# Electromagnetics and vector calculus

A friend of mine claims that vector calculus was invented to do electrodynamics. I'm dubious. I know that Maxwell first wrote down the so-called Maxwell's equations in scalar form and only later converted them into their vector forms. Because all of classical electrodynamics is derivable from Maxwell's equations, I wouldn't see the need to develop a branch of mathematics to simply condense the equations of a theory that was pretty much done.

Wikipedia oddly does not have a "History" section for vector calculus. So can anyone either confirm or deny this? What is the history of vector calculus? Why was it invented (/discovered)?

EDIT: I specifically want to know about how E&M may (or may not) have affected the development of vector calculus.

An example of something that might be addressed: Apparently Heaviside's The Electrian was written a couple of decades before Gibbs' Vector Analysis -- which apparently is the book that codified the modern notations. How influential was The Electrian on Gibbs's work?

• possible duplicate of How were vector quantities developed? – Alexandre Eremenko Mar 23 '15 at 3:30
• hsm.stackexchange.com/questions/1925/… – Alexandre Eremenko Mar 23 '15 at 3:32
• hsm.stackexchange.com/questions/773/… – Alexandre Eremenko Mar 23 '15 at 3:33
• I read each of those you suggest as a duplicate @Alexandre. The first one is closest. But it does completely answer my question. In fact it just leads me to more questions like: How influential was Heaviside's The Electrian to modern vector analysis? Was vector calculus written in its entirely modern form in Gibb's Vector Analysis -- or was it just mostly complete? If the latter, what was missing? What did Hamilton use quaternions for? Was it solving E&M problems? What are the earliest uses of vector/ quaternion calculus for E&M? (Is there such a thing as "quaternion calculus"?) – user1992 Mar 23 '15 at 3:45
• You can obtain the answers to most of these questions by reading the references given in those answers. – Alexandre Eremenko Mar 23 '15 at 12:26

## Before the vector

Before vector calculus was introduced, a few landmarks have to be considered to understand vector history. These are:

1. complex numbers and their geometrical interpretation
2. Leibniz's work on the geometry of position
3. the parallelogram representation of force and velocity

The first one can be traced back to Cardan's Ars Magna published in 1545 as the guy was the first to introduce roots of negative numbers for themselves. One has to wait for two centuries before witnessing approval of this strange writing.

Leibniz, in a letter to Huygens, express the will to give position a mathematical expression, just as amplitude has one. This is exactly what vectorial analysis is about, right? Unfortunately, I don't know what Leibniz' answer to his own question was.

Now let's quote a genius

"A body, acted on by two forces simultaneously, will describe the diagonal of a parallelogram in the same time as it would describe the sides by those forces separately."

Newton in 1678, did not have the idea of a vector, but this does look like the sum of two vectors, right?

Between 1799 and 1828, three pairs of two authors simultaneously and independently worked on the geometry of complex numbers. Wessel and Gauss in 1799 wrote independently on how to represent direction analytically, Buéé and Argand follow up in 1806 with geometrical interpretations for complex numbers, and Warren and Mourey both published in 1828 extensive books describing such representations.

## Going 3-D: the quaternion

Hamilton (we're getting closer to electromagnetism) published a philosophical-ish paper in 1837 in which he expresses his hope to come up with a "theory of triplets" to describe 3-D geometry. In 1843 he finally comes up with a multiplication operation on what is now known as quaternions. He was very happy with that and proceeded to spend the rest of his life writing on quaternions.

Hamilton (him again) introduced in a subsequent paper (1846) the terms scalar and vector, to describe the real and imaginary parts of his quaternions. The vector part of the quaternion product of two purely vectorial quaternions is equal to the opposite of what is know the vector/dot product, and the scalar part is what is now known as the cross product.

Hamilton died but left a successor to his cause, Tait. Tait published numerous papers on quaternions, including extensive description of the use of operator Nabla which earned him Maxwell's praise as the "Chief Musician upon Nabla".

## Maxwell, or answering the original question

In 1873, Maxwell published his Treatise on Electricity and Magnetism, a paper that had a huge impact on 19th century science. In this paper, Maxwell presents many of his results not only in the then-usual cartesian form, but also in their quaternionic forms. Maxwell defended and advertise the use of quaternions, not only as a practical tool (he seemed to be more at ease with cartesian geometry), but as a more effective way to think space-related quantities.

For what I know, Maxwell did not contribute anything to vector calculus, but his endorsement of the then praised, but not used, quaternions in a breakthrough paper allowed vector calculus to become a widespread object in physics. Modern (post-Hamilton) Vector Analysis is mostly based on Gibbs and Heaviside work at the turn of the 20th century, but Maxwell sure contributed a lot to vector calculus by using them in what is, perhaps, the most read paper of the 19th century.

This answer is based both for structure and content on a talk that I invite you to read as it's a very entertaining/informative piece of writing.

• The second paragraph on quaternions mixes up dot and cross products, e.g., "the scalar part is now what is known as the cross product." – KCd Dec 25 '17 at 15:48

According to Thomas Hankin's biography of William Rowan Hamilton, Michael Faraday met with Hamilton in Dublin in 1834. Faraday is famous for his electromagnetic experiments and read about the field ideas of Boscovich. Faraday's experiments suggested to him that they could best be explained by the idea of fields. Faraday realized that field treatment must involve geometry and numerical analysis. Hamilton was known as a mathematical genius and Faraday convinced him of the need for a geometric Algebra.

