An Overview of Calculus: Foreword, Important Concepts, and Learning Resources
Let’s talk about measure and change for a bit.
This article is part of my series in Foundations of Understanding Through Mathematics where I layout and presents an overview of some of the most important concepts in Mathematics [with links to learning resources] that are held with strong relevance to Artificial Intelligence and Computer Science.
For this article, we will have to take a look at the theory of Infinitesimal Calculus — the study of continuous change.
“ The calculus was the first achievement of modern mathematics and it is difficult to overestimate its importance. I think it defines more unequivocally than anything else the inception of modern mathematics, and the system of mathematical analysis, which is its logical development, still constitutes the greatest technical advance in exact thinking.” — John von Neumann.
Overview
Calculus is a branch of mathematics focused on the notion of limits, functions, derivatives, integrals, infinite sequences and series.
Calculus is mainly divided into differential and integral calculus. Differential calculus is mainly concerned about instantaneous rates of change while Integral calculus regards the accumulation of quantities of which indicates the measure between a bounded domain. These two branches are related to each other by the fundamental theorem of calculus. Both make use of the fundamental notions of convergence of infinite sequences and infinite series to a well-defined limit [1].
Calculus is one of the ideas that cannot be well understood without geometry and algebra.
Notes on The Fundamental Theorem of Calculus (FTC)
The fundamental theorem of calculus is a theorem that links the concept of differentiating a function with the concept of integrating a function. There are two parts of the theorem: the first part implies the existence of antiderivatives (integrals) for continuous functions; the second part which provides a general idea to numerical integration.
The first part of the theorem, states that one of the antiderivatives, say F, of some function f may be obtained as the integral of f with a variable bound of integration. This implies the existence of antiderivatives for continuous functions.
Conversely, the second part of the theorem, states that the integral of a function f over some interval can be computed by using any one, say F, of its infinitely many antiderivatives. This part of the theorem has key practical applications because explicitly finding the antiderivative of a function by symbolic integration avoids numerical integration to compute integrals. This provides generally a better numerical accuracy.
Why study Calculus?
Calculus has widespread applications in science, economics, and engineering. It offers an alternative approach to solving many problems for which algebra alone is deemed insufficient.
Applications of differential calculus include computations involving velocity and acceleration, the slope of a curve, and optimization. Whereas integral calculus is applied to computations involving area, volume, arc length, the center of mass, work, and pressure. Hence, in physics and engineering calculus provides a mathematical means of analyzing phenomena. Not only in the field of physics and engineering but the breadth of the subject also provides further applications in Social Science, Economics & Business just to name a few.
Some Personal notes before you get started
Personally, the reason that has got me into studying Calculus was my keen interest in starting out with Machine Learning. Although I have not been very well when it was first introduced to me in high school, I spent months reading books, finishing online courses, and doing exercises that have built a good foundation for understanding optimization problems in economics and backpropagation algorithms in neural networks. It may seem daunting at first but practice and your goal into getting the concepts right will pay off.
Spend your time wisely.
One of the lessons I have learned was that there will only be a handful of ideas that benefits you in the long run: save yourself some time. Don’t waste your time practicing how to integrate a function by hand as fast as possible, because in the real world these algorithmic procedures can be automated and if all you had learned was how to numerically integrate functions by hand, then you will not be as effective as you would like into approaching real-world problems that actually matter to you.
The thing is, you only need a couple of problems, in varying degrees of difficulty, to have a fine understanding of the concept. Test your understanding by sketching a proof of some theorem or solving word problems — because they involve more than carrying through an algorithmic procedure of number-crunching. Word problems require you to break-down the problem, translate, and seek for solutions. Follow Polya’s problem-solving techniques and master them.
Historical notes
The idea of calculus, according to some mathematicians and historians — such as Carl Boyer — have been motivated to solve one of the most famous problems in philosophy called the Zeno’s Paradoxes — these are set of philosophical problems, as proposed by the Greek philosopher Zeno of Elea, that supports the doctrine contrary to the evidence of one’s senses, the belief in plurality and change is mistaken and in particular that motion is nothing but an illusion [2].
In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started so that the slower must always hold a lead.
Infinitesimal calculus was developed independently in the late 17th century by Isaac Newton and Gottfried Wilhelm Leibniz.
