Need to Measure Change
So here we arrive at the central idea. Every modern invention, from rockets to robots to power grids, runs on one idea i.e. measuring tiny changes. That idea is calculus. While talking about Limits and Continuity, we understood that real world is not static. It keeps moving, flowing and evolving continuously. Before calculus, there was Arithmetic to count and compare, Geometry to describe shapes and Algebra to relate quantities. These tools worked well for simple measurements and machines. But there was the limitation that only states could be described with the available mathematical tools and not processes.
Classical geometry was excellent at describing shapes but it had limitations. It could not study motion or questions related to change like how fast, how steep etc. Algebra was a major leap. It introduced symbols, variables and equations. With this we could tell how are two quantities related and the effect of change in value of one over another. We could relate the quantities with equations and solve for the unknow quantities. But algebra too had limitations. Only finite differences in quantities could be compared and not the changes happening in motion. How change is happening at a particular point on a curve (i.e. at a particular moment in a physical system) could not be known. Algebra worked with completed changes and could not work on intermediate changes happening every moment.
Going Beyond Algebra’s Limitation
We move from ‘What is’ to ‘What is happening’. By shrinking the difference between two points to infinitesimally small, calculus enabled calculation of rate of change at an instant, finding slope at a single point, velocity and acceleration at a moment. Suddenly algebraic relationships grew from static to dynamic! The critical questions about change, not just values, could now be answered. Earlier mathematics could only measure average change. Earlier, distance travelled in a fixed time gave the average speed but could not describe the speed or acceleration at an exact second. It is because at a single point, time does not pass and distance does not change. So change at a point seemed impossible to define. But nature doesn’t act in averages. It acts moment by moment in a smooth continuous evolution. So the concept of limits and continuity evolved which showed that when we magnify and focus into time duration to the level of near zero (0.000000……1) second, what the exact picture is. Taking the example of speed, we could now see how the change was happening at near zero interval and know the exact instantaneous speed. Same concept was applied to other physical quantities like acceleration, force, flux etc.
Wow! This ground-breaking concept enabled us to get precision, prediction and control. Change was now measurable! Motion, trajectory and forces could be predicted and calculated. Astronomy evolved by leaps and bounds as with the understanding of gravitational force, orbits could be predicted. Engineering became scientific when humans learned to predict how things would behave actually before building them. Engineers could now calculate and predict the forces in a system and know whether it would work or not. For example, motion of pistons could be predicted for acceleration, inertia and load. Thus engine parts can be sized before manufacturing rather than building on the basis of trial and error. Similarly, vibrations could be analysed, efficiency could be estimated and point of failure in structures could be predicted mathematically. We could finally calculate forces and instantaneous values, we could predict and optimize, make efficient designs and quantify safety. It made engineering predictable and precise. With Calculus, engineers could compute stress variation along beams, predict buckling of columns, analyse bending of bridges, study fluid pressure on dams, control systems, analyse stability and what not. The fact that change can be measured gave rise to new fields of knowledge and application such as fluid dynamics, thermodynamics, electromagnetism, elasticity theory, control systems, signal processing and found various other applications. This made possible the development of autopilots, power grids, industrial automation, robotics and modern electronics among others.
In Part II of this blog, we will discover how Calculus is applied in real machines, technologies, and real-world problems.
Leave a Reply