It took Hamilton 9 years, in 1843 he came up with Quaternions and in 1844 biquaternions (complex quaternions)[proceedings of Royal Irish Academy]. Maxwell used quaternions learning from his boyhood friend Peter Guthrie Tait. Maxwell wrote to Tait that he wanted to let quaternions "leaven electromagnetism",[Michael Crowe History of Vector Analysis] Unfortunately, Maxwell died young of cancer and Electromagnetism was kidnapped by telegraph operator Oliver Heaviside who hated quaternions, titling one of his book's sections "On the abstrusity of quaternions and the advantages gained in ignoring them". Heaviside with the Help J. Willard Gibbs, truncated the quaternion to make vector analysis. This lead Physics down a wrong path that still exists today in spite of a small number of people championing quaternions since Hamilton's time.

Ludwik Silberstein was first to publish the (bi)quaternion version of Maxwell's Equation where all 4 Maxwell's are written as one biquaternion equation in the same paper he gave the quaternion Lorentz transformation [L.Silberstein "The Quaternion form of Relativity" , Philosophical Magazine,1913] The quaternion form of Dirac's equation was discovered by Cornelius Lanzcos 1n 1929.

The original 1864 paper of James Clerk Maxwell, "A dynamical theory of the electromagnetic field" contained the whole electrodynamics in the form of 20 nonvectorial equations in 20 variables.

Following this paper, efforts were made for casting these equations into a more systematic form. At those times, algebraic quaternions were the unique mathematical instrument available to that aim. J.C. Maxwell tried to work with quaternions, but was not very successful. Anyway, such efforts were merely considered for the beauty of the gest, because the complete theory was already available from Maxwell's original paper.

Eventually, Josiah Gibbs and Oliver Heavyside came up with vector analysis, and what today is called "Maxwell's Equations" is in fact a reformulation by Oliver Heavyside of the original equations proposed by J.C. Maxwell. So far for the engineering part of the story.

On the other hand, it has rarely been noticed that quaternions are the "square root" of an algebraic number identity: Leonhard Euler's 4-squares identity:

\begin{align} (a_0 + ia_1 + ja_2 + ka_3)\times(b_0 + ib_1 + jb_2 + kb_3) &= (a_0b_0 - a_1b_1 - a_2b_2 - a_3b_3)\\ &+ i(a_0b_1 + a_1b_0 + a_2b_3 - a_3b_2)\\ &+ j(a_0b_2 - a_1b_3 + a_2b_0 + a_3b_1)\\ &+ k(a_0b_3 + a_1b_2 - a_2b_1 + a_3b_0) \end{align}

The square of this product is, $$(a\times b)\times (a\times b)'' = a\times(b\times b'')\times a''$$

and with the following $$q\times q'' = q''\times q = (q_0^2 + q_1^2 + q_2^2 + q_3^2)$$

We have Leonhard Euler's 4-squares-identity: \begin{align} (v_0^2 + v_1^2 + v_2^2 + v_3^2)(w_0^2 + w_1^2 + w_2^2 + w_3^2) &= (v_0w_0 - v_1w_1 - v_2w_2 - v_3w_3)^2\\ &+ (v_0w_1 + v_1w_0 + v_2w_3 - v_3w_2)^2\\ &+ (v_0w_2 - v_1w_3 + v_2w_0 + v_3w_1)^2\\ &+ (v_0w_3 + v_1w_2 - v_2w_1 + v_3w_0)^2 \end{align}

Electrodynamics using quaternions has thus the advantage of being metaphysically backed by an algebraic number identity: everything is conserved under such transformation.

A further advantage is the easy and transparent formulation. The gradient, $\delta\times A$, of the vector potential $A$ is not a 2-form, but a vector, as expected for a physical quantity supposed to represent a force or a movement, having a direction in space:

\begin{align} (\delta_0 + i\delta_1 + j\delta_2 + k\delta_3)\times(A_0 + iA_1 + jA_2 + kA_3) &= (\delta_0A_0 - \delta_1A_1 - \delta_2A_2 - \delta_3A_3)\\ &+ i(\delta_0A_1 + \delta_1A_0 + \delta_2A_3 - \delta_3A_2)\\ &+ j(\delta_0A_2 + \delta_2A_0 - \delta_1A_3 + \delta_3A_1)\\ &+ k(\delta_0A_3 + \delta_3A_0 + \delta_1A_2 - \delta_2A_1) \end{align}

Under Lorenz gauge, the first line vanishes identically, and the remaining lines yield the electromagnetic field in source $(E/c)$ and $\operatorname{curl} (B)$:

$$\delta\times A = i(E_1/c + B_1) + j(E_2/c + B_2) + k(E_3/c+ B_3)$$

Source and curl are the two parts of a 4-dimensional isoclinic double-rotation around a point; a quaternion being an operator for performing general isoclinic double-rotation & stretching operations in 4-dimensional space.

The square $(\delta\times A) \times (\delta\times A)''$ is equal to $(E⁄c)^2 + B^2$, i.e. there are no mixed terms involving $E$ and $B$, because both parts of an isoclinic double rotation are orthogonal to each other.

William Rowan Hamilton invented the Del or Nabla operator when working with quaternions. ee https://books.google.com/books?id=iuoZSkSOBQsC&pg=PA142&dq=%22William+Rowan+Hamilton%22+and+Nabla&hl=en&sa=X&ei=u78_VaGAGsWEsAWevoGoCg&ved=0CDkQ6AEwBA

• Could you perhaps expand your answer a bit, making it more self-contained? Perhaps try to summarize what can be found in the books you linked? On Stack Exchange sites we strive to avoid link-only answers, and currently this answer is not very far away from being one. – Danu Apr 29 '15 at 7:12