Criticisms on the First principles of Calculus
In modern mathematics, the foundations of calculus are formalized in the field of real analysis, which holds full definitions and proofs of the theorems used in calculus. Foundations refer to the rigorous development of the subject from axioms and definitions that sought to have been deduced and logically justify conjectures.
In early calculus, the use of infinitesimal quantities was thought unrigorous and was fiercely criticized by a number of authors, most notably Michel Rolle and Bishop Berkeley.
Working out a rigorous foundation for calculus occupied mathematicians for much of the century following Newton and Leibniz, and is still to some extent an active area of research today.
Several mathematicians, including Maclaurin, tried to prove the soundness of using infinitesimals, but it would not be until 150 years later when, due to the work of Cauchy and Weierstrass, a way was finally found to avoid mere “notions” of infinitely small quantities [3].
The foundations of differential and integral calculus had been laid. In Cauchy’s Cours d’Analyse, we find a broad range of foundational approaches, including a definition of continuity in terms of infinitesimals, and a (somewhat imprecise) prototype of an (ε, δ)-definition of limit in the definition of differentiation.
Weierstrass, on the other hand, formalized the concept of limit and eliminated infinitesimals. Following the work of Weierstrass, it eventually became common to base calculus on limits instead of infinitesimal quantities, though the subject is still occasionally called “infinitesimal calculus”. Bernhard Riemann used these ideas to give a precise definition of the integral. It was also during this period that the ideas of calculus were generalized to Euclidean space and the complex plane.
Today, the field is equipped with foundations that have been carried on — developed — for ages. However, there still remains a question that is yet to be solved making the area of mathematical analysis, in general, an active research domain.
Final Remarks
Note that the terms I used here may be different from a more common definition of Integral since the term of a more common approach is only limited to a single variable i.e. the area under the curve. It is, therefore, important to realize that, in the real world, there is often not a single variable involved in forms of analysis — the generalization of single-variable Calculus is extended through a series of Linear Transformations (which benefits to another area of mathematics called Linear Algebra). Fundamentally, forms of generalizations are conceptually akin to the theorems proven in single-variable Calculus although it is important to keep in mind that it is concerned not only in 2-dimensions.
To your learning journey, I wish you luck. Cheers!
Learning resources:
http://www-math.mit.edu/~djk/calculus_beginners/
http://tutorial.math.lamar.edu/…/CalcI/DerivativeProofs.aspx
https://ocw.mit.edu/…/18-01sc-single-variable-calculus-fal…/
https://ocw.mit.edu/…/18-02-multivariable-c…/video-lectures/
https://ocw.mit.edu/…/res-18-001-calculus-online-…/textbook/
Helpful site to begin with the symbolic language used in the parlance of analysis: https://mathvault.ca/hub/higher-math/math-symbols/calculus-analysis-symbols/
YouTube Playlist and video lectures
These playlists develop your intuition and visual understanding of the essential concepts introduced in Calculus. Note that a more rigorous discourse of this subject is another field in mathematics called Analysis.
The essence of Calculus by 3 Blue 1 Brown: https://www.youtube.com/watch…
Calculus 1 and 2 by Khan Academy: https://www.youtube.com/watch…
Multivariable Calculus by Khan Academy: https://www.youtube.com/watch…
References
[1] DeBaggis, Henry F.; Miller, Kenneth S. (1966). Foundations of the Calculus. Philadelphia: Saunders. OCLC 527896.
[2] Papa-Grimaldi, Alba (1996). “Why Mathematical Solutions of Zeno’s Paradoxes Miss the Point: Zeno’s One and Many Relation and Parmenides’ Prohibition” (PDF). The Review of Metaphysics. 50: 299–314.
[3] Russell, Bertrand (1946). History of Western Philosophy. London: George Allen & Unwin Ltd. p. 857.
[4] Wikipedia contributors. (2019, July 14). Calculus. In Wikipedia, The Free Encyclopedia. Retrieved 07:54, July 26, 2019, from https://en.wikipedia.org/w/index.php…
[5] Boyer, Carl B. (1959). The History of the Calculus and its Conceptual Development. New York: Dover. OCLC 643872
[6] DeBaggis, Henry F.; Miller, Kenneth S. (1966). Foundations of the Calculus. Philadelphia: Saunders. OCLC 527896
