Oliver Heaviside: The Self-taught Pioneer of Electromagnetism and Vector Calculus

January 21, 2023 by Biljana Ognenova

From the coaxial cable to the bounce-back effect of radio waves against the ionosphere, Heaviside’s contributions have stood the test of time.

At a time when it was not unusual for engineers to be self-taught and share their work via letter correspondence, Oliver Heaviside (1850–1925), a self-made mathematician, physicist, and electrical engineer, continues to inspire other engineers to venture off the beaten path. 


A portrait of Oliver Heaviside

A portrait of Oliver Heaviside. Image courtesy of Smithsonian Libraries [CC Public Domain]


Heaviside is credited with bringing complex numbers to circuit analysis, creating a new technique to solve differential equations, single-handedly devising vector calculus, and rewriting James Clerk Maxwell's equations in the forms used today.


Heaviside Launches His Career As Telegrapher

Heaviside's contributions to operational calculus were a sharp contrast to his detest of geometry, a subject that caused him to leave school at 16 and start learning independently, according to some accounts. Other historians speculate that Heaviside dropped out of school as a teenager because his parents didn’t have the means to continue his education.

Having learned the Morse code, Danish, and German by himself, Heaviside traveled to Denmark, where he became a telegrapher. A family member, Sir Charles Wheatstone, helped Heaviside get his first job as a telegrapher. He was praised as a quick learner, often ranked among the top five in his class of 500 students. His talent in telegraphy brought him back to England, where he worked at the Great Northern Telegraph Company. 

At the time, transatlantic telegraphy was at the peak of interest in the electrical engineering community, but it was ridden with failures. Multiple scientists worked on solving the problems with signal transmission and cable landing along submarine lines. Heaviside took the credit for many of its successful improvements, including his work on the “Telegrapher’s Equations.”         


Heaviside Revolutionizes the Field of Electromagnetism  

Heaviside experienced hearing loss due to the Scarlet fever he suffered as a child, and because of his deafness, he left telegraphy to study electricity.

Heaviside's focus shift was also heavily influenced by his fascination with “Maxwell's New Treatise on Electricity and Magnetism.” After reading it, Heaviside became engrossed by electricity and delved deeper into the math behind Maxwell’s theories. He reduced Maxwell's 20 equations in 20 variables to four equations in two variables. He also developed the step function and the cover-up method


The four Maxwell equations simplified by Heaviside

The four Maxwell equations simplified by Heaviside embossed at the end of George Street in Edinburgh. Image courtesy of Wikimedia Commons


The unusual derivation methods he used were later proven by Thomas Bromwich. Until then, they existed without rigorous conventional mathematical proof. Heaviside believed mathematics should not be subject to such rigor. This lack of scientific rigor caused him many problems in the academic community. (One of the subtitles in Chapter 5 of his book “Electromagnetic Theory” is called “Rigorous Mathematics is Narrow, Physical Mathematics Bold and Broad.”) Among his fellows, Heaviside was known as a sarcastic man who wasn’t afraid to lock horns and defend his stance. 


Heaviside Delves Into Operational Calculus

Heaviside created operational calculus based on the Laplace transforms used for simplifying linear differential equations that can be later solved as algebraic equations. 

He initially became interested in vector analysis by collaborating with another scientist, J. Willard Gibbs, but it was Heaviside who was responsible for operational calculus applied to circuits. 

This operational calculus has many uses in electrical engineering today: 

  • Analyzing electronic circuits
  • System modeling
  • Digital signal processing
  • Nuclear physics (analyzing radioactive decay)
  • Process control (analyzing the effect of altered variables) 

Laplace transforms are also used for analyzing temporary oscillatory current created in a circuit because of sudden electromagnetic disturbance. Transient current can be caused by lightning strikes, faults, unfiltered electrical equipment, electromagnetic coupling, EMI/RFI noise, and the opening or closing of a switch. Much of Heaviside’s work in this area has enabled clear signal transmission over long distances without signal interference and data loss. 


"Journey to the Heaviside Layer"

Heaviside patented the coaxial cable with the core copper layer wrapped in a plastic insulator and a subsequent layer of copper mesh used to transmit home cable TV. His work on inductance was later applied to the induction coil and patented by Michael (Mihajlo) Pupin and George Campbell of ATT. 


Induction coil from England

Induction coil from England between 1910–1930. Image courtesy of UK Science Museum


He was the first to theorize the existence of the ionosphere, the conducting atmospheric layer capable of returning radio waves past the Earth’s rounded surface instead of traveling forward to space. At the same time, Arthur E. Kennelly, an American electrical engineer, made a similar prediction about the ionosphere. Therefore, its initial name was the Kennelly-Heaviside layer for many years.

In “Electromagnetic Theory” (written and published in several volumes from 1893 to 1912), originally published in full in 1922, Heaviside hinted at Einstein’s special theory of relativity. He hypothesized that an electric charge would increase in mass as its velocity increases.

Not everyone doubted his eccentric approach to scientific discovery.  Sir Oliver Lodge, another pioneer with little formal training, recognized the powerful theoretical implications of Heaviside's postulates about electromagnetic waves. 


A plaque by the Torquay Civic Society placed in honor of Oliver Heaviside

A plaque by the Torquay Civic Society placed in honor of Oliver Heaviside. Image courtesy of Linda Hall Library


In 1891, Heaviside was elected a Fellow of the Royal Society. This was one of the highest official honors he received. One pop culture recognition of Heaviside’s work is mentioned in Andrew Loyd’s Webber musical “Cats” where his prediction of the ionosphere is used in the song “Journey to the Heaviside Layer” as a metaphor for heaven.



Featured image from Hackaday.