Working Miracles with the Differential Operator - II
16th March 2014
[This article is excerpted from the FEM Lectures by Rahul Bhattacharya. This is the second of the three part lecture on the topic of Operational Calculus.
I cannot ascribe the following quote to any one particular person, because, my old boss, Justin P., who taught me options trading, said to me once that someone who had taught him the tricks of the trade had, in turn, heard it from someone else in those yo-yo days of Wall Street in the mid-1980s where he worked at a bulge bracket brokerage house. The words have remained with me ever since:
||If you want to learn anything about finance just talk to a (options) trader...the foot soldier that wins the war or gets killed...just like if you need to understand the goddamn math, you got to talk to an engineer who suffers because of it.
In the spirit of the above quote then let's move on with this topic of Differential Operators. Let's for the moment not go any further into the territory of Quantum Mechanics, a subject that many of us here, including myself, don't understand very well. The subject is complex, abstruse, mathematically tedious, absolutely essential for an understanding of how our universe functions and where it was formed, and yet utterly fascinating. But we'll leave it for another day.
We need to concern ourselves with the notion and theory of differential operators in this lecture. Before we proceed any further with we need to clearly define what we mean by an operator. In our previous lecture we left this definition sort of hanging in the air, even though when we talked about the differential operator, , we more or less hinted at what an operator actually does. A mathematical "Operator" is nothing but a function, like any other function, but whose domain is a set of functions (and, not a set of real or complex numbers). For example, multiplication of a function by the number 10 is an operator, and we can write it as, say, . In fact, the more appropriate way of expressing this operator would be . This shows that the multiplication operator works on function for all values of . Similarly, we can say that squaring a function is also an operator and we can represent this as and this can be written as .
In the same manner, a differential operator, that does differentiation with respect to a variable. We have already seen that we write a differential operator as and it can be symbolically expressed as which shows that we are differentiating with respect to another variable. If that variable is then we can also express the differential operator as D[y](x)=y^' (x). This tells us that we are taking the first mathematical derivative of y with respect to x. Differential operators can also be used to represent second, third or higher order mathematical derivatives. For example, we can write, for all values of x which means
In fact, we can combine all these three operators that we have just talked about, i.e. multiplication operator, square operator and the differential operator into one single, complicated operator. For example, we can write the following
In the above, is one single, complex operator; and, as we can clearly see, the above operator is basically a differential equation written as follows:
Even though, you may think that the operation of differentiation - i.e. taking the mathematical derivative of one variable, or a function, with respect to another variable - should be treated in the same manner as the operation of multiplication or taking the square of a variable, because after all, differentiation is a mathematical process involving two variables or one variable and a constant, yet there is something utterly fascinating about this differential operator (and indeed in many fundamental ways it behaves quite unlike other operators). As we saw in the previous lecture, this differential operator contained within it the mathematical properties that would, in the early twentieth century, be able to extract the new physics of quantum mechanics from the classical Newtonian mechanics.
Anyway, we shall delve into the myriad mysteries of this differential operator in later lectures but for now let's just wonder how did this actually come about? Trivial as it may seem to many of us today, recognition of the "operator" like properties of the term, required a pure stroke of genius. Who was this man, who thought that could be used and manipulated as an operator just like other mathematical operators, thereby unlocking great secrets of mathematical physics and electrical engineering?
As it turns out, there was not one man but in fact there were three who realized the enormous power and beauty inherent in the operator, or , as the case may be, which lies at the heart of differential calculus.
The key to differential operators lies in understanding that algebra is the route to calculus. Now, our high school teacher, and indeed even some of our freshman college professors might find this statement quite absurd and extreme. You may even think the same. After all, algebra is algebra and calculus is calculus. In the early 19th century such was the conventional thinking for sure. The subject of mathematical analysis, i.e. differential and integral calculus and the theory of differential equations, which was making great waves in the world of physics, was a discipline as far away from algebra as imaginable. Until, of course, a young Cambridge mathematician thought otherwise.
Duncan F. Gregory (1813-1844), a young Cambridge mathematician, editor of the Cambridge Mathematical Journal, who died prematurely at a tender age of 30, was the first to recognize a revolutionary idea in mathematics. He realized - based on his analysis of a significant body of symbolic algebra, a large part of which was shared with mathematicians such as William Rowan Hamilton (1805-1865) and De Morgan - that if we can separate the "symbols of operation" from those of the "quantities" on which these operations were performed then we can in one stroke connect algebra with calculus. When we write , the "operation" becomes taking the derivative and y is the "quantity". Hence, the expression can simply be written as or even as .
However, it would be left to George Boole, the famous English mathematician and the inventor of Boolean Algebra, which lies at the heart of all computer science today, to improve upon Gregory's work and develop Gregory's work in the field of differential operators and refine and improve it. His seminal book Treatise on Differential Equations was published in 1859 and in this book, for the first time, we get a glimpse of detailed treatment of differential operators using rules of symbolic algebra.
Despite Gregory's earlier attempts at the subject, it was Boole who pointed out the striking parallels between the differential operators, , and the rules of algebra and how one is embedded in the other. It was Boole, who looked at a differential equation purely in terms of operator logic (the kind that we talked about just a little while ago).
