better. One of my physics teachers at the secondary level, Mal. Ibrahim Physics would introduce all Physics topics to us like what I did above and I can still remember vividly, over 29 years now, his specific practical examples and explanations of physics principles to this day. His explanations created a strong bond between Physics and me to this day.

Salihu Lukman is an Assistant Professor of Civil Engineering at the University of Hafr Al Batin, Saudi Arabia