Differential and integral calculus are two of the most important mathematical discoveries of the last few centuries. Since the work of Isaac Newton in 1665, through Lagrange and Cauchy, to formalize this branch of mathematics—calculus has served a crucial role in most fields of human inquiry. Anyone who’s done even basic geometry and algebra would appreciate the importance of calculus in the natural sciences and the more sophisticated forms of physics.
The question for us is, at the moment, what exactly is calculus? You can further ask whether there are limitations to the techniques and how they can be improved if at all.
What is Calculus?
There are two primary forms of calculus—integral and differential calculus that are concerned with different mathematical problems. Differential calculus is associated with determining rates of change, playing a crucial role in continuity analyses while integral calculus helps determine size, volumes, and areas.
As I said before, differential calculus is used to calculate rates of change—this pertains to issues like calculating velocities, acceleration, and solving optimization problems. Using the principles of differential calculus, it’s possible for us to create predictive models for how certain variables will behave—the changes in variable values and so on.
Integral calculus is primarily used to calculate volumes and mass contained within physical spaces. The integration operation is also called the anti-derivative, used in mathematical analysis in tandem with differential calculus to solve mathematical problems.
What Does AGT Have to Do With any of This?
One of the most important contributions of Algebraic General Topology is an alternative representation of continuity—a central concern in calculus. It further uses its representation of generalized limit to present a new version of differentiability that can be applied to a broader range of functions—even arbitrary discontinuous functions.
AGT moves away from infinitesimal calculus to use novel concepts like staroids, funcoids and, multifuncoids, to help pave the way for a new approach to mathematical analysis. While the fundamental ideas remain the same, my AGT is a much more powerful mathematical theory that encompasses mathematical concepts that are inexpressible in general or algebraic topology—as well as calculus.
If you’re interested in Algebraic General Topology and wish to see how it can help you become a better mathematician, you should read Algebraic General Topology book Volume 1. If you have any questions, you should get in touch with me today—my doors are always open for new debate on my math research ideas.