Uncovering the Origins of Vector Analysis in Maxwell's Equations

[cooev769]

I'm quite interested in the history behind vector analysis especially Curl and Divergence and gradient operators etc. When James Maxwell derived Maxwell's equations of electromagnetism where these sorts of operations well known and commonly used, or are they modern fabrications. Did Maxwell actually derive his equations using the methods we use today or some other methods?

 

[micromass]

You should like this book: https://www.amazon.com/dp/0486679101/?tag=pfamazon01-20

 

[jedishrfu]

There's a book of the subject:

http://en.wikipedia.org/wiki/A_History_of_Vector_Analysis

and at Amazon as a Dover publication:

https://www.amazon.com/dp/0486679101/?tag=pfamazon01-20

Vector analysis grew out of the relative complexity of using quaternions. Hamilton was trying to extend complex numbers into a new kind of number and that's where the i,j,k unit vector idea came from: a + bi + cj + dk

For the divergence there's a history of the divergence theorem:

http://www.sciencedirect.com/science/article/pii/0315086078902124

The curl must have been recognized and came out of similar theorems.

 

[SteamKing]

I'm quite interested in the history behind vector analysis especially Curl and Divergence and gradient operators etc. When James Maxwell derived Maxwell's equations of electromagnetism where these sorts of operations well known and commonly used, or are they modern fabrications. Did Maxwell actually derive his equations using the methods we use today or some other methods?

You can see for yourself. Maxwell's treatise on electricity and magnetism can be found at the
Internet Archive:

http://archive.org/search.php?query=creator:"Maxwell, James Clerk, 1831-1879"

Maxwell worked with differentials and integrals, as modern vector notation had not been developed. Most of what Maxwell used in his work were quaternions. Modern vector analysis was developed by Josiah Willard Gibbs in the 1870s to simplify having to deal with quaternions and to provide more facility for his students in understanding vector concepts (When was the last time a professor went out of his way to simplify things?)

 

[D H]

Maxwell worked with differentials and integrals, as modern vector notation had not been developed. Most of what Maxwell used in his work were quaternions.

The first sentence is correct, but the second is not. It has become an internet meme, favored by crackpots who like to imagine that the vector notation used at intermediate college level hides some of Tesla's key discoveries (whatever!).

Maxwell's original work, published in 1861 and 1862 as On Physical Lines of Force did not use vectors or quaternions. Nor did his Dynamical Theory of the Electromagnetic Field, published in 1864 and 1865. His original development of his theory followed the painstaking style of only using scalar equations that were in near universal use at the time. The mathematics to simplify the notation didn't exist.

He did mention quaternions in passing in his final book on the subject, A Treatise on Electricity and Magnetism, published in 1873. But that was in passing, where he discussed interesting developments by others in extending his work. Maxwell himself did not use quaternions in the development of his theory.

 

[SteamKing]

Perhaps I was over generous in describing the extent of the use of quaternions by Maxwell in Electricity and Magnetism, but he did provide equivalent expressions for some of his results using quaternions.

That modern crackpots may have a bone to pick because Maxwell did or did not use quaternions in the manner they believe he should have used them is neither here nor there.

 

[D H]

Yes, Maxwell did demonstrate the use of quaternions on some of his results in A Treatise on Electricity and Magnetism, but that is not what he used to develop them. He published that book near the end of his life. He spent a good deal of his life developing electrodynamics, and he did that without the help of quaternions.

Aside: There is an even more elegant way to represent Maxwell's equations, which is with 4-vectors. That approach also wasn't available to Maxwell.To answer the question raised in the opening post, "Did Maxwell actually derive his equations using the methods we use today or some other methods?" He did it tediously, one scalar equation at a time. Most physics texts and journal articles written in the late 19th century and earlier are tedious to read because they didn't have the aid of higher mathematical structures.