For example, if we have a differential equation such as:
Then, we can write this in the form of one complex operator, , as . Here, the operator comprises the operators, and and we can express as: .
Boole insight was that the operator was nothing but a simple polynomial in algebra and by symbolically transforming the equation with terms such as and to an equation with terms such as and we effortlessly move from the realm of differential equations to algebraic equations, which can solved much easily. In his paper "On the Integration of Linear Differential Equations with Constant Coefficients" he improved and enhanced Gregory's method for solving such differential equations, an improvement based on a standard tool in algebra, the use of partial fractions.
The theory of differential operators will have to wait for almost another 50 years for a bright electrical engineer to take it to a whole new plane and give it a new meaning. Indeed, from this point onwards the subject of Operational Calculus was almost fifty years in the making.
That bright electrical engineer's name was Oliver Heaviside (1850-1925). To say that Heaviside was bright may still be an understatement. This highly original English electrical engineer, who we all know from our study of Heaviside function, was simply a mathematical genius.
Even if he had done nothing else in his life, the Heaviside function would have ensured a place for him in history. This mathematical function shows up in many important applications in physics, engineering and quantitative finance. Heaviside is the same man who, independently (and simultaneously with another American engineer named Arthur Kennelly) theorized and conjectured about the existence of a layer of an ionized gas approximately 100 kilometers above the earth's surface. This layer is now called the Heaviside Layer, or Kennelly-Heaviside Layer (In 1924, another British physicist, Edward Appleton conclusively proved that such a layer existed through his experiments and for which he was awarded the 1947 Nobel Prize in physics). [Many of you may have heard of the song "Journey to the Heaviside layer" from the musical CATS by Andrew Lloyd Webber. Webber took this song from a poem by the same name written by the famous poet T.S. Eliot. Heaviside layer was perhaps a form of heaven that Eliot dreamed of.]
Heaviside did much, much more than just this. He wrote two seminal books in the field of electrical engineering titled, Electrical Papers (1873-1891) and Electromagnetic Theory, 3 volumes (1893-1912); he extensively reworked on Maxwell's field equations, the basis of electromagnetic theory, in terms of electric and magnetic forces and energy flux (so much so that many think that Maxwell's equations should, in fact, be called Maxwell-Heaviside equations); he worked on transmission lines and in 1880 he developed the coaxial cable. He was possibly the brightest electrical engineer that has ever lived.
Heaviside contributions in the field of mathematics were equally astounding. His ground breaking work in the field of mathematical analysis gave rise to the entire discipline of Operational Calculus which is today an essential ingredient in the study of many important areas of electrical engineering.
Mathematicians think in abstract terms and they are not necessarily concerned with the problems in physics or engineering, even though more or less everything they work on eventually finds its way in these disciplines. It is left to the physicists and engineers - mostly the engineers - to figure out how all this dense and esoteric mathematics can be broken down, twisted, sliced up and added together, solved numerically and with great precision, how all this abstraction in thought can be crystallized within the working of machines that they build. Heaviside was one such engineer, an end user of mathematics. But rather than suffering the subject - as the great stock broker said in his famous quote (the one that I mentioned at the beginning of this lecture) - he set out to re-construct it, in a way that, above all he and numerous other engineers like him, could understand and then simply apply in solving problems (mostly in electrical engineering).
His work on differential operators went way beyond Boole's work. His work, however, was motivated by problems in the electric circuit theory, where most problems involve the use of ordinary and partial differential equations containing the differential operator . Heaviside called this operator, p, and by comparing the operational formulas of a specific problem with their known explicit solutions, he was able ascribe a specific meaning and significance to the operator, .
I am not going to go into details of Heaviside's operational calculus here - and frankly speaking, it will take a good part of a semester to cover this subject area, let alone a few sessions - and will try to cover some important parts of it in other lectures, towards the end of the program, if, of course, time permits. Suffice it to say, now, that Heaviside detailed exposition of the meaning, significance and the rules for using and algebraically manipulating the differential operator opened up new vistas for the physicists and one of the areas where it found greatest application was in the field of Quantum Mechanics (a brief glimpse of which we got in the previous lecture).
So much for history lesson! In the next lecture, we'll actually take up solution to problems in mathematics and electrical engineering and if, we are brave enough, quantum mechanics using the differential operator.
- Stanford Encyclopedia of Philosophy, April 21, 2010
- Heaviside's Operational Calculus, Ernst Julius Berg, McGraw Hill, 1936
- Electrical Circuit Theory and the Theory of Operational Calculus, Lectures by John R. Carson, Moore School of Electrical Engineering, University of Pennsylvania, 1952;
- An Operational Standpoint in Electrical Engineering, Frederic Rotella and Irene Zambettakis, Electronics, Vol. 17, No. 2, December 2013.
- A treatise on differential equations, George Boole, McMillan, 1859;
- Symbolical Algebra as a Foundation for Calculus:D. F. Gregory's Contribution, Patricia R. Allaire, Department of Mathematics and Computer Science, Queensborough Community College, CUNY, New York and Robert E. Bradley, Department of Mathematics and Computer Science, Adelphi University, New York
- Linear Operators and Linear Differential Equations, David Lerner, University of Kanasa;
- The Road to Reality, Roger Penrose, Vintage Books, 2